ARXIV-2608.01396Complete proof (three conjectures)
For a finite simple graph \(G\), let \(t(G)\) be the largest order of an induced tree and let \(g(G)\) be the girth. We prove three consecutive conjectures of DeLaViña's Graffiti.pc program. First, writing \(\ell(v)\) for the independence number of the subgraph induced by the neighbourhood of \(v\), we prove \(t(G) \)\({}\ge \lfloor g(G)\)\(/2 \rfloor \)\({}- 1 \)\({}+ \max_{v \in V(G)} \ell(v)\). Second, if \(\mathrm{Per}(G)\)…
ARXIV-2607.22301Complete proof (at most five parts)
For partitions \(\lambda,\mu,\nu\) the Littlewood–Richardson coefficient \(c^{\nu}_{\lambda\mu}\) stretches to a function \(P(t)\)\({}=c^{t\nu}_{t\lambda,t\mu}\) which, by a theorem of Derksen and Weyman, is a polynomial in \(t\). King, Tollu and Toumazet conjectured that \(P\) has no negative coefficient. The conjecture is known only for \(c^{\nu}_{\lambda\mu}\)\({}\le 2\) and is otherwise open. We…
ARXIV-2606.28041Complete proof for N ≤ 200 (computer-assisted)
Erdős conjectured that every triangle-free graph on N vertices can be made bipartite by deleting at most N²/25 edges; the bound would be sharp, attained by the balanced blow-up C₅[N/5]. Writing β(G) for the minimum number of edges whose deletion makes G bipartite and a(N) = max{β(G) : G triangle-free on N vertices}, the conjecture is a(N) ≤ N²/25, and for N…
ARXIV-2606.18462Partial answer (scoped theorems)
Dirac asked in 1970 whether for every \(k\ge 4\) there is a \(k\)-vertex-critical graph without critical edges; Jensen settled all \(k\ge 5\), and only \(k=4\) remains open. Following Skottová and Steiner, call a graph \(G\) a \((4,1)\)-graph if \(\chi(G)=4\), \(\chi(G-v)=3\) for every vertex \(v\), and \(\chi(G-e)=4\) for every edge \(e\); they proved \(\delta(G)\ge 6\) and \(\lambda(G)\ge 6\) for every \((4,1)\)-graph and asked whether a 6-regular…
EP-278Complete proof
Let \(A\)\({}=\{n_1\)\({}<\)\(\cdots\)\({}<n_r\}\) be a finite set of positive integers. Choose one residue class \(a_i\bmod n_i\) for each modulus and maximize the natural density of their union. This is the unsettled maximum-density half of Erdős Problem 278. We give an exact uniform characterization whose state space depends on \(r\), not…
AIM-GEOMETRY-0175Complete negative answer
An AIM problem asks whether complex sectional curvatures remain uniformly bounded below in a collapsing circle Cheeger deformation. At a fixed point whose normal circle representation contains rotation blocks of speeds \(a,b>0\), we derive an exact fixed-point formula. A totally isotropic complex…
AIM-TOPOLOGY-0102Complete counterexample
Bubenik and Milićević defined cubical singular homology theories of Čech closure spaces from an interval and either the product or inductive product. Excision is known for the product theories and was left open for the three inductive theories. We give a four-point counterexample for the directed…
AIM-DYNAMICAL_SYSTEMS-0005Complete negative answer
An AIM problem asks whether the arithmetic or geometric Galois group of the generic iterated-preimage tower of a rational map over a p-adic field determines its Julia set. We give a uniform negative answer. For every prime p, the quadratic polynomials z² + 1 and z² − p⁻⁶ over ℚ_p have arithmetic…
AIM-COMBINATORICS-0233Complete proof
Let \(h\geq 2\) be fixed, let \(f\) be a natural-valued polynomial of degree \(d\geq 2\) , and suppose that \(A\subseteq\N\) satisfies \(f(\N)\subseteq hA\) . We prove…
AIM-DYNAMICAL_SYSTEMS-0095Complete proof
We classify the abstract weighted ramification portraits of rigid complex Lattès maps. For a Lattès map induced by an affine torus endomorphism of degree \(d\), the complete portrait is determined by its action \(\phi\) on the finite branch-value set and by a uniform fiber formula. Reducing the affine map on…
AIM-GEOMETRY-0263Complete proof
Let \((M,g)\) be a compact semi-Riemannian manifold of indefinite signature whose null geodesics are complete. We prove that no \(C^1\) one-form \(\eta\) can have the property that, along every nonconstant affinely parametrized null geodesic \(\gamma:\R\to M\), the function \(\eta(\dot\gamma)\) is affine with nonzero slope. This gives a negative…
AIM-COMBINATORICS-0230Complete counterexample
Fix \(K>1\) . We construct finite nonempty sets \(A,B\)\({}\subseteq\mathbb Z\) with \(|A|>|B|\) and \(|A+B|<K|A|\) for which \(B\) is not contained in any generalized arithmetic progression of bounded rank and size \(O_K(|A|)\) . More strongly, for every prescribed rank bound \(d\) and size constant \(C\) , one counterexample defeats…
AIM-GEOMETRY-0274Complete counterexample
We construct a closed complete flat pseudo-Riemannian manifold of dimension \(16\) and signature \((8,8)\) carrying a nonzero parallel self-adjoint endomorphism \(N\) with \(N^2=0\), although neither the manifold nor any connected double cover admits a nonzero parallel vector field. This gives a negative answer, as…
AIM-ALGEBRAIC_NUMBER_THEORY-0109Complete counterexample
An AIM problem asks whether every positive-rank elliptic curve over ℚ has a prime p for which its rational points are dense in its p-adic points. We give a negative answer. For E: y² = x³ − 1516563 and P = (6403/9, 511280/27), an unconditional full 2-descent and saturation certify E(ℚ) = ℤP. A…
AIM-FUNCTIONAL_ANALYSIS-0027Complete proof
We answer an AIM question about whether the minimal or maximal \(C^*\)-tensor product commutes with forcing. We use the standard convention that a ground-model \(C^*\)-algebra is replaced in the extension by the metric completion of its old normed \(*\)-algebra. Under this convention, both \(\otimes_{\min}\) and \(\otimes_{\max}\) commute…
AMR-011-0025Complete proof
An explicit question of Abért asks for a free-spanning-forest proof that the first \(L^2\)-Betti number is multiplicative under passage to a finite-index subgroup. We give such a proof using the free uniform spanning forest (FUSF). For \(H\le\Gamma\) of index \(k\), lift a spanning tree of the finite Schreier…
AIM-PROBABILITY-0126Complete proof
We characterize the functions arising as Fuglede–Kadison determinant transforms of bounded commuting tuples in a fixed \(\mathrm{II}_1\) factor. After recovering the log-modulus formula intended by an AIM range question, we show that such functions are exactly the logarithmic potentials of compact probability…
AIM-ANALYSIS-0015Complete proof
For \(\alpha\in\mathbb R\) , let \(\ell^2_\alpha\) be the sequence space with squared norm \(\sum_{n\geq0}|x_n|^2(n\)\({}+1)^\alpha\) . We determine the spectrum of the classical Hilbert matrix \(H=(m+n+1)^{-1}\) on these spaces, answering a problem from the 2024 AIM workshop on Riemann–Hilbert problems and Toeplitz matrices. The matrix is bounded exactly when…
AIM-GEOMETRIC_GROUP_THEORY-0027Complete proof
An AIM problem asks for families of free-by-cyclic groups with first Betti number greater than two and many connected components of the Bieri–Neumann–Strebel invariant, even after quotienting by the outer automorphism group. For every \(m\geq2\) we construct a linearly growing UPG automorphism of \(F_{2m+1}\)…
AIM-GEOMETRY-0195Complete proof
An AIM problem asks for a characterization of the face-angle and dihedral-angle data of a triangulated polyhedral surface in \(\mathbb R^3\) and conjectures that the realizable data have dimension \(E-1\) in every genus. We use the realization convention later made explicit by Hempel: a labeled, simplexwise-linear,…
AIM-TOPOLOGY-0203Complete proof
An AIM problem asks for an infinitely generated Fuchsian group of the first kind whose critical exponent is nonconstant on its quasiconformal Teichmüller space. We give an explicit affirmative construction. For a closed surface \(S_g\) , let \(K\) be the kernel of the epimorphism \(\pi_1(S_g)\to F_g\) that kills a cut…
AMR-011-0004Complete proof
Let \(\Gamma\) be a countable subgroup of \(\G(\mathbb Q_p)\) containing no noncentral element of trace \(2\) or \(-2\). We prove that, outside a Haar-null set of \(g\in\G(\mathbb Q_p)\), the generated group \(\langle\Gamma,g\rangle\) has the same property. Thus adjoining one random element preserves the absence of parabolics almost surely, answering Question…
AIM-DYNAMICAL_SYSTEMS-0011Complete proof
Let \(\D=\Z[1/2]/\Z\) be the dyadic circle and let \(T\) be Thompson's orientation-preserving circle group. We prove that, for every nonempty finite set \(A\subset\D\) , its setwise stabilizer is a maximal proper subgroup of countably infinite index in \(T\) . If \(|A|=k\) , then…
AIM-PROBABILITY-0111Complete negative answer
Let a nonempty finite bounded selfadjoint tuple have finite joint nonmicrostates free Fisher information, and let \(A\) be the positive selfadjoint operator associated with the closure of its polynomial free-gradient form. We prove that \(e^{-tA}\) does not converge uniformly to the identity in \(L^2\) on the operator-norm…
AIM-ARITHMETIC_GEOMETRY-0067Complete affirmative answer
We give an explicit integral projective curve whose Hilbert scheme of eighteen points has a rational component of dimension seventeen. This answers affirmatively Problem 20 in the 2010 AIM workshop list Components of Hilbert Schemes, even with the curve required to be integral and projective. The…
AIM-ARITHMETIC_GEOMETRY-0078Complete proof
We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic zero. The support…
AIM-ALGEBRAIC_GEOMETRY-0125Complete proof
For every prime \(p\) , integer \(n\geq2\) , and degree \(d\geq n+1\) , we construct a geometrically smooth hypersurface \(X\subset{\mathbb{P}}^n_{{\mathbb{F}_p}}\) of degree \(d\) such that \(\#X({\mathbb{F}_p})\not\equiv1\pmod p\) . The construction stays over the specified prime field in every characteristic and degree. A finite-field moment polynomial supplies nonzero coefficients…
AIM-REPRESENTATION_THEORY-0023Complete negative answer
Let \(E\) be the standard object in the generic polynomial Hecke category over \(\mathbb C(q)\) , and let \(U\) be an ordinary finite-dimensional multiplicity space. We determine every homogeneous component of the Berenstein–Zwicknagl quantum symmetric algebra \(A\)\({}=S_\sigma(E\)\({}\otimes U)\) : in degree \(n\geq2\) , only the one-row and one-column…
AIM-SEVERAL_COMPLEX_VARIABLES-0010Complete proof (stated special case)
We construct a homotopy through proper rational holomorphic maps from the unit ball in \(\mathbb{C}^2\) to the unit ball in \(\mathbb{C}^4\) , joining \((z,w)\)\({}\mapsto(z^3,\sqrt{3}zw,w^3,0)\) to \((z,w)\)\({}\mapsto(z,w,0,0)\) . The construction answers the explicit target-four question in the AIM list on the Cauchy–Riemann equations. A mixed-term polynomial…
KOU-21.68Complete counterexample
M. Kida conjectured that every finite semi-abelian group is monomial (J. Group Theory, 2025; Kourovka Notebook, Problem 21.68). We show that the conjecture is false. If a finite group \(W\) acts on a finite abelian group \(N\), a linear character of \(N\) has stabiliser \(T\) in \(W\), and \(T\) has an irreducible…
OWR-12861-021Complete negative answer
At the 2014 Oberwolfach workshop on combinatorics, P. Heinig asked whether, for every odd \(n\ge 7\), every \(n\)-vertex graph with minimum degree at least \(\lceil n/2\rceil\) contains a spanning copy of the graph obtained from the square of an \(n\)-cycle by deleting every other edge on the periphery until exactly three consecutive…
KOU-21.76Complete affirmative answer
An elementary net (carpet) of order \(n\) over a field \(K\) is closed if its elementary net group contains no new elementary transvections, and it is completable if its diagonal can be supplemented to a full net. Completable nets are closed. Koibaev gave closed nets that are not completable over fields of…
OWR-16164-019Complete negative answer
At the 2018 Oberwolfach workshop on enumerative combinatorics, A. Sportiello compared two families of objects attached to a digitally convex shape \(\lambda\) (a convex polyomino): the permutations whose graph lies in \(\lambda\) and that avoid a \(123\)-pattern whose two corner cells also lie in \(\lambda\), counted by \(A_\lambda\), and the…
OWR-1782-009Complete counterexample (all-n form)
Rödl, Ruciński and Szemerédi stated the following conjecture, which they attribute to Katona and Kierstead: every \(k\)-uniform hypergraph on \(n\ge k+1\ge 4\) vertices in which every \((k-1)\)-set lies in at least \(\lfloor (n\)\({}-k\)\({}+3)\)\(/2\rfloor\) edges has a tight Hamiltonian cycle. They proved it for \(k=3\) and all sufficiently large…
OWR-14299577-018Complete proof
Let \(A\)\({}\subset\{-n,\)\(\dots,n\}\)\({}\setminus\{0\}\) contain exactly one of \(k\) and \(-k\) for every \(1\le k\le n\). At the 2025 Oberwolfach workshop on analytic number theory, C. Bernert and N. Arala Santos asked two questions about such sets. First, must \(A-A\) contain \((1-o(1))n\) elements of \(\{1,\dots,n\}\)? Second, for every \(B\)\({}\subseteq\{1,\)\(\dots,n\}\) with \(|B|\ge\varepsilon n\), must the…
AIM-LOGIC-0084Complete counterexample (specified presentation)
We study computability of sections of the restriction map \(S_1(N)\to S_1(M)\) for countable stable models \(M\preccurlyeq N\). Types are represented by characteristic functions on formulas with named parameters, and the two elementary diagrams are separately decidable. Classical stable definability gives a continuous…
AIM-LINEAR_ALGEBRA-0012Complete theorem (D+a branch)
For independent indeterminates \(a,x_1,\)\(\ldots,x_N,y_1,\)\(\ldots,y_N\), we prove that the sequence \(b_k\)\({}=a e_k(X)\)\({}+\sum_i y_i e_k(X\)\({}\setminus x_i)\) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients of \((uD_z\)\({}+v)\prod_i(1\)\({}+x_i z)\), coefficientwise…
AIM-COMBINATORICS-0177Complete counterexample (fixed label order)
We give two loopless directed graph layers on six vertices, each with constant outdegree two, having no rainbow directed cycle and hence no increasing rainbow cycle. For the fixed label order \(1<2\), the sizes of the sets reachable by possibly trivial increasing paths are \(5,5,5,5,5,4\). Their average…
AIM-REPRESENTATION_THEORY-0102Complete obstruction (three explicit families)
Consider the supercharacter theory of \(S_n\) whose superclasses are the \(S_n\)-conjugacy classes contained in \(A_n\), together with the single block \(S_n\setminus A_n\). We prove that its superclass-function spaces, summed over \(n\) with one-dimensional components in degrees zero and one, admit no connected graded Hopf algebra…
AIM-COMPUTATION-0073Complete scoped theorems (closed model)
We give a positive integer dataset on \(S_4\) whose likelihood on the closed two-component Birkhoff model has at least three distinct strict local maxima in distribution space. The dataset has \(2013\) observations and all three probability vectors have full support. One component of each displayed mode is a…
AIM-PROBABILITY-0108Complete scoped polynomial theorems
We isolate an elementary highest-degree obstruction to polynomial changes of variables preserving unit free-Brownian diffusion. For every unital moment functional and every nonconstant noncommutative polynomial of degree \(d\), its free quadratic-variation polynomial has degree exactly \(2d-2\). The…
AIM-REPRESENTATION_THEORY-0007Complete all-rank fixed-vector proof
For every prime \(p\) and every rank \(r\geq1\), let \(H_r\)\({}=\{\operatorname{diag}(I_r,aI_r)\)\({}:a\)\({}\in\mathbb Z_p^\times\}\) inside \(\operatorname{GSp}_{2r}(\mathbb Q_p)\). We give an elementary proof that every smooth character of \(H_r\) occurs in every irreducible infinite-dimensional smooth complex representation of this group. In particular, such a representation has a nonzero vector…
AIM-PROBABILITY-0027Complete finite-system comparison theorem
We compare the stationary occupancy law of a finite join-the-shortest-queue system with its finite-buffer and sampled-routing approximations. For \(N\) unit-rate exponential servers, Poisson arrival rate \(0<\lambda<N\) and \(\rho=\lambda/N\), regeneration gives \(H\)\({}=((1\)\({}-\rho)^{-N}\)\({}-1)\)\(/\lambda\). With capacity \(b\) and blocking probability…
AIM-ANALYSIS-0138Complete zero-location theorems
For positive-definite Hermitian matrices \(A,B\), we study the zeros of \(P_m(z)\)\({}=\operatorname{Tr}((A\)\({}+zB)^m)\). Loewner bounds \(lB\le A\le uB\) give a sharp dimension-independent zero region: the union of two explicit closed disks. Every point of this region is attained by a commuting two-dimensional pair, while a nonreal boundary…
AIM-ANALYSIS-0089Complete symmetric-weighing theorem
For real symmetric zero-diagonal weighing matrices \(W\) with \(W^2=dI\), we deduce induced-\(\ell_p\) \(\varepsilon\)-paving into at most \(\lceil36\varepsilon^{-q}\rceil\) parts, where \(q=\min(p,p')\) and \(q=1\) at the endpoints. The exponent \(q\) is optimal uniformly over symmetric conference matrices. The elementary compression estimate \(\|W[S]\|_p\)\({}\le\|W[S]\|_2^{2/q}\) transfers…
AIM-ANALYSIS-0164Complete normalized-covariance criterion
Let \(R\) be a twice continuously differentiable, positive semidefinite complex kernel with unit diagonal on a connected plane domain. We give a necessary and sufficient condition for \(R\) to be the normalized covariance of a centered proper Gaussian analytic function with positive variance everywhere. A…
AIM-ALGEBRAIC_NUMBER_THEORY-0111Complete low-degree Hilbert-locus theorem
For the degree-\(d\) Fermat fourfold in complex projective 5-space, and for \(\delta\ge 2\) with \(d\gt 10\delta-6\), we describe the integral degree-\(\delta\) Hilbert locus as a disjoint union of \(15d^3\) smooth open plane-form spaces. The point classification applies Salberger's diagonal-curve inequality after a minimal-support…
AIM-COMBINATORICS-0093Exact law and five-cycle counterexample
We give an exact finite-time distribution for single-source internal diffusion limited aggregation driven by an arbitrary positive nearest-neighbour chain on the integers. The probability of each occupied interval is a B-spline value on the harmonic-scale knots of the chain. Lifting a cycle until…
OPG-46575Complete negative answer
The valency-variety \(w(G)\) of a graph \(G\) is the number of distinct vertex degrees of \(G\). In a problem recorded by Vizing in 1968 and later listed by Jensen and Toft, Melnikov conjectured that every graph \(G\) with \(n\ge 2\) vertices satisfies \(\chi(G)\)\({}>\lceil\lfloor w(G)\)\(/2\rfloor\)\(/(n\)\({}-w(G))\rceil\). We show that the conjecture is false. The…
OWR-17474-010Complete negative answer
Consider the Edwards–Anderson spin glass at zero temperature on an infinite, connected, locally finite graph, with i.i.d. absolutely continuous couplings, and let \(\mathcal{G}(J)\) be its set of ground states. Bäumler proved that on every locally finite tree, for coupling laws of linear growth, \(|\mathcal{G}(J)|\) is almost…
OWR-15428-015Complete negative answer and characterization
In 2017 Kahle, Kubjas, Kummer and Rosen asked whether, for every pattern of observed entries, the set of nonnegative partial tensors that cannot be completed to the joint distribution of independent discrete random variables is convex. The answer is no, and a negative answer is already implicit in…
OWR-4138-001Complete proof
The Bernoulli displacement model is the random Schrödinger operator \(h_{\omega,\lambda}\)\({}=h_0\)\({}+V_\omega\) on \(\ell^2(\mathbb{Z})\) in which each cell of two neighbouring sites carries one single-site potential \(\lambda\neq0\). The potential sits on the left or on the right site of the cell according to an independent Bernoulli variable \(\omega_k\). Nichols and…
OWR-730-009Conditional resolution (ETH)
In the \(\varepsilon\)-Dense Steiner Tree problem of Karpinski and Zelikovsky, every terminal is adjacent to at least an \(\varepsilon\)-fraction of the non-terminals, and a Steiner tree with the fewest edges is sought. For every fixed \(\varepsilon\gt 0\) the problem has a polynomial-time approximation scheme. At an Oberwolfach problem session…
OWR-14298808-012Complete proof
For \(2k\gt d+1\), the centred and normalized total \(k\)-volume of a stationary Poisson process of \(k\)-planes in hyperbolic space \(\mathbb H^d\), observed in a growing ball, converges to a non-Gaussian infinitely divisible law \(Z_{d,k}\). Bühler and Hug asked how the standardized law \(Z^*_{d,k}\)\({}=Z_{d,k}\)\(/(\operatorname{Var} Z_{d,k})^{1/2}\) behaves…
OWR-17290-002Complete proof (complete setting)
Oberwolfach Report 47/2019 records the following question, posed in the abstract of L. Rizzi's talk (joint work with V. Franceschi, D. Prandi and M. Seri): is the Laplace–Beltrami operator of the almost-Riemannian structure on \(\mathbb{R}^2\) with orthonormal frame \(X_1=\partial_x\), \(X_2\)\({}=x(x^{2\ell}\)\({}+z^2)\partial_z\) essentially self-adjoint…
OWR-14298367-003Complete proof
Let \(T_f\varphi\)\({}=\varphi\circ f\) be a dissipative composition operator on \(L^p(X,\mathfrak{B},\mu)\), \(1\le p\lt \infty\), induced by a bimeasurable bijection \(f\) such that \(\mu\circ f\) and \(\mu\circ f^{-1}\) are bounded by a multiple of \(\mu\). D'Aniello, Darji and Maiuriello proved that if \(f\) has bounded distortion, then \(T_f\) has the shadowing property if and only if it is generalized…
AIM-OTHER-0001Complete rooted-forest linear-space algorithm
For every finite rooted forest with minimal roots, we compute the full spaces of linear order-ideal and antichain statistics that have zero average on every rowmotion orbit. The algorithm retains the orbit through the empty ideal and two affine hulls, each represented by at most \(n+1\) vectors,…
AIM-COMPUTATION-0025Explicit exponential multimodality bounds
The likelihood of the crossed bivariate Gaussian seemingly unrelated regression model can have several local modes. Its almost-sure eventual uniqueness is known. We give an explicit finite-sample bound: for fixed noncollinear regressor vectors of correlation \(r\), with \(d=1-|r|\) and \(k=n-2\geq 2\), the probability…
AIM-COMBINATORICS-0209Effective density characterization
Fix \(s\geq 3\) and consider increasing integer sequences whose every \(s\) consecutive terms contain no nontrivial three-term arithmetic progression. We show that every admissible gap sequence can be decreased coordinatewise to an admissible sequence with all gaps at most \(B_s=1+2\binom{s}{3}\). Consequently the unrestricted…
OPG-37325Logarithmic bounds and negative answer
An equivalence graph is a vertex-disjoint union of cliques. We give an explicit cover of every noncomplete cycle power by a logarithmic number of equivalence subgraphs. More precisely, for integers \(k\geq1\) and \(n\geq2k+2\),…
AIM-PROBABILITY-0009Exact forest and cycle inference counts
For a binary restricted Boltzmann machine on a fixed labeled bipartite graph, consider the pair of conditional maximum-a-posteriori maps in both directions, with the same weights used in the two maps and no ties. We count such pairs exactly on forests and on even cycles. On a forest, local…
AIM-OTHER-0060Finite induced subgraph realization
Every finite bipartite simple graph occurs as a vertex-induced unrooted subgraph of the principal graph of an irreducible finite-depth inclusion of hyperfinite type \(\mathrm{II}_1\) factors. For part sizes \(m\) and \(n\), an explicit neighborhood-multiplicity parameter \(q\) gives index \(q3^n\), with \(q\) at most \(m+1\), and depth…
OWR-14299088-013Complete counterexample
A real polynomial is multi-affine if it has degree at most one in each variable. Basu and Perrucci asked whether the number of connected components of the common real zero set of two multi-affine polynomials of degree at most \(d\) is bounded in terms of \(d\) alone, independently of the number of…
OWR-11578-001Complete counterexample
In the unified variational formulation of the parabolic-elliptic eddy-current equations due to Arnold and Harrach, the electric field is written as \(E=A+\nabla\varphi_A\), where \(A\) is divergence free and \(\varphi_A\) solves \(\operatorname{div}(\sigma\nabla\varphi_A)\)\({}=-\operatorname{div}(\sigma A)\). To solve this equation, Arnold and Harrach assume a conductivity \(\sigma\) that is bounded…
OWR-13498-011Complete proof
Two disks of radius \(1\) centred at \((\pm1,1)\) and the \(x\)-axis enclose a curvilinear triangle. In the open problem session of the 2015 Oberwolfach workshop on discrete differential geometry, Propp and Kenyon asked whether, among all packings of this triangle by disks that touch the \(x\)-axis, the greedy packing…
AMR-090-0002-0003Proof of two Ramassamy conjectures
Let \(E_n\) be the Euler up/down numbers, \(\sum_n E_n x^n\)\(/n! \)\({}= \sec x \)\({}+ \tan x\), and let \(e_{n,i}\) be the Entringer numbers, which form the Seidel–Entringer–Arnold triangle. Arnold observed, without proof, that the least \(2\)-adic valuation \(m_i\) of the entries on the \(i\)-th diagonal of this triangle is weakly increasing in \(i\), and…
AMR-096-0015Sharp asymptotics and range counterexample
Let \(N_i\) be the number of excursions of length \(i\) from a fixed vertex in a uniformly random Eulerian circuit of the complete graph \(K_n\) in which every edge is replaced by two opposite arcs. Aldous and Yu (Open Problems in Mathematics, 2014) stated as the natural conjecture that \(\mathbb{E} N_i\) is asymptotic to \(e^{-i/n}\).…
OWR-14299911-029Complete corner-peeling counterexample
Let \(G\) be a finite connected subgraph of the grid \(\mathbb{Z}^3\), not necessarily induced, and let \(X(G)\) be the cube complex whose cells are the squares and \(3\)-cubes of \(\mathbb{Z}^3\) all of whose vertices and edges lie in \(G\). A corner is a vertex that lies in exactly one maximal cell, and a corner peeling is an ordering \(v_1,\dots,v_n\)…
OWR-1323-008Partial answer (scoped theorems)
In the report of the 2006 Oberwolfach mini-workshop on hypercyclicity, H. Petersson asked two questions about a (hereditarily) hypercyclic operator \(T\) on a separable Fréchet space \(X\). First, does \(\mathrm{HC}(T)\)\({}\cup\{0\}\) contain the range of an injective operator \(S\in L(X)\)? Second, for such an \(S\), is every linearly independent…
OWR-15436-004Partial answer (scoped theorems)
In an Oberwolfach abstract of 2017, Rendall remarked that the kinetic-proofreading module of the Altan-Bonnet–Germain model of T-cell activation appears to have deficiency one even under the strongest simplifying assumptions, noted that it is unknown whether it admits multiple steady states, and…
OWR-15428-014Partial answer (degree in classified cases)
Kahle, Kubjas, Kummer and Rosen showed that for a set \(E\) of observed entries of a \(d_1\)\({}\times\)\(\dots\)\({}\times d_n\) tensor with \(|E|=\sum_j(d_j-1)\) that meets every maximal slice and has a full-dimensional completable region (we call such \(E\) admissible), the algebraic boundary of the set of partial tensors that are restrictions…
AMR-021-0015Complete counterexample
The coefficient-parity conjecture of Forsgård and Shapiro uses the indices at which \(a_k^2-a_{k-1}a_{k+1}\) is nonnegative to bound the number of real zeros of a polynomial with positive coefficients. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative explicit degree-\(77\)…
AMR-021-0014Complete counterexample
Forsgård and Shapiro proposed bounding the number of real zeros of a positive-coefficient polynomial by the parity changes among all indices at which \((k\)\({}+1)a_k^2\)\({}-ka_{k-1}a_{k+1}\) is positive. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative counterexample…
AMR-021-0016Complete criterion
We give a necessary and sufficient finite criterion for prescribed numerical positions of all derivative roots of a real-rooted polynomial-like function. Polynomial-like degree \(n\) means that the nth derivative is nowhere zero; the function need not be an ordinary polynomial of degree \(n\). Eliminating…
OWR-13678-010Complete answer (exact dimension formula)
A bipyramid cell is a tetrahedral partition with one interior vertex \(v_0\), \(n\) boundary vertices that are coplanar with \(v_0\) and surround it, and two further boundary vertices on opposite sides of this base plane. Colvin, DiMatteo and Sorokina determined the dimension of the space \(S^1_d(\Delta)\) of \(C^1\) splines of…
OWR-12697711-015Complete answer (all loopless matroids)
Let \(M\) be a loopless matroid on a finite set \(E\) with rank function \(r\), and let \(\Sigma(M)\)\({}\subseteq\mathbb{R}^E\) be the support of its matroid fan. Draisma, Eggleston, Pendavingh, Rau and Yuen defined \(\operatorname{adim}(M)\) as the minimum of \(2\dim(\Sigma(M)\)\({}+R)\)\({}-\dim R\) over rational subspaces \(R\)\({}\subseteq\mathbb{R}^E\); for the matroid of a complex linear space this is the dimension…
AMR-050-0041Complete proof (least period N>4)
For a noncircular elliptic billiard with a nondegenerate confocal elliptic caustic, we derive closed formulas for the signed area ratios of the caustic contact polygon and the boundary tangent polygon to their discrete evolutes. The formulas hold for every physical periodic orbit of least period…
OWR-2489-009Complete counterexample
For a bipartite graph \(G=(U\cup V,E)\), a set \(I\subseteq U\cup V\) and rationals \(b_{ij}\), let \(S(G,I)\)\({}=\{x\)\({}\in\mathbb{R}^{U\cup V}\)\({}: x_i\)\({}+x_j\)\({}\ge b_{ij}\ (ij\)\({}\in E),\ x_i\)\({}\in\mathbb{Z}\ (i\)\({}\in I)\}\), and let \(k\) be the least positive integer with \(kb\) integral. In the 2008 Oberwolfach report on combinatorial optimization, Conforti conjectured that \(\operatorname{conv} S(G,I)\) is the intersection of the…
OWR-1452-024Negative answer (Q̄ case open)
In Oberwolfach Report 01/2007, Kraft and Russell stated that two morphisms of varieties over an algebraically closed field of infinite transcendence degree, whose fibres over all closed points are isomorphic, become isomorphic after a dominant étale base change, and asked whether this holds over…
OWR-13750332-001Complete proof (both conjectures)
The star transform integrates a function on the plane along \(m\) rays with a common vertex, with directions \(\gamma_1,\)\(\dots,\gamma_m\) and nonzero weights \(c_1,\dots,c_m\). In the inversion formula of Ambartsoumian and Latifi, the directions \(\psi\) at which \(\sum_j c_j\prod_{i\ne j}\langle\psi,\gamma_i\rangle\) vanishes, called singular directions of Type 2, cause instability.…
OWR-1703876-006Partial answer (question 1 solved, sharp)
In an Oberwolfach report from 2020, Paták considered closure operators on topological spaces whose closures have at most \(b\) path-connected components. He noted that \(\binom{b+1}{2}(k\)\({}-1)\)\({}+b\)\({}+1\) points suffice for a constrained drawing of the star \(K_{1,k}\), conjectured that \(kb+1\) points suffice, and asked two further…
OWR-15208-008Complete negative answer
Let \(M_0=\mathbb{C}\) and \(M_1=S^2\), with complete metrics \(g_\kappa\) of constant curvature \(\kappa\in\{0,1\}\). Every complete smooth metric of nonnegative curvature on \(M_\kappa\) can be written uniquely as \(\varphi^*(e^{-2u}g_\kappa)\), where \(\varphi\) is an orientation-preserving diffeomorphism fixing \(0\) and \(1\) (and \(\infty\) if \(\kappa=1\)) and \(u\) is a smooth function. Work of…
OWR-14298580-008Complete negative answer (computer-assisted)
Alfieri and Binns say that an \(\mathbb{F}[U]\)-module \(M\) satisfies the strong geography restriction if it has a direct summand \(\mathbb{F}[U]\)\(/U^\ell\)\({}\oplus\mathbb{F}[U]\)\(/U^{\ell-1}\)\({}\oplus\)\(\dots\)\({}\oplus\mathbb{F}[U]\)\(/U\), where \(\ell\) is the least integer with \(U^\ell M_{\mathrm{red}}\)\({}=0\). They showed that \(\mathit{HF}^-(Y)\) satisfies it when \(Y\) is surgery on a knot in \(S^3\) or large surgery on a link, and asked…
OWR-4798-013Complete affirmative answer
For \(0\le i\le d-2\), Klee and Novik defined \(B(i,d)\) as the subcomplex of the boundary of the \(d\)-dimensional cross-polytope generated by the facets whose \(xy\)-words have at most \(i\) switches, and asked whether \(B(i,d)\) is a combinatorial triangulation of \(S^i\times B^{d-i-1}\). We show that it is, for all \(0\le i\le d-2\). Klee and Novik…
OWR-3389-016Complete negative answer
Let \(E_1\le E_2\le\cdots\) be the eigenvalues of the Dirichlet Laplacian on a bounded open set \(\Omega\)\({}\subset\mathbb{R}^n\), and let \(M_p(J)\)\({}=\big(\frac{n+2p}{n}\frac1J\sum_{j\le J}E_j^p\big)^{1/p}\). Harrell and Stubbe proved that \(M_1(J)^2\)\({}-M_2(J)^2\)\({}\ge\frac14(E_{J+1}\)\({}-E_J)^2\), and in a problem list of the 2009 Oberwolfach workshop on low eigenvalues of Laplace and Schrödinger operators they…
OWR-1195-004Complete negative answer
At an Oberwolfach problem session in 2006, T. Schick asked the following. Fix closed Riemannian manifolds and a homotopy equivalence \(f\). Do the signature operators twisted by a bundle \(E\) and by \(f^*E\) have isomorphic kernels once the curvature of \(E\) is small enough? By a theorem of Hilsum and Skandalis,…
OWR-3385-008Complete proof
For points \(A_1,\)\(\dots,A_n,A'_1,\)\(\dots,A'_n\) in \(K^2\), where \(K\) is a field, write \([XY]\) for the \(2\times2\) determinant and let \(\alpha(\pi)\) be the cyclic quotient \(\prod_k[A_{\pi_k}A_{\pi_{k+1}}]\)\(/\prod_k[A'_{\pi_k}A'_{\pi_{k+1}}]\), with indices read cyclically. Below, Krummeck and Richter-Gebert conjectured in 2003 that \(\sum\operatorname{sgn}(\pi)\,\alpha(\pi)\)\({}=0\), where the sum runs over the permutations…
OWR-9790358-016Partial answer (quadrilaterals complete)
Mean value coordinates of a planar polygon \(P\) with \(n\) vertices define a map from \(P\) to \(\mathbb{P}^{n-1}\). At an Oberwolfach mini-workshop in 2022, F. Sottile asked for the homogeneous equations of the Zariski closure \(S_P\) of its image. We answer this question for quadrilaterals and give partial results for larger…
AMR-046-0027Partial answer (existence and small cases)
Let \(f\in\mathbb{C}[x]\) have at most \(k\) monomials and put \(M_n\)\({}=\int_0^1 f(x)^n\,dx\). A. Gasull asked whether there is a number \(N(k)\) such that \(M_1=\dots=M_{N(k)}=0\) forces \(f=0\), and, if so, for its value or a good upper bound. We show that \(N(k)\) exists for every \(k\). The proof treats the exponents as parameters, applies Hilbert's basis…
OWR-17135-036Partial answer (scoped theorems)
F. Santos asked in an Oberwolfach problem session (2019) whether every combinatorial \(d\)-sphere \(S\) with \(n\ge d+3\) vertices is a subcomplex of a combinatorial \((d+1)\)-sphere with the same \(n\) vertices. We give partial results. If \(S\) has an edge \(vw\) with \(\operatorname{lk}(v)\)\({}\cap\operatorname{lk}(w)\)\({}=\operatorname{lk}(vw)\), equivalently an edge that lies in no…
AMR-014-0019Partial answer (scoped theorems)
For even \(n\) let \(B'_{n,m}\) be the supremum of the number of real projective zeros of a sum of squares of real forms of degree \(n\) in \(m\) variables, taken over those with finitely many real zeros. Fröberg, Lundqvist, Oneto and Shapiro conjectured that \(B'_{n,m}=(n/2)^{m-1}\); Choi, Lam and Reznick had proved the…
AMR-050-0048Counterexample and corrected formula
The experimental invariant list of Reznik, Garcia and Koiller prints the value \(2\) for the focal-inverse area of the outer tangent polygon divided by that of a four-periodic elliptic billiard orbit. We give a rational counterexample to this value and a self-contained area calculation showing that the…
AMR-050-0025Complete proof
Let a rectangle be circumscribed about a noncircular nondegenerate ellipse with center \(O\) and a focus \(F\). We show that its antipedal quadrilateral with respect to \(F\) has vertex centroid \(O\) and area centroid \(O+(F-O)/3\), independently of the rectangle's orientation. The quadrilateral is always finite and…
AMR-050-0026Complete proof
Fix a noncircular ellipse and a nondegenerate confocal elliptic caustic supporting a family of billiard orbits of effective even period. Form the outer polygon by intersecting consecutive boundary tangents, then take its antipedal with respect to either focus. We prove that its vertex centroid is…
OWR-14299906-003Negative answer (general case)
For a primitive \(0\)–\(1\) matrix \(A\), Freslon, Gerontogiannis and Skalski computed the quantum isometry group \(\mathbb{G}_A^{\infty}\) of the log-Laplacian spectral triple on the Cuntz–Krieger algebra \(\mathcal{O}_A\). They showed that the quantum automorphism group \(\operatorname{QAut}(A)\) is a quantum subgroup of a larger quantum group \(\mathbb{G}_A^{1}\) and asked if…
OWR-1275-010Complete answer
At an Oberwolfach problem session in 2006, V. Fock asked for the cohomology and the number of components of the space \(B_n\) of maps \(f\colon\mathbb{Z}\)\(/n\)\({}\to\mathbb{RP}^1\) with \(f(i)\ne f(i+1)\) and at least three values, modulo \(\mathrm{SL}(2,\mathbb{R})\). We answer both questions for all \(n\ge3\). The space \(B_n\) is a real-analytic \((n-3)\)-manifold with exactly…
OWR-782-001Complete proof
For even \(k\ge4\) let \(f_k\)\({}=1\)\({}+O(q^{\dim M_k})\) be the extremal modular form of weight \(k\) for \(\mathrm{SL}_2(\mathbb{Z})\). Bannai, Koike, Shinohara and Tagami conjectured that if \(k=12\mu\) and \(f_k\bmod p\) is a nonconstant power series in \(q^p\) (their Case (2)), then \(f_k(\tau)\)\({}\equiv g(p^r\tau)\pmod p\) for an extremal modular form \(g\) of smaller weight and some \(r\ge1\). We…
OWR-17294-017Negative answer (counterexamples)
For a set-theoretic solution \((X,\sigma)\) of the Yang–Baxter equation, Farinati and García Galofre showed that a comparison map from braided (co)homology to the Hochschild (co)homology of the structure algebra \(A=kM(X,\sigma)\) can be chosen to be the quantum symmetriser of \(-\sigma\), and that it factors through a complex…
OWR-13497-006Complete answer
Four agents in the plane move by the standard distance-based gradient law \(\dot p_i\)\({}=\sum_{j\ne i}e_{ij}(p_j\)\({}-p_i)\), \(e_{ij}\)\({}=\|p_i\)\({}-p_j\|^2\)\({}-\bar d_{ij}^{\,2}\), where all six desired distances come from one planar target. In an Oberwolfach report of 2015, B. D. O. Anderson asked whether, as for rectangular targets, all incorrect equilibria…
OWR-11786-023Complete counterexample
Let \(\bar f(2,n)\) be the maximum diameter of the dual graph of a triangulated surface with boundary on \(n\) vertices. At a 2012 Oberwolfach workshop, Santos asked whether \(\bar f(2,n)\le n-3\). We show that the answer is no. A triangulation \(M_7\) of the real projective plane minus two disjoint open discs has \(7\) vertices,…
AMR-050-0068Complete scoped proof
For an elliptic billiard with a nondegenerate confocal elliptic caustic and effective even period, we prove that the pedals of the two unit-focus-inverted orbit polygons are congruent and have equal ordered signed areas. Equality does not imply nonvanishing. We construct a continuous family of…
AMR-050-0069Complete proof
Invert the vertices of a triangle about a point and form the pedal triangle of its inverse with respect to that same point. We derive its ordered signed area in terms of the original circumcircle power and the three vertex distances. The formula proves that every finite isogonal-conjugate pair off…
AMR-050-0001Complete counterexamples
We give two exact counterexamples using one connected family of convex, primitive four-periodic billiards in a noncircular ellipse with a fixed confocal elliptic caustic. Write \(A\) for the orbit's ordered signed area, \(A'\) for the area of its consecutive tangent-intersection polygon, \(A''\) for the area…
AMR-050-0019Complete counterexample
For a periodic elliptic billiard, let \(A'\) be the area of the polygon formed by consecutive tangents to the ellipse, and let \(A^*_M\) be the antipedal area of the original orbit with respect to a fixed point \(M\). We give an exact counterexample to the claimed constancy of \(A'/A^*_M\) for periods divisible by four,…
AMR-050-0021Complete scoped proof
For a genuine elliptic billiard orbit of least period divisible by four, we prove that the signed area of its pedal polygon at either focus times the signed area of its antipedal polygon at the same focus is constant throughout the fixed-ellipse, fixed-confocal-caustic family. Convex and coprime…
AMR-050-0012Complete scoped proof
We prove constancy of the product of the signed area of an elliptic billiard orbit and the signed area of its pedal polygon at the common center, for a fixed nondegenerate confocal elliptic caustic and least period odd or divisible by four. Primitive star trajectories are included. A…
AMR-050-0014Complete scoped proof
For a fixed confocal elliptic-caustic billiard family with least period congruent to two modulo four, we prove that the signed area of the outer tangent polygon times the signed area of its pedal polygon with respect to any fixed point is constant. Coprime star trajectories are included. The…
AMR-050-0027Complete scoped proof
For an odd-periodic billiard in an ellipse with a fixed confocal elliptic caustic, let \(C\) be the Steiner curvature centroid: the weighted average of the vertices with weights \(\sin(2\theta_i)\), where \(\theta_i\) is the internal reflection angle. Let \(A\) be the signed area of the original chord polygon and \(B_C\) the signed…
AMR-050-0007Complete scoped proof
We prove an explicit positive, phase-independent formula for the product of Euclidean distances from either focus of an ellipse to the consecutive boundary-tangent intersections of an elliptic billiard. The theorem assumes a nondegenerate confocal elliptic caustic and a least period divisible by…
AMR-050-0023Complete scoped proof
We establish explicit, phase-independent vertex-centroid formulas for antipedals of the original polygon in elliptic billiards with a nondegenerate confocal elliptic caustic and even least period. The pole is the ellipse center or either focus, and all consecutive antipedal intersections are…
AMR-050-0029Complete scoped proof
For an odd-periodic elliptic billiard with a fixed nondegenerate confocal elliptic caustic, consider the polygon of caustic contact points and its own Steiner curvature centroid. We prove that this centroid is finite at every real phase and that the signed area of its pedal polygon is a…
AMR-050-0036Complete scoped proof
For a periodic elliptic billiard, form the polygon of intersections of consecutive boundary tangents, then its antipedal with respect to either focus. Classical central symmetry implies equality of the two signed antipedal areas for effective even periods. Their quotient is therefore one where…
AMR-050-0044Complete scoped proof
For an elliptic billiard with a fixed nondegenerate confocal elliptic caustic, we prove that the original signed shoelace area times the signed area of its unit focal inversion is a positive phase-independent constant when the genuine least period is divisible by four. Both foci give the same…
AMR-050-0046Complete scoped proof
For every nondegenerate confocal elliptic-caustic billiard of genuine least period \(n\ge 6\) with \(n\) congruent to 2 modulo 4, we prove that the original ordered signed area divided by the signed area after unit inversion about either boundary-ellipse focus is a positive phase-independent constant, identical…
AMR-050-0062Complete proof
For every odd-periodic real billiard family in an ellipse, we prove that the product of the trajectory-order signed areas of the two polygons obtained by unit inversion of the original vertices about the two original boundary foci is a positive phase-independent constant. The result includes every…
AMR-050-0053Complete adjacent-conjecture counterexample
We construct an explicit fixed circle and strictly nested noncircular ellipse admitting a Poncelet family of primitive convex pentagons with constant sum of squared side lengths, even though the circle center is neither the ellipse center nor either focus. If \(t\) is the unique root in \((1/10,1/5)\) of…
AIM-GEOMETRIC_GROUP_THEORY-0134Exact limit and finite-CW counterexample
We compute the exact finite determinant limit for Kammeyer's element \(f=1-2a+2b\) in the integral group ring of the discrete Heisenberg group. Along the congruence tower modulo powers of any fixed odd prime \(p\), the normalized Fuglede-Kadison determinant converges to \(2^{1/(p+1)}\), whereas the infinite determinant…
OWR-15212-005Complete negative answer
In an Oberwolfach report of 2017, Wolansky introduced a domination order between vector-valued measure spaces \((X,\sigma)\) and \((Y,\eta)\) with \(k\) components, defined by one Markov kernel that transports every \(\sigma_i\) to \(\eta_i\), and asked whether two atomless spaces that dominate each other must admit deterministic maps…
OWR-12723-008Complete answer (on the square)
For a partition of \([0,1]^2\) into \(N\) measurable pieces \(\Omega_i\), let \(\mathcal F\)\({}=\sum_i|\Omega_i|\,\mathcal A(\Omega_i)\)\({}+\sum_i(|\Omega_i|\)\({}-\min_j|\Omega_j|)\), where \(\mathcal A\) is the Fraenkel asymmetry with respect to discs. Steinerberger proved \(\mathcal F\ge1/60000\) for large \(N\) and asked, in Oberwolfach Report 49/2013, whether the extremal configuration is a hexagonal tiling,…
OWR-2040-002Counterexamples (Conjecture 2 first refuted by Kriebel)
In an Oberwolfach abstract, Knudson considered the discrete vector field \(V_P\) formed by the incident pairs of the persistence pairing \(P\) of a simplexwise filtration. He stated that \(V_P\) is a discrete gradient (his Theorem 1), conjectured a single gradient path from \(\tau\) to \(\sigma\) in \(V_P\) for every non-incident pair…
OWR-16766-004Complete counterexample
Kadu and van Leeuwen proposed to reconstruct a binary image from line sums by solving the Lagrange dual of the least-squares problem and reading off the sign of its numerically computed solution, where zeros mark undetermined pixels. From exhaustive experiments with images of up to \(4\times4\) pixels and two…
OWR-15428-003Complete negative answer
For a real tensor of format \(2\times\dots\times2\) with \(n\) factors, let \(d_i\) be the determinant of the Gram matrix \(M_iM_i^{T}\) of its \(i\)-th flattening \(M_i\). Seigal called the set of tuples \((d_1,\dots,d_n)\) of tensors in the Frobenius unit ball the Gram locus, described it for \(n=3\), and conjectured that for \(n\ge4\) it is cut out of the cube…
AIM-GEOMETRY-0036Counterexample and corrected fixed-point proof
We give counterexamples to the uniqueness assertion for maximum potentials in Theorem 4.7 of Ishibashi's Fixed point theorem for cluster modular groups, arXiv:2603.14338v2. Both a rank-one example and a full-rank type \(A_2\) example satisfy its filling conditions and have nontrivial finite modular…
AMR-103-0226Complete affirmative answer
By a congruence of Murasugi, the determinant and the signature of a knot \(K\) satisfy \(\det K\)\({}\equiv(-1)^{\sigma(K)/2}\pmod 4\); moreover, a knot of determinant \(1\) has signature divisible by \(8\), because its symmetrized Seifert form is even and unimodular. In 1971 Shinohara realized all pairs \((\det K,|\sigma(K)|)\) allowed by these conditions,…
AIM-GEOMETRY-0250Counterexample and sufficient descent criterion
A minimal closed geodesic need not have a simple projection under a finite Riemannian cover. We give explicit flat double covers and primitive minimal geodesics whose primitive projections have exactly \(m\) double points, for every positive integer \(m\). Both manifolds can be chosen orientable in every…
AMR-099-0003Complete affirmative answer
Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described…
AIM-PROBABILITY-0131Bounded-tracial theorem
We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to scalar Fourier series. The Fourier shadows obtainable from free sums form a proper subset of products of such shadows in every dimension at least two. A…
AIM-ALGEBRAIC_GEOMETRY-0289Presentation-independence theorem
Let R=S/I be an integral algebra of finite type over a perfect field of characteristic p>0, with S polynomial and I of height c. We give a direct proof that restriction to R of degree-e Cartier maps with coefficients in Ic(pe−1) is independent of the polynomial presentation, in every degree. A dummy-variable…
AMR-022-7062Complete negative answer
Problem 7.62 in Hayman and Lingham's Research Problems in Function Theory, posed by A. Hinkkanen, asks whether one can, without assuming the Continuum Hypothesis, associate with every countable ordinal \(\alpha\) a convergent series \(\sum_n x_{n,\alpha}\) with non-negative non-increasing terms such that \(x_{n,\alpha}\)\(/x_{n,\beta}\)\({}\to0\)…
AMR-022-2045Complete affirmative answer (computer-assisted)
L. Zalcman asked whether the equation \(J_0(z)=1\), where \(J_0\) is the Bessel function of order zero, has at most one solution on each ray from the origin. The question is Problem 2.45 in Hayman and Lingham's Research Problems in Function Theory, where it is noted that an affirmative answer would show that…
AMR-066-0018Complete counterexample
In his list of questions on scalar curvature, Gromov states that a complete Riemannian manifold \(X\) with \(\mathrm{Sc}(X)\ge0\) which is isometric at infinity to a complete flat manifold \(X_{\mathrm{fl}}\) is flat, provided there is a homomorphism \(\pi_1(X)\)\({}\to\pi_1(X_{\mathrm{fl}})\) compatible with the isometry at infinity. He conjectures that this…
AIM-LINEAR_ALGEBRA-0007Regular-loop logarithm counterexample
We construct an explicit family of regular, entire, two-periodic totally nonnegative loops with determinant one whose inverses are also entire. Each loop is the product of two positive exponential phases. We prove that this product admits an entire complex matrix logarithm if and only if the phase…
AMR-036-0044Partial answer (new negative cases)
A real linear system \(\dot x=Ax+Bu\), \(y=Cx\) with \(n\) states, \(p\) inputs and \(m\) outputs is stabilizable by static output feedback if, for some real matrix \(K\), all eigenvalues of \(A+BKC\) have negative real part. Generic systems are stabilizable when \(n\lt mp\), and when \(n=mp\) and the degree \(d(m,p)\) of the Grassmannian…
KP-1.19Affirmative answer (written proof of a recorded result)
Problem 1.19 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026) asks: if \(Y=Y_1\#Y_2\) is a connected sum of \(3\)-manifolds, neither of them \(S^3\), and the twist \(\Phi\) about the connected-sum sphere is not isotopic to the identity, is there a knot \(K\subset Y\) such that…
AMR-042-0001Complete answer (M = 6 for every prime)
Problem A of T. Ward's list “Six problems in algebraic dynamics” concerns the polynomial \(f\)\({}=1\)\({}+u_1u_2\)\({}+u_1^2u_2\)\({}+u_1^3u_2\)\({}+u_1^4\)\({}+u_2^2\)\({}+u_1^4u_2^2\) and the algebraic \(\mathbb{Z}^2\)-action attached to the module \(\mathbb{Z}[u_1^{\pm1},u_2^{\pm1}]\)\(/\langle p,f\rangle\), where \(p\) is a prime for which \(f\) is irreducible modulo \(p\). Its order of mixing \(M\) was known to satisfy…
AIM-TOPOLOGY-0276Explicit Witt-family worked example
We give four explicit matrices defining a genus-two surface-group representation over \(\mathbb{R}[s,t,1\)\(/(st(1\)\({}+t^2))]\). Every real specialization has zero Witt class and zero Toledo number. Over \(\mathbb{R}(s,t)\), however, the Witt class is the anisotropic Pfister form \(\langle 1,-(1\)\({}+t^2)\rangle\)\({}\otimes\langle 1,-s\rangle\), of additive order two. A nonzero…
AIM-COMPUTATION-0020Sharp conditional-ratio theorem
Fix the support of a binary \(d\)-way probability table, its feasible interior one-way margins, and every conditional pairwise odds ratio whose four cells remain positive. The largest possible dimension of the resulting family is \(\lceil 2^{d+1}\)\(/3\rceil\)\({}-d\)\({}-1\) for \(d\ge2\), even with uniform margins. For \(d\ge3\), equality holds…
AIM-COMPUTATION-0095Sharp bound and unrestricted-exponent counterexample
For a prime power \(q\) and integers \(m\ge n\ge2\), let \(D_q(n,m)\) be the least degree bound on a polynomial \(g\in\mathbb F_q[X]\) for which \(X^m+g(X)\) has an irreducible factor of degree \(n\). We give the elementary sharp identity \(\max_{m\ge n}D_q(n,m)\)\({}=n\)\({}-1\): equality holds whenever \(m\)\({}\equiv\)\({}-1\pmod{q^n-1}\). More generally, the residue \(m\equiv-j\) forces \(D_q(n,m)\ge n-j\)…
AMR-011-0009Normal-subgroup intersection counterexample
We construct a family of finite-index normal subgroups of a fixed finitely generated free group which has property \((\tau)\), whereas its prefix-intersection chain does not. The construction uses Kassabov's bounded-degree alternating-group expanders. Two quotient maps have identical expanding generators…
AMR-103-0120Complete counterexample; classification by gauge group
Guadagnini and Pilo conjectured that for a closed oriented 3-manifold \(M\) and a compact gauge group \(G\) the absolute value of the quantum invariant \(\tau^G_r(M)\), when it does not vanish, depends only on the fundamental group of \(M\); this is Conjecture 7.5 of Ohtsuki's problem list. They proved it for lens spaces…
AMR-011-0049Complete scoped theorem
We give the exact probability distribution of every group word, with no constants and any number of variables, on the full unitriangular group \(\operatorname{UT}_4(\mathbb F_q)\), for every finite field. A nonzero exponent sum modulo the characteristic makes the word map uniform. Otherwise the distribution is determined by…
AIM-PROBABILITY-0093Negative answer at the stated scale
For a standard Brownian excursion of duration \(T\), exact area \(\rho T\), and maximum \(M_T\), with fixed \(\rho>0\), we prove…
AIM-CONVEX_GEOMETRY-0001Minimum-perimeter theorem
For every integer \(n\ge3\), a central hyperplane section of the side-one real cube satisfies…
AIM-CONVEX_GEOMETRY-0045Octahedron minimum-volume theorem
We determine the minimum volume of the intersection of a regular octahedron with a centered slab of every prescribed width, including all equality directions. For the octahedron with vertices \(\pm e_i\) and slab half-width \(t\), the minimizing normal is diagonal below a single threshold \(\tau\)\({}=0.4740349423895888\)\(\ldots\)…
AIM-OTHER-0011Corner-factor theorem
We construct an explicit binary-word factor of Striker–Williams superpromotion on alternating sign matrices of order \(n\). The membership indicator of a specified corner of the ASM poset returns after \(3n-2\) steps. Its itinerary of that length therefore intertwines superpromotion with cyclic rotation.…
AIM-SEVERAL_COMPLEX_VARIABLES-0028Zero-curve theorem and counterexample
Let \(p\) be a nonzero homogeneous plurisubharmonic polynomial on \(\mathbb C^2\). We prove that every irreducible affine algebraic curve avoiding the origin on which \(p\) vanishes is exactly a nonzero level of a homogeneous holomorphic polynomial of degree at most \(\deg p\). No smoothness or rationality assumption is needed.…
AIM-COMBINATORICS-0112Scoped theorem and orientation counterexample
A fitting orientation of a simplicial rooted forest specifies an unsigned chip-firing rule on its nonroot ridges. In every dimension \(d\ge2\), a six-facet rooted forest with relative boundary determinant one admits cyclic sandpile groups of orders \(d^6-d^3-d\) and \(d^6-2d\) for two fitting orientations. When…
AIM-ARITHMETIC_GEOMETRY-0050Scoped scheme-theoretic theorems
We construct canonical closed subschemes representing relative Frobenius factorization of a morphism from a smooth projective curve family, including over nonreduced bases. For Gauss maps of smooth plane curves of degree d in characteristic p, we identify these subschemes with explicit linear…
AIM-COMBINATORICS-0257Scoped lower-bound theorem
For each \(a\in\{1,3\}\), infinitely many primes \(p\equiv a\pmod4\) admit a set of exactly \(p\) points in \(\mathbb F_p^2\) with at least \(p^{1.0058}\) unordered pairs at quadratic distance one. We transfer Sawin's number-field unit-distance construction to prime fields using a field-uniform quantitative Chebotarev estimate of Zaman.…
PE-1012Complete theorems and a new constant
Two players choose numbers from \(\{1,\dots,n\}\). Equal numbers draw, an odd difference is won by the smaller number, a nonzero even difference is won by the larger number, and a win with the number \(m\) pays \(2m-h\), where \(0\le h\lt 2\) is fixed. For \(h=1\) this is the game of Problem 1012 of Project Euler; for \(h=0\) the payment is proportional…
AIM-PROBABILITY-0013Scoped critical-point theorem
We classify the finite critical points of exact empirical log likelihood for binary restricted Boltzmann machines in which every hidden unit has at most two visible neighbors. All allowed weights and biases are independent real parameters. A pairwise exponential-family representation and its…
AIM-TOPOLOGY-0043Scoped quantum-torsion theorem
We extract an explicit infinite family of torsion elements from the two-solid-torus presentation of Bakshi, Le and Przytycki. Over \(R\)\({}=\mathbb{Z}[q,q^{-1}]\), put \(B=R[x_1,x_2]\) and \([m]\)\({}=(q^{2m}\)\({}-q^{-2m})\)\(/(q^2\)\({}-q^{-2})\). For every \(m\ge 2\) we construct an element whose exact \(B\)-annihilator is \(([m])\), and these elements embed the direct…
AMR-068-0001Complete four-point classification
We classify the stress-sign stratification of all labeled four-point configurations in \(\mathbb{R}^3\) for the complete graph \(K_4\), including coincident points and every affine-rank degeneration. There are exactly 65 connected strata: two of affine rank three, 25 of affine rank two, 37 of affine rank one, and the…
AIM-PROBABILITY-0042Complete logarithm-removal theorem
Let \(G\) be a connected loopless Eulerian directed multigraph with \(n\) vertices and \(m\) arcs, and let \(X\) be its half-lazy simple random walk. For every deterministic target sequence \((u_t)\), we prove that the expected hitting time is at most \(t_{\mathrm{unif}}(1\)\(/4)\)\({}+10m(n\)\({}-1)\)\({}=O(mn)\), uniformly in the starting vertex. This removes…
AIM-COMBINATORICS-0263Complete one-point classification
Let \(p\) be an odd prime, \(G\) a proper multiplicative subgroup of \(\mathbb{F}_p^*\), and \(e\) outside \(G\). This preprint classifies every unrestricted self-sumset \(A+A=G\) union \(\{e\}\), including diagonal sums. They occur exactly when \(p\ge 5\), \(G=\{-1,1\}\), and \((e,A)\) is \((0,\{-1/2,1/2\})\), \((3,\{-1/2,3/2\})\), or \((-3,\{-3/2,1/2\})\). A differential…
AMR-014-0007Complete geometric count for (4,2)
A general complex binary octic has exactly 76 decompositions as a sum of three fourth powers of quadratic forms, modulo permutations and independent fourth-root-of-unity rescalings. This preprint gives a geometric proof of the count, previously obtained numerically by Kowalczyk and Vill.…
AIM-GEOMETRY-0171Complete spherical convergence proof
We prove smooth convergence of the normalized inverse Gauss curvature flow on the two-sphere for every smooth initial conformal metric of positive Gauss curvature. No smallness or symmetry assumption is imposed, and convergence takes place in the original conformal coordinates. The…
KP-4.124Computer-assisted theorem (related result; the problem stays open)
Problem 4.124 of the K3 problem list asks whether a hyperbolic integer homology 4-sphere exists, whether an arithmetic one exists, and more generally whether there is a closed hyperbolic 4-manifold with Euler characteristic 2. By a theorem of Belolipetsky, an orientable arithmetic example would be…
AIM-TOPOLOGY-0275Complete split F4 product-surface theorem
Let \(\mathsf F\) be the connected linear split real group of type \(F_4\), and let \(Q_\rho\) be its canonical oriented three-plane bundle, obtained from a maximal compact reduction of a flat \(\mathsf F\)-bundle. We determine the range of the characteristic number \(\langle p_1(Q_\rho),[\Sigma_g\)\({}\times\Sigma_h]\rangle\) over all representations of a product of two closed…
OPG-57403Complete answers (a proof and a counterexample)
A reloid from a set \(A\) to a set \(B\) is a filter on \(A\times B\). Reloids were introduced by V. Porton as a common generalisation of binary relations and uniformities. We settle two problems on reloids from Porton's book "General Topology as Ordered Semigroup Actions" which are also listed in the Open Problem Garden.…
AIM-GEOMETRY-0038Complete finite-type comparison theorem
We compare two positive subsets of a braid variety: the positive locus of its weave cluster structure and the geometric positive locus defined by the joint nonnegative closure in a twisted product of flag varieties. For every positive braid word of finite crystallographic simple type, with a fixed…
KP-2.28Partial answer (explicit classes of graphs; the problem stays open)
Problem 2.28 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026), recorded by T. Koberda, asks: if \(\Gamma\) is a finite graph that is not a join, is there an injective homomorphism from the right-angled Artin group \(A(\Gamma)\) to the mapping class group of a…
AIM-GEOMETRIC_GROUP_THEORY-0044Complete proof
We prove that an Artin group abstractly isomorphic to an Artin group of spherical type is itself of spherical type. No condition is imposed on the isomorphism or on the rank of the source presentation. The main observation is that, for an irreducible nonspherical Artin group, the center has zero…
OWR-15427-019Complete dense-regime theorem
Let \(F(n,m)\) be the minimum number of unordered collinear triples in an \(m\)-point subset of \([n]^3\). We prove that \(F(n,m)\) has order \(m^4/n^6\) whenever \(12n^2\le m\le n^3\), with absolute constants. The lower bound removes the logarithmic loss in the Balogh-Solymosi supersaturation estimate. Its proof counts all short primitive…
AMR-067-0008Partial answer (three further cases; one regime stays open)
Gil-Medrano asked whether, in the real projective space \(\mathbb{RP}^{2n+1}\) with a Berger metric \(g_\mu\) (the round metric with the Hopf fibres rescaled by \(\sqrt\mu\)), the projective subspaces coming from equatorial \(k\)-spheres are minimal, and whether they are the only volume-minimising \(k\)-cycles in their homology class. A recent…
AMR-022-5039Partial answer (exact for p ≤ 2; bounds for p > 2)
Let \(g\) be subordinate to \(f\) in the unit disc, that is, \(g=f\circ\varphi\) with \(\varphi\) analytic, \(|\varphi|\lt 1\) and \(\varphi(0)=0\). Goluzin proved in 1951 that \(M_2(r,g')\)\({}\le M_2(r,f')\) for \(r\le 1/2\), and that \(M_p(r,g')\)\({}\le M_p(r,f')\) for every \(p\gt 0\) when \(r\le\sqrt2-1\). Problem 5.39 of Hayman and Lingham's Research Problems in Function Theory, posed by P. L.…
AMR-044-0001Partial answer (periodicity criterion; periods in special cases)
Dragović and Radnović posed the following problem (Arnold Math. J. 1 (2015)). A billiard table is bounded by two concentric half-circles of radii \(R_1>R_2\), lying on opposite sides of a common diameter, and by the two segments of this diameter between them. Consider the trajectories tangent to a…
AIM-GEOMETRY-0111Complete scoped 3D theorem
For any packing of unit balls in three-dimensional Euclidean space, we construct two fixed modifications preserving its upper and lower soft-density profiles at every fixed outer radius. One modification has an unbounded vacant component at every finite radius. The other has infinitely many…
AMR-096-0008Partial answer (negative answers to two items as literally posed)
The list of open problems of D. Aldous contains the following question, dated March 2009. On a finite Cayley graph let \(\mu_p\), \(0<p<1\), be the law of \(X(T_p-1)\), where \(X\) is the random walk started at the identity and \(T_p\) is a geometric time with parameter \(p\); let \(\mu_0\) be the uniform law; and let \(\tau(p)\) be the…
AIM-GEOMETRY-0112Complete lattice theorem
For a Euclidean lattice whose nonzero vectors have length at least two, consider the density of the union of balls of radius \(1+\lambda\) centered at its points. We prove that, in every fixed dimension \(d\ge 2\) and for all sufficiently small \(\lambda\gt 0\), the maximizing lattices are exactly the densest hard-sphere lattices…
AMR-049Seven conjectures decided (four proved, one proved in corrected form, two disproved)
In a problem contribution (Arnold Math. J. 7 (2021)), D. Fuchs stated seven conjectures on the billiard trajectories in a regular \(n\)-gon which join two vertices (short trajectories) and on their \(n-2\) types \(A_0,\dots,A_{n-3}\), on closed trajectories, and on geodesics on the regular dodecahedron. We decide all…
AIM-OTHER-0067Complete crossed-product theorem
Let a finite group \(G\) act on an additive braided rigid monoidal category by additive strong monoidal autoequivalences. Every tensor-nilpotent object of the crossed product has nilpotency index at most the order of the subgroup generated by its nonzero homogeneous degrees. The bound applies to…
AIM-ANALYSIS-0146Complete finite-operator theorem
We classify the nonzero finite-order differential operators with real polynomial coefficients that preserve hyperbolicity and retain a positive uniform fraction of additive zero mesh on the whole Laguerre-Polya class: they are exactly the nonzero scalar multiples of the identity. For every other…
OWR-8415343-001Answered: the estimate holds for every smooth G (explicit constants)
P. T. Chruściel asked (Oberwolfach Reports 18 (2021)) whether the classical composition estimate in the Sobolev space \(H^s(\mathbb{T}^2)\), \(s>1\), survives when the functions depend on a parameter \(t\) and the supremum over \(t\) is taken for each Fourier coefficient separately, inside the weighted sum. With \(\mathcal{N}_s(u)\)\({}=\sum_{\ell\in\mathbb{Z}^2}(1\)\({}+\lvert \ell\rvert)^{2s}\sup_t\lvert u_\ell(t)\rvert^2\):…
OWR-16415-018Partial answer (four-holed spheres, genus 2 only)
In the problem session of the Oberwolfach workshop “New Trends in Teichmüller Theory and Mapping Class Groups” (2018), D. Gekhtman asked: given a finite cover \(f\colon S_h\to S_g\) of closed orientable surfaces with \(h>g\ge2\), is there always a four-holed sphere \(H\subset S_g\) with essential boundary components such that…
AIM-SEVERAL_COMPLEX_VARIABLES-0021Complete counterexample
We construct a meromorphic germ in \(\mathbb{C}^3\) that is holomorphic from an attached wedge into the open unit ball, extends continuously to its real-algebraic generic edge with values on the unit sphere, and nevertheless has no holomorphic extension to a full neighborhood of the edge point. The edge has…
AIM-DYNAMICAL_SYSTEMS-0042Complete proof (stated special case)
We determine joint Galois images for finite families of normalized Chebyshev maps with nonconstant affine targets over function fields. Over \(\mathbb{C}(t)\), endpoint graphs and degree parity determine the geometric image; least common multiples merge repeated endpoint pairs, while greatest common divisors…
AMR-046-0017Complete special-case proof
We prove that a planar two-zone piecewise-affine system with a straight switching line, two strict off-line foci, and at least one virtual focus has at most three isolated crossing limit cycles. The estimate includes nonhyperbolic cycles and every switching offset. Known three-cycle examples show…
AIM-COMBINATORICS-0183Complete special-class proof
Let \(D\) be a finite simple oriented graph with indegree and outdegree three at every vertex. A Seymour vertex has at least as many exact second out-neighbors as first out-neighbors. We prove that at least \(\lceil 3|V(D)|\)\(/5\rceil\) vertices are Seymour vertices. If \(B\) is the set of the remaining vertices, the sum…
OWR-8415347-019Answer for every N (no for N = 2, 3; yes for N ≥ 4); nilpotent residual of O(O_N^+) determined
In the problem session of the Oberwolfach workshop “Quantum Groups – Algebra, Analysis and Category Theory” (Report 44/2021), Freslon, reporting on joint work with Franz and Skalski, asked three questions about the nilpotent residual \(I^\infty\)\({}=\bigcap_nI^n\) of the augmentation ideal \(I\) of a Hopf algebra \(A\): for which…
OWR-16163-011Partial answer (structure of the measure; no closed formula)
M. Gerhold asked for a concrete description of the vacuum distributions \(\mu_t\) of the operators \(\lambda_t^*\)\({}+\lambda_t\)\({}+\rho_t^*\)\({}+\rho_t\) on the monotone Fock space, that is, of the sum of the two faces of bi-monotone Brownian motion (Oberwolfach Rep. 15 (2018)); the same question closes his paper on bi-monotone Brownian motion,…
AIM-GEOMETRY-0315Complete scoped proof and example
Let \(K\) be the dual boundary complex of a compact simple polytope, and let an integral matrix define a coordinate-torus quotient, with nonsingular restriction at each vertex. We show that restriction from coordinate-torus equivariant cohomology to the kernel subgroup is surjective over the integers…
OWR-14299905-036Partial answer (two constants determined, with the extremal grids; computer-assisted)
For a word \(w\) and a toroidal grid of letters in dimension \(d\), an appearance of \(w\) is a cell together with one of the \(3^d-1\) directions in \(\{-1,0,1\}^d\)\({}\setminus\{0\}\) along which the letters spell \(w\), and \(C_d(w)\) is the supremum, over all grids, of the number of appearances per cell. We prove that \(C_2(\mathsf{ABBB})\)\({}=8\)\(/5\). This answers…
OWR-13494-011Complete negative answer (classification theorem)
For a finite union \(L\) of lines in \(\mathbb{P}^3\) with homogeneous ideal \(I\), let \(\alpha(I)\) be the least degree of a non-zero form in \(I\), and let \(I^{(2)}\) be the second symbolic power of \(I\), the ideal of the forms that vanish to order at least two along every line of \(L\); one always has \(\alpha(I^{(2)})\ge\alpha(I)+1\). Janssen proved (J. Pure…
OWR-1703876-012Complete counterexample for every n ≥ 6; corrected criterion
A closed polygon in \(\mathbb{R}^3\) has at each vertex a turning angle in \((0,\pi)\), and a sequence \(A=(\alpha_0,\dots,\alpha_{n-1})\in(0,\pi)^n\) is called realizable if it is the sequence of turning angles of such a polygon. Efrat, Fulek, Kobourov and Tóth conjectured (Oberwolfach Rep. 17 (2020); J. Graph Algorithms Appl. 26 (2022)) that a…
OWR-3385-009Complete negative answer
Problem 7 of the problem list of the Oberwolfach workshop “Discrete Differential Geometry” of 2009 (collected by G. Rote; Oberwolfach Reports 6, p. 108), posed by I. Izmestiev, asks whether there is a convex 4-polytope whose 1-skeleton is linked as a graph in the boundary of the polytope, a…
AMR-103-0016Problem answered in print by others; sharper bound and short proofs
Problem 1.16 of Ohtsuki's list (E. Ferrand, A. Stoimenow) asks whether every alternating link \(L\) satisfies \(\sigma(L)\)\({}\ge\operatorname{min\,deg}_l P_L\)\({}\ge\operatorname{min\,deg}_a F_L(a^{-1},z)\), where \(\sigma\) is the signature, \(P_L(l,m)\) the HOMFLY polynomial and \(F_L(a,z)\) the Kauffman polynomial. Both inequalities are in print: the second is a corollary of…
AIM-ANALYSIS-0055Complete fixed-double-pole theorem
We give an all-parameter degree calculation for the birational maps \(F(x,y)=(\beta/x+\gamma/x^2-\delta y,x)\) over the complex numbers, with \(\gamma\delta\ne0\).
AMR-036-0017Complete counterexample to the proposed alternatives
Let \(P\) be the compact-open closure of the polynomials whose zeros lie in the closed lower half-plane. The entire function \(e^{iz}+z+2i\) has every zero of every derivative in that half-plane, but lies neither in \(P\) nor in the exceptional families.
AMR-036-0023Auxiliary counterexample and conditional theorems
For a locally finite configuration of positive masses in \(\mathbb R^n\), \(n\geq3\), we study spheres on which the Newtonian potential is uniformly small. Unit masses at the positive integers contradict the dimensional extension of an auxiliary lemma of Clunie, Eremenko and Rossi for every \(n\geq4\); the same example nevertheless has infinitely many nondegenerate equilibrium points.
AIM-TOPOLOGY-0097Coxeter higher-A vanishing theorem
We prove that the standard Cayley graph of every finite-rank Coxeter system has trivial \(A\)-homotopy groups in all degrees \(j\geq2\). The argument replaces the Coxeter group by the right-angled Coxeter group retaining exactly its commuting relations.
AIM-GEOMETRIC_GROUP_THEORY-0042Sharp multipartite spectral theorem
Partition at least four vertices into at least two nonempty parts. Assign Coxeter exponent \(2\) within each part, a fixed exponent \(m\geq5\) between parts, and replace selected cross-part exponents by \(3\).
OWR-12697693-003Complete negative answer (already for left braces)
An element \(x\) of a skew brace \(B\) has property (s), in the sense of Colazzo, Ferrara and Trombetti, if it has only finitely many conjugates of each of the kinds \(x*b\), \(b*x\), \(b+x-b\) and \(b\circ x\circ b^{-1}\). In her abstract in Oberwolfach Report 9/2023, Colazzo asks whether the elements with property (s) of a skew brace…
AMR-022-7031Complete answer: comparison theorem and characterization
Problem 7.31 of Hayman and Lingham's collection Research Problems in Function Theory, proposed by W. K. Hayman, concerns sequences with \(a_1>0\) and \(0\le a_n\le n\) and their partial sums \(b_n=a_1+\dots+a_n\), \(c_n=b_1+\dots+b_n\). It states that \(\sum(a_n/c_n)^\alpha<\infty\) for \(\alpha>\frac12\), and asks for which functions \(f\) the series \(\sum f(c_n/a_n)\) converges, and…
AIM-GEOMETRY-0201Singular example and minimality theorem
We give an explicit planar framework with five distinct vertices spanning the plane and six bars whose locally one-dimensional configuration set has two analytic branches at a reflection-symmetric configuration. Both outgoing branches are expansive and immediately break the reflection. Every…
AIM-PROBABILITY-0118Attributed continuation theorem
We derive preservation of symmetric weak unimodality under every real free additive convolution power \(t\ge 1\), without support, moment or density assumptions. The essential input is the Cauchy-transform inequality in Anonymous's unrefereed preprint Symmetric unimodality under free additive convolution…
AIM-PROBABILITY-0053Complete weighted Gibbs theorem
We consider uniform random-scan coordinate Gibbs sampling on \([0,1]^d\) for a density proportional to \(\exp(-a\sum_{i<j}c_{ij}(x_i-x_j)^2)\), with symmetric nonnegative weights. For every fixed nonzero weight matrix, we prove a worst-start total-variation upper bound of order \(a\)…
OWR-15993-009Five steady states for chain lengths N ≥ 651 (computer-assisted); at most three for N ≤ 4
Rendall asked, in an abstract on joint work with Sontag (Oberwolfach Report 8/2018), whether the ODE model for the initiation of T cell activation of François, Voisinne, Siggia, Altan-Bonnet and Vergassola has parameter values with more than three steady states. In the form studied by Rendall and…
OWR-12697710-006Explicit counterexamples in five and in seven variables (the first negative answer is due to a prior preprint)
For an even degree \(m\) and an odd exponent \(2k+1\), let \(\Sigma_{n,m}(2k+1)\) be the set of real forms \(f\) of degree \(m\) in \(n\) variables such that \(f^{2k+1}\) is a sum of squares of forms. B. Reznick asked (Oberwolfach Report 14/2023, p. 779) whether this set is a closed convex cone. It is a closed cone; whether it is convex…
OWR-16164-020Complete proof: divisibility by the Catalan power of det A for every n
Teufl and Wagner posed the following problem (Problem 1 of their contribution to Oberwolfach Report 23/2018, pp. 1457–1458). Let \(A=(a_{ij})\) be an \(n\times n\) matrix of indeterminates, and let \(T=T(A)\) be the matrix, indexed by the set partitions of an \(n\)-set, whose entry \((P,Q)\) is the sum of the weights of the…
AIM-PROBABILITY-0028Complete scoped counterexample
We give a fixed, smooth, sub-Gaussian noise distribution for which the bias of the Gaussian Stein unbiased risk estimate does not vanish after division by dimension, even for ordinary Haar soft thresholding. For every dyadic dimension \(d\ge2\), threshold \(1/8\) and signal constant on each finest coordinate…
OWR-793-006Complete negative answer (an explicit primitive group of twisted wreath type)
Let \(G\) be a finite primitive permutation group of twisted wreath type, with socle \(T^k\) and point stabiliser \(P\), and let \(H\) be its primitive component: the group induced on one coordinate of the product decomposition \(\Delta^k\), where \(|\Delta|=|T|\). Praeger and Seress (Oberwolfach Report 12/2005, on joint work with…
AMR-029-0044Complete proof: the path condition suffices for all roots and all convex sets
Let \(D\) be a finite acyclic digraph with root-nodes \(r_1,\dots,r_k\), not necessarily distinct, and convex node sets \(U_1,\dots,U_k\) with \(r_i\in U_i\). The EGRES Open collection lists as open the question, proposed by S. Fujishige, N. Kamiyama and N. Katoh, whether there are independent \(r_i\)-arborescences \(F_i\) with \(V(F_i)=U_i\) if…
AMR-109-0082Observation: negative answer (unbounded multiplicity for every term of the Johnson filtration)
Farb asked (Question 7.6 of Some problems on mapping class groups and moduli space) whether \(\operatorname{spec}(\mathcal{I}_g(k))\), the set of the logarithms of the dilatations of the pseudo-Anosov elements of the \(k\)-th term of the Johnson filtration of the mapping class group \(\operatorname{Mod}_g\), has bounded multiplicity for \(k\ge3\); the…
AIM-ANALYSIS-0041Complete scoped theorems
Coordinatewise completeness imposes an arithmetic restriction on the dimension of unweighted \(L^p\)-Bergman spaces. For every positive rational \(p\), such a space on a complete Reinhardt domain in any complex dimension is either zero or infinite-dimensional. In \(\mathbb C^2\), the only possible finite positive…
AIM-ANALYSIS-0056Complete scoped criteria
For a class of holomorphic skew products tangent to the identity, we give necessary and sufficient conditions for an open attracting domain and for an open domain attracted along the base axis. The transverse leading coefficient is a fixed matrix with real spectrum and arbitrary Jordan blocks. If…
AIM-PHYSICS-0055Complete planar theorem
We prove that a collision-free noncollinear planar Newtonian three-body motion with constant triangle area divided by barycentric moment of inertia is necessarily a Lagrange homographic motion. The masses are arbitrary positive numbers, and the conclusion holds on any open time interval.…
OWR-4990374-015Complete proof of the conjecture in every rank (characteristic zero)
Ş. O. Tohăneanu conjectured (Oberwolfach Report 5/2021, p. 284; Conjecture 4.1 of the preprint arXiv:1707.07091v3) that every central free hyperplane arrangement over a field of characteristic zero whose exponents are \(1\)'s, \(2\)'s and at most one \(3\) is supersolvable. We prove this conjecture in every…
AIM-ALGEBRAIC_GEOMETRY-0211Complete scoped toric theorem
An ordered projective anchor simplex and \(n\) further marks in its complement determine a configuration torus modulo diagonal translation. Choosing a pivot mark identifies this quotient with a product of tori, which has a product-permutohedral compactification. We construct a pivot-independent normal…
AMR-020-0202Complete scoped counterexample
Consider the projection of the four-torus onto its last two coordinates, equipped with a constant magnetic symplectic form with real parameter c. We compare global symplectic equivalence with equivalence preserving the Lagrangian fibration. Fibre-preserving equivalence identifies exactly parameters…
AMR-011-0031Complete counterexample
We construct a connected seven-regular Cayley graph of \(\left(C_5*\left(C_2\times C_2\right)\right)\)\({}\times C_2\) with chromatic number three but no translation-invariant random proper three-coloring. A central involution partitions two finite subgroup graphs into pairs. Every proper three-coloring has at least four equal-colored pairs…
AMR-050-0043Complete scoped proof
Fix an ellipse and a nondegenerate confocal elliptic caustic admitting a periodic billiard family. We prove that inversion of the vertices about either focus preserves a phase-independent sum of internal-angle cosines, provided the genuine least period is not four. The proof includes every…
OWR-741-001Complete counterexample from rank 6 on; proof up to rank 5
Knutson and Tao proved that a nonempty hive polytope has an integral vertex. De Loera and McAllister conjectured that this extends to arbitrary integral right-hand sides: if \(B\) and \(R\) are the matrices of the boundary equations and of the rhombus inequalities of hive patterns of rank \(r\), every nonempty…
AIM-COMBINATORICS-0007Complete quantitative counterexample
We construct smooth, real, mean-zero, 1-bounded functions on high-dimensional spheres whose average product over zero-sum unit triangles tends to 7/16384, while their correlation with every function of \(m\) linear coordinates is \(O(\sqrt{m/d})\) when \(m=o(d)\). The bound is uniform over all choices of coordinates…
AIM-PROBABILITY-0086Complete fixed-parameter proof
We give a spatial mollifier construction of the O'Connell-Warren multilayer stochastic heat equation directly from nonintersecting Brownian bridges with coalesced endpoints. Both the fully Wick-ordered exponential of the total path energy and the product of separately Wick-ordered path weights…
AIM-GEOMETRIC_GROUP_THEORY-0059Complete scoped corollary
Let \(A_S\) be a finitely generated Artin group such that the sum of the reciprocals of the three labels on every triple of generators is strictly less than one, with \(1/\infty=0\). This note records a direct consequence of the Martin-Przytycki coned-off Deligne construction and the domination criterion…
AIM-SEVERAL_COMPLEX_VARIABLES-0011Complete scoped theorem
For a fixed integer \(k\ge2\), let \(N_k(t)\) be the least positive integer \(n\) for which \(((1+x)^{2k}-tx^k)^n\) has nonnegative coefficients. Known positivity theorems imply that \(N_k(t)\) is finite for \(0\)\({}\le t\)\({}\lt T_k\)\({}=2^{2k-1}\), whereas no exponent works at \(t=T_k\). We give an explicit necessary lower bound as \(t\) approaches this boundary.…
AIM-ALGEBRAIC_NUMBER_THEORY-0077Complete scoped corollary
For \(\ell\in\{11,13\}\) and every \(m\geq1\), we identify the semisimple \(\ell\)-torsion field of the Jacobian of the connected principal modular curve \(X(\ell^m)\) over \(K_m\)\({}=\mathbb Q(\zeta_{\ell^m})\). It is the compositum of \(K_m\) with the field cut out by the residual Ramanujan Delta representation, and its Galois group over \(K_m\) is \(\operatorname{SL}_2(\mathbb F_\ell)\). The…
AIM-ANALYSIS-0148Complete scoped theorem and counterexample
Let \(S\) be a finite-codimensional subspace of \(\mathbb R[x]\) with a positive definite moment inner product, and let \(q_n\) be its degree-ordered monic orthogonal basis. For a fixed nonzero polynomial \(a\) of degree \(d\), an eventual scalar band recurrence for \(a q_n\) exists exactly when multiplication by \(a\) preserves \(S+S^\perp\),…
AIM-TOPOLOGY-0101Complete scoped nonuniqueness counterexample
We give an explicit nonuniqueness example for the (J1, categorical-product) Eilenberg-Steenrod axioms on Cech closure spaces. Applying the known clique and closed neighborhood complex constructions to the mutual singleton graph gives two integral relative homology theories, with subspace complexes…
AMR-040-0001Complete scoped counterexample and theorem
For a finite metric \(d\), its fundamental polytope is the convex hull of \((e_i-e_j)/d_{ij}\). We answer the universality clause of Vershik's question about Euclidean metrics: not every abstract combinatorial type is Euclidean. The four-cycle metric gives a cube, whereas four Euclidean points give a fundamental…
OWR-4136-005Complete construction-specific counterexample
Rademacher proposed a half-ball joined to a cone, with a small tip truncation, as a counterexample to monotonicity of the expected volume of a random tetrahedron under inclusion. We prove that this specific construction works in dimension three, with cone height \(10^{-5}\) and truncation height \(10^{-10}\). An…
OWR-15957-003Complete scoped theorem
The cp-plus-rank is the smallest number of columns in a strictly positive Gram factor. We prove that, in every order \(n\), its largest finite value equals the maximal cp-rank \(p_n\). A strictly positive factor stays positive under a sufficiently small contraction along the Perron eigendirection. Together with…
OWR-15221-002Complete scoped theorem
For a Poisson support-boundary experiment with fixed known bins and independent positive locally log-Lipschitz priors, we prove posterior equivalence to the product-exponential reference when \(\sum_j(n\Delta_j)^{-2}\to0\), uniformly over bounded true heights. Equal bins yield \(K=o(n^{2/3})\).…
OWR-1111-001Complete answer: NP-hard, with and without edge costs
In the problem Blocking Shortest Paths one is given an undirected graph with edge costs and some pairs of vertices, and asks for an edge set of minimum cost whose deletion increases the distance of each pair by at least \(1\); that is, the set must meet every shortest path between the two vertices of…
AIM-ANALYSIS-0095Complete scoped theorem
For a complex Parseval frame of \(m\) vectors with squared norms at most \(\alpha\), we construct an entire coordinate subcube of good signings at the historical MSS bound \(\sqrt{8\alpha}+2\alpha\).…
AIM-PROBABILITY-0133Complete scoped counterexample
For every \(s_0\gt 0\), \(q\gt 1\) and finite integer \(K\geq 1\), we construct a continuum of distinct affiliated \(R\)-diagonal laws with identical joint star-moments and identical first \(K\) alternating cumulants of \(c(1+s c^*c)^{-1/2}\) at every scale \(s=s_0q^k\), \(k\in\mathbb{Z}\).…
AIM-GEOMETRIC_GROUP_THEORY-0126Complete scoped proof
We give an explicit generating pair for the torsion-free infinite simple group in Corollary IV(ii) of Radu’s paper on lattices in products of trees. The pair is \(x_1x_3\) and \(x_1y_3\). A short group calculation using only the first five displayed equations recovers every original generator. Thus this…
AIM-GEOMETRY-0095Complete scoped proof
We prove that, for every sufficiently large real exponent \(s\), the unique five-point minimizer of chordal Riesz \(s\)-energy on the unit two-sphere, up to orthogonal congruence and relabeling, is a square pyramid.
AMR-109-0020Negative answer to the example question: not unique
Problem 2.19 of Farb's list of problems on mapping class groups asks for properties of automorphism groups that determine a unique point of the moduli space \(\mathcal M_g\), and asks as an example whether there is a unique Riemann surface of genus \(g\ge2\) whose automorphism group is nilpotent and of the largest…
OWR-13494-009Containment proved for every n = 3m (cluster reading)
In an Oberwolfach report of 2015 Harbourne considered the configurations \(C_n\) of \(\frac43(n^2-3)\) plane cubics due to Roulleau and Urzúa, defined when \(3\) divides \(n\), and asked whether \(I^{(3)}\not\subseteq I^2\) for the ideal \(I\) of the singular points of \(C_n\). He reported that for \(C_3\) the containment \(I^{(3)}\subseteq I^2\) holds, with \(I\) the ideal…
AMR-105-0051Complete answer: yes, explicit examples
Problem 51 of the list of unsolved problems in virtual knot theory and combinatorial knot theory by Fenn, Ilyutko, Kauffman and Manturov (2014) asks to prove or disprove the conjecture that some free link has several minimal representatives, that is, non-isomorphic diagrams with the least number of…
OWR-793-006Negative answer: an infinite family attains the bound
Let \(G\) be a finite primitive permutation group of twisted wreath type, with socle \(T^k\) and primitive component \(H\). Giudici, Li, Praeger, Seress and Trofimov (2006) proved that the least non-trivial subdegree of \(G\) satisfies \(\operatorname{MinSubDeg}(G)\)\({}\le k\)\({}\cdot\operatorname{MinSubDeg}(H)\), and asked whether this inequality is always strict…
OWR-1738-011Complete answer: both bounded, by 4
Gonçalves and del Río asked two questions in the problem session of Oberwolfach Report 55/2007 (Problem 28); the first is also the first half of Question 5.4 of their paper in J. Group Theory 11 (2008). A nonzero complex number \(z\) is a free point if the matrices \(\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)\)…
AMR-014-0004Counterexamples: the conjecture is false
For a complex binary form \(f\) of degree \(kd\), the \(k\)-rank of \(f\) is the least number of \(k\)-th powers of binary forms of degree \(d\) whose sum is \(f\). Fröberg, Lundqvist, Oneto and Shapiro conjectured (Arnold Math. J. 4 (2018), Conjecture 1.9) that for all \(k\) and \(d\) the maximal \(k\)-rank of binary forms of degree \(kd\)…
AMR-099-0065Partial answer: the recursive constant exceeds the BRW constant
In Question 9.1 of his survey “Euclidean vs. Graph Metric” (2013), Benjamini considers first-passage percolation on the hierarchical graphs \(G_n\) obtained by replacing, recursively, every edge by one edge followed by two parallel edges. With independent positive edge weights of law \(\xi\), the source–sink distance…
AMR-096-0023Part (a) proved; parts (b), (c) false in a parameter range
In rough notes of 2007 on a spatial model of city growth, D. J. Aldous stated a three-part conjecture about the unbalanced regime. Immigrants arrive one at a time at uniform random points of the unit square; a city of population \(n\) exerts the influence \(c_0n^\alpha r^{-\beta}\) at distance \(r\); an immigrant founds…
AMR-005-0008Complete proof
For a Lorentz ellipsoid, we compute the positive unreduced rotation lift of the equator return map formed by alternating null geodesics.
AIM-PROBABILITY-0113Complete proof of the CAR subquestion
Let \(C_{m,n}\) be an \(m\times n\) array of \(mn\) distinct self-adjoint Clifford generators, each squaring to one. Its squared operator norm is \(2(m+n-2)\) for \(m,n\ge2\), and \(\max(m,n)\) when \(\min(m,n)=1\).
AIM-TOPOLOGY-0147Complete proof for the stated entropy model
Let \(\mu_g\) be normalized Weil–Petersson measure on the moduli space of closed hyperbolic surfaces of genus \(g\). Positive mass on infinite-genus surfaces therefore requires superlogarithmic surface relative entropy.
OWR-15587-006Counterexamples: lattice hulls of balls need not be normal
Michałek asked in 2017 (Oberwolfach Report 44/2017, Conjecture 1) whether the lattice hull of a Euclidean ball \(B\subset\mathbb{R}^n\), the convex hull of the lattice points in \(B\), is always a normal polytope, and whether there is a very ample, non-normal simple polytope, or even simplex. We answer the first question…
OWR-723-005Negative answer for the expected p-th power
In an Oberwolfach report of 2004, E. Novak (joint work with A. Hinrichs) considered the generalized lattices with shift \(M_n^{z,\Delta}\)\({}=\{jz\)\({}+\Delta \bmod 1\)\({}:\ 0\)\({}\le j\)\({}<n\}\), with \(z\) and \(\Delta\) independent and uniformly distributed in \([0,1]^d\), and asked whether, for even \(p\), the expectation of \(\operatorname{disc}_p^p(M_n^{z,\Delta})\), the \(p\)-th power of the \(L_p\)…
OWR-13677-003Complete answer: yes, even co-meager
Durante, Fernández-Sánchez and Trutschnig (Oberwolfach Report 20/2015) asked whether the class of purely singular \(d\)-dimensional copulas is of second category in \((\mathcal{C}_d,D_1)\), where \(D_1\) is the metric of Trutschnig defined through the conditional distributions (Markov kernels) of the copulas. The answer…
OWR-14299292-013Answer: yes, an elementary invariant separates the pair
At the problem session of the Oberwolfach workshop Computational Group Theory (June 2025), M. Stanojkovski asked whether an invariant simpler than the order of the automorphism group can show that two groups \(G_1\) and \(G_2\) of order \(p^8\), class \(2\) and exponent \(p\) (\(p\) an odd prime) are not isomorphic. They are…
AIM-ALGEBRAIC_GEOMETRY-0078Complete proof of ambient multiplicity formula
Let \(T_{n,\beta}\) be the Hilbert scheme of \(n\) points on the thickened line defined by \(y^\beta=0\) over \(\mathbb C\). For the component indexed by a partition \(\lambda\) of \(n\) with largest part at most \(\beta\), we compute its ambient generic multiplicity as \(\prod_i\binom{\beta}{\lambda_i}\).
OWR-14601-005Complete answer: the cone is Gorenstein
In his abstract in Oberwolfach Report 19/2016, Michałek asked whether the affine cone over the second secant variety \(\sigma_2\) of the Segre product \(\mathbb{P}^1\)\({}\times\mathbb{P}^1\)\({}\times\mathbb{P}^1\)\({}\times\mathbb{P}^1\)\({}\times\mathbb{P}^1\) is Gorenstein. We prove that it is. The homogeneous coordinate ring \(R\) of \(\sigma_2((\mathbb{P}^1)^5)\)\({}\subset\mathbb{P}^{31}\), the variety of binary tensors of border rank at…
OWR-4428-011Partial answer: irrational algebraic values
In an Oberwolfach report of 2010, Ł. Grabowski asked three questions about the set \(\mathcal{C}(G)\) of \(\ell^2\)-Betti numbers arising from a group \(G\), that is, of von Neumann dimensions of kernels of square matrices over \(\mathbb{Q} G\). Question 4 asks whether \(\mathcal{C}(G)\not\subseteq\mathbb{Q}\) holds exactly when \(G\) contains a wreath product \(\mathbb{Z}\)\(/p\mathbin{\wr} K\) with…
AIM-COMPUTATION-0021Complete proof of scoped augmentation results
Given an observed support in a finite Cartesian table, we minimize the number of distinct new cells needed to make every maximal-clique projection of a chordal graphical model surjective. For a clique path, the missing clique configurations form a partially ordered set…
AIM-ALGEBRAIC_GEOMETRY-0190Complete proof of scoped calibration theorem
For an independent rational 3-by-3 matrix pencil, we give an exact procedure that counts its real projective rank-two members compatible with two positive finite focal lengths, or reports that there are infinitely many. Principal points are known, skew is zero and aspect ratio is one. A…
AIM-TOPOLOGY-0106Complete fixed-test classification
Fix a nonempty finite graph \(T\), with loops permitted, and a nonzero commutative coefficient ring \(R\) with identity. We classify when ordinary relative homology of \(\operatorname{Hom}(T,-)\) satisfies excision for every induced graph cover. This occurs exactly when \(T\) dismantles to a single looped vertex. Exactness of the degree-zero…
AIM-FUNCTIONAL_ANALYSIS-0012Complete finite-frequency theorem
We prove a quantitative nonlinear origin-symmetry theorem for convex bodies in three dimensions whose radial functions are close to one in \(C^2\) and whose odd parts have bounded spherical harmonic degree. The even part is unrestricted. An explicit bound controls the odd radial part by derivatives at zero…
OWR-4428-012Partial answer: the square root of two is an l2-Betti number
Grabowski asked whether \(\sqrt2\) is an \(\ell^2\)-Betti number arising from the lamplighter group \(\mathbb{Z}\)\(/2\mathbin{\wr}\mathbb{Z}\), that is, the von Neumann dimension of the kernel of a matrix over its integral group ring (Groups Geom. Dyn. 2016, Problem 4). We show that it is. For an explicit matrix \(T'\)\({}\in M_2(\mathbb{Q}[\mathbb{Z}\)\(/2\mathbin{\wr}\mathbb{Z}])\) with \(8T'\)…
AIM-ALGEBRAIC_GEOMETRY-0035Complete external obstruction
Let p be odd, \(h=(p-1)p^{r-1}m\) with \(r\ge1\) and \(m\gt1\), and \(H=C_{p^r}\) a subgroup of the height-h Morava stabilizer. For Honda Morava E-theory over \(\mathbb F_{p^h}\), put \(A=E^{hH}\). We prove that no finite p-local spectrum F satisfies A smash F equivalent to E as A-modules.
AMR-103-0231Complete counterexample
We give counterexamples to the sufficiency assertion in Turaev's Conjecture 12.26 in Ohtsuki's problem list. If a closed connected oriented 3-manifold has first homology \(\mathbb Z\oplus T\) and its Alexander polynomial is nonzero modulo a prime p, its order of vanishing at t=1 modulo p is at least \(\dim_{\mathbb F_p}(T/pT)\).
AIM-ALGEBRAIC_GEOMETRY-0184Complete scoped theorem
Suppose projective camera matrices are available and the two diagonal focal scales are known in every view, while skew and principal point may vary independently between views. We give explicit sharp generic thresholds for admissible rank-three metric upgrades over the complex numbers: four views give a finite fiber of degree at least two, whereas five compatible views give a unique upgrade.
OWR-14298804-012Partial answer: fare zones under bounded congestion, settled up to polylog(k)
At the Oberwolfach workshop on Combinatorial Optimization in November 2024, Peis, with Pabst, Kauther and Fiorini, posed the following problem (Oberwolfach Report 50/2024, p. 2992). Commodities on a tree have paths, weights and budgets. A set of tree edges, the zone borders, is to be chosen; every…
OWR-4428-012-BPartial answer: genuine witnesses and irrational values for every p
Grabowski asked what the set \(\mathcal{C}({\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}})\) of \(\ell^2\)-Betti numbers arising from the lamplighter group is, that is, of von Neumann dimensions of kernels of square matrices over its rational group ring, and whether \(\mathcal{C}(\mathbb{Z}\)\(/p\mathbin{\wr}\mathbb{Z})\) contains irrational algebraic numbers (Invent. Math. 2014). Ara and Claramunt…
OWR-14299904-015Complete answer: the maximum is determined for all n, k, d
Konvalinka asked (Oberwolfach Report 2/2026, p. 150) for the maximum of \(\Delta(\lambda,\mu,\nu)\)\({}=\eta(\lambda)\)\({}-\eta(\mu)\)\({}-\eta(\nu)\), where \(\eta(\lambda)\)\({}=|\lambda|^2\)\({}+\frac{d(d-1)(2d-1)}{6}\)\({}-\sum_{i=1}^{d}(\lambda_i\)\({}-i\)\({}+1)^2\), over the triples of partitions with \(|\lambda|=n\), \(|\mu|=k\), \(|\nu|=n-k\), at most \(d\) parts and nonzero Littlewood–Richardson coefficient \(c^{\lambda}_{\mu\nu}\). By the work of Klep, Štrekelj…
OWR-15215-006Partial answer: the optimal constant grows exponentially
Let \(V_K\) be the volume of the simplex spanned by \(d+1\) independent uniform random points in a convex body \(K\subset\mathbb{R}^d\). Meckes asked, as recorded by Reitzner in Oberwolfach Report 10/2017, whether there is a constant \(c(d)\) with \(\lim_dc(d)^{1/d}\)\({}<\infty\) such that \(\mathbb{E} V_K\)\({}\le c(d)\,\mathbb{E} V_L\) whenever \(K\subseteq L\), and the report asks for a lower…
AIM-OTHER-0057Complete cyclic-family theorem
For every cyclic group \(C_n\) and every associator class \(\omega_t\), this preprint explicitly computes the Brauer-Picard group of \(\operatorname{Vec}_{C_n}^{\omega_t}\) and its action on all indecomposable module categories. Module labels are the divisors of \(\gcd(n,t)\). For \(p^k\) exactly dividing…
EP-963Partial answer: the conjecture holds for all n up to 31
Let \(f(n)\) be the largest \(k\) such that every set of \(n\) real numbers contains a dissociated subset of size \(k\), that is, a subset whose \(2^k\) subset sums are pairwise distinct (the empty sum included). Erdős noted that \(f(n)\)\({}\ge\lfloor \log_3 n\rfloor\) and asked whether \(f(n)\)\({}\ge\lfloor \log_2 n\rfloor\) for every \(n\) (Erdős Problem #963). The general question…
AMR-050-SYM7Complete answer: seven identities proved, exceptional sets stated
Reznik, Garcia and Koiller listed many experimentally found invariants of the families of \(N\)-periodic billiard trajectories in an ellipse. Seven of their rows, \(k_{406,a}\), \(k_{607}\), \(k_{609}\), \(k_{610}\), \(k_{905}\), \(k_{906}\) and \(k_{907,b}\) (records AMR-050-0024, -0035, -0037, -0038, -0064, -0065 and -0067 of the UnsolvedMath…
OWR-9790358-011Partial answer: an explicit rule for n = 3, no symmetric rule for n = 4
In Oberwolfach Report 7/2022 Y. Xu asked whether Gaussian cubature rules of degree \(2n-2\), with \(\dim\Pi^3_{n-1}\)\({}=\binom{n+2}{3}\) real nodes, exist for the product Chebyshev weight of the second kind on the cube \([-1,1]^3\); existence is known only for \(n=2\) and \(n=3\). (The source prints the formula of the weight of the first kind…
OWR-15986-005Negative answer to Question 1, by a short argument
In Oberwolfach Report 1/2018, Becher and Slaman ask whether some Martin-Löf random real \(x\) has discrepancy \(D_N((2^nx)_{n\ge1})\)\({}=O((\log N)\)\(/N)\) for all large \(N\). Read as asking for constants \(C\), \(N_0\) with \(ND_N\le C\log N\) for all \(N\ge N_0\), the question has a negative answer, also for Schnorr, computably and Kurtz random \(x\). More precisely,…
AMR-050-0011Complete answer: three pedal-area identities for every point
Reznik, Garcia and Koiller listed many quantities that are experimentally constant along the families of \(N\)-periodic billiard trajectories in an ellipse whose caustic is a confocal ellipse. We treat three of their rows on the signed areas of pedal polygons with respect to an arbitrary point \(M\). Let \(A\)…
OWR-11695865-007Complete tree singular-locus theorem
We compute the coefficient-map Jacobian determinant of every bidirected-tree linear compartmental model whose sole input, output and leak are at one root. The determinant is a product of explicit edge powers and squared resultants of characteristic polynomials of sibling branches.
AIM-ANALYSIS-0032Complete convex sharp-constant theorem
Larson proved that the boundary-mean Hermite-Hadamard inequality on bounded convex domains in \(\mathbb{R}^n\) has sharp constant \(n\) for nonnegative subharmonic functions. We construct explicit nonnegative convex near-extremizers on collapsing simplices, proving that restricting the test functions to the convex class does not improve this universal constant.…
AMR-050-0018Disproof as printed; the focal form of the row is proved
Row \(k_{401}\) of Reznik, Garcia and Koiller's list of experimentally found invariants of \(N\)-periodic billiard trajectories in an ellipse (record AMR-050-0018 of the UnsolvedMath collection) states that for \(N\equiv2\pmod 4\) and every point \(M\) the product \(A'A^*_M\) is constant along the family; here \(A'\) is the signed area…
AIM-ANALYSIS-0019Complete weak-coupling theorem
We give a quantitative comparison for the first Robin eigenvalue among triangles of fixed area.
AIM-ANALYSIS-0016Complete explicit counterexample
For the atomic singular inner functions \(u_{\xi,\alpha}(z)=\exp[-\alpha(\xi+z)/(\xi-z)]\), we construct an explicit four-point functional annihilating all individual nonnegative powers at singularities \(1,-1,i\) with mass \(3\pi\), while detecting \(z^2\).
AMR-050-0022Complete answer: five focal area identities proved
Reznik, Garcia and Koiller listed many quantities that are experimentally constant along the families of \(N\)-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We prove five of their rows on areas of polygons built from the trajectory and the two foci, for trajectories of…
AMR-050-0056Complete answer: five inversive-object rows decided
Reznik, Garcia and Koiller listed many quantities found experimentally to be constant along the families of \(N\)-periodic billiard trajectories in an ellipse. We treat five rows of their table of inversive objects (records AMR-050-0056, -0057, -0058, -0059 and -0061 of the UnsolvedMath collection),…
AMR-050-0049Complete answer: four inversive area rows proved for every N
Reznik, Garcia and Koiller listed many experimentally found invariants of the families of \(N\)-periodic billiard trajectories in an ellipse. Let \(A\) and \(A'\) be the signed areas of the trajectory and of its outer polygon, and \(A^\otimes\) and \(A'^\ominus\) those of the polygons obtained by inverting the vertices of the trajectory…
AIM-ALGEBRAIC_GEOMETRY-0271Complete scoped proof
Suppose lognormal data are fitted by an ordinary normal model and a shape-scale gamma model.
AMR-050-0002Corrected form proved; two rows proved on their domains
Reznik, Garcia and Koiller listed many quantities that they found experimentally to be constant along the families of \(N\)-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We treat four rows of their tables (records AMR-050-0002, -0028, -0039 and -0032 of the UnsolvedMath…
AIM-CONVEX_GEOMETRY-0019Complete proof (classical corollary)
The product of the normalized solid angles of a simplicial cone in \(\mathbb R^n\) and its positive dual is at most \(4^{-n}\), with equality exactly for orthogonal images of the positive orthant.
AIM-GEOMETRY-0388Complete theorem (product family)
For a fixed closed connected hyperbolic surface \((\Sigma,h)\) of curvature \(-1\), we determine the full Paneitz spectrum for \(g_t=g_{S^2}\oplus t^{-1}h\), where the round sphere has curvature \(1\) and \(t\gt0\).
OPG-59927Partial answer: exact values for K_{3,b} with b at most 8
For integers \(a,b\ge2\) let \(\varphi(a,b)\) be the least \(t\ge0\) such that the join \(K_{a,b}+K_t\) of the complete bipartite graph \(K_{a,b}\) with the complete graph \(K_t\) is chromatic-choosable, that is, \((t+2)\)-choosable. This is the Ohba number \(\tau_0(a,b)\) of \(K_{a,b}\); to determine it is a question of the Open Problem Garden, attributed…
OWR-15988-004Complete theorem (two-dimensional case)
The scaling-rotation distance between positive-definite matrices minimizes the distance between their eigendecomposition fibers.
AIM-COMBINATORICS-0153Complete theorem (recursive construction)
The classical recursive construction for tight-cycle-free three-uniform hypergraphs has optimal edge count \(R(n)=\max_{0\leq m<n}\{m\binom{n-m}{2}+R(m)\}\). We give an exact algorithm for a single input \(n\) that visits only \(O(\log(n+2))\) states.
OWR-14298158-012Negative answer: a proper subclass of the separable permutations whose generating function lies outside the field
Question 5 of R. Brignall's abstract (joint work with M. Bóna, C. Defant, M. Opler and V. Vatter) in Oberwolfach Report 6/2024, p. 292, asks whether every proper subclass of the separable permutations Av(2413, 3142) has a generating function in K = Q(z, √(1−4z), √(1−6z+5z²)). The answer is no. The…
AMR-022-4023Partial answer: connected lemniscates in every dimension lie in a ball of radius 2
For points \(x_1,\dots,x_n\) in \(\mathbb{R}^m\) let \(E\)\({}=\{x\)\({}\in\mathbb{R}^m\)\({}:\prod_{i=1}^n|x\)\({}-x_i|\)\({}\le1\}\). A question of Erdős, posed in Hayman's collection of problems of 1974 and recorded as Problem 4.23 by Hayman and Lingham, asks how large the volume of \(E\) can be and whether a connected \(E\) can be covered by a ball of radius \(2\); for \(m=2\) the covering…
AIM-REPRESENTATION_THEORY-0033Complete theorems (specified models)
We study explicit ways to replace the symmetric-group automorphisms of the category of finite sets and injections by their Schur double covers. If the central involution acts freely on a finite lift fibre of an injection of codimension \(d\), then the fibre cardinality is divisible by \(2^{\lfloor d/4\rfloor+1}\).
OWR-3385-011Partial answer: a star-shaped spherical heptagon whose development self-intersects
Problem 13 in G. Rote's list of open problems from the 2009 Oberwolfach workshop Discrete Differential Geometry, posed by J. O'Rourke, asks when a simple piecewise-geodesic curve on the sphere, developed onto the plane with the same edge lengths and turning angles, does not intersect itself. The…
AMR-014-0005Complete classification (cube rank)
Let \(M=\prod_i x_i^{a_i}\) have degree \(3d\gt 0\), and let \(o\) be the number of odd exponents. We determine the least number of cubes of homogeneous degree-\(d\) forms whose sum is \(M\). Over \(\mathbb{C}\), this number is one if every exponent is divisible by three, three if \(M\) is not a cube and \(o\le d\), and four if \(o\gt d\). The same formula holds…
AIM-COMBINATORICS-0085Complete theorem (layered colorings)
Let \(f:\mathbb{R}\to\{-1,1\}\) be measurable and color the plane by \(\chi(x,y)=f(y)\). Call an equilateral side length essentially avoided when, for each fixed orientation, the set of monochromatic translations has planar measure zero.
OPG-369Complete theorem (four weight classes)
Let a finite matroid be labeled by an arbitrary abelian group, and let \(S\) be the set of its basis-weight sums. We prove the Schrijver-Seymour stabilizer bound whenever the nonloop labels occupy at most four cosets modulo the stabilizer of \(S\).…
AMR-096-0036Solved: S(1) of a scale-invariant random spatial network is not a tree
D. Aldous asked for a proof that in a scale-invariant random spatial network (SIRSN) the subnetwork \(S(1)\), the union of the routes between the points of a Poisson process of rate one, cannot be a tree, even with Steiner points (Electron. J. Probab. 19 (2014), Open Problem 10; Open Problem 36 of the…
AMR-020-0402Solved: the algebraic Birkhoff conjecture for outer billiards on the sphere and the hyperbolic plane
Bolsinov, Matveev, Miranda and Tabachnikov (Phil. Trans. R. Soc. A, 2018) asked to prove the algebraic version of the Birkhoff conjecture for outer billiards on a non-Euclidean surface of constant curvature. We observe that it follows from published theorems of A. Glutsyuk. Let \(\gamma\) be a simple closed…
AMR-051-0010Complete theorem (smooth maps and flows)
A point of an invertible plane dynamical system is one-sided trapped if its forward semiorbit is bounded and its backward semiorbit is unbounded.
OPG-37656Complete proof (K5 concentration)
Let \(L_n\) be the random \(n\)-lift of \(K_5\) obtained by independent uniform perfect matchings over its ten edges. We…
OWR-14299911-020Partial answer: the travelling polytope constant of every product of trees
The travelling polytope constant \(\operatorname{TP}(X)\) of a graph \(X\) compares the travelling salesman problem through a finite set \(F\) of vertices with the shortest way to sweep \(F\) by convex hulls which grow or shrink by one adjacent vertex at a time. A. Genevois introduced it in his thesis (2017), as the travelling…
OWR-1782-017Partial answer: real-rooted integer polynomials satisfy a g-theorem (Conjecture 2 proved)
In his abstract Conjectures on \(f\)-vectors in Oberwolfach Report 1/2008, V. Welker recorded the following conjecture of F. Brenti and himself (Conjecture 8 there). Let \(h(t)\)\({}=1\)\({}+h_1t\)\({}+\)\(\dots\)\({}+h_dt^d\) have positive integer coefficients and only real roots. Then \((h_0,\dots,h_d)\) satisfies a \(g\)-theorem: it increases strictly…
OWR-12007-010Partial answer: no nontrivial word has vanishing rotation number under every circle action
In an Oberwolfach report of 2012, D. Calegari, reporting on joint work with A. Walker, asked whether there is a nontrivial word \(w\) in the free group \(F=\langle a,b\rangle\) such that the lifted rotation number \(\widetilde{\mathrm{rot}}(\rho(w))\) vanishes for every homomorphism \(\rho\) from \(F\) to the universal central extension \(\mathrm{Homeo}^+(S^1)^{\sim}\) of…
OWR-4531-016Partial answer: the cubic Fermat equation has nontrivial solutions in some model spaces
For an inner function \(\theta\) on the unit disc and \(1\le p\le\infty\), let \(K^p_\theta\)\({}=H^p\)\({}\cap\theta\,\overline{H^p_0}\) be the associated model (star-invariant) subspace of the Hardy space \(H^p\). K. M. Dyakonov asked (Oberwolfach Reports 7 (2010); Integral Equations Operator Theory 78 (2014)) whether “Fermat's last theorem” holds in these spaces: does…
AIM-GEOMETRY-0110Complete scoped proof (finite stability)
We give explicit one-parameter families of saturated unit-ball packings in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) with trivial translational stabilizer and exact square or cubic grid contact graphs.
AMR-020-0507Solved: quadratic Poisson brackets without polynomial integrable systems
Bolsinov, Matveev, Miranda and Tabachnikov asked whether each quadratic Poisson bracket on a vector space admits a polynomial integrable system, that is, \(\frac12(\dim\)\({}+\operatorname{corank})\) Poisson-commuting polynomials whose differentials are linearly independent almost everywhere (Question 5.9 in the numbering of arXiv:1804.03737v2;…
GRAPH-049Complete connected follow-up proof with induced minimality
For every odd integer \(m\ge 5\), let \(G_m\) be the Mycielski graph of the cycle \(C_m\) with an additional edge joining the shadow copies of two adjacent cycle vertices. We prove…
AIM-ARITHMETIC_GEOMETRY-0079Complete equidegree-family theorem
Let \(\mathcal{H}_{t,a}\) be the Hilbert component of curves in projective three-space defined by the maximal minors of a \((t+1)\)-by-\(t\) matrix of degree-\(a\) forms, over an algebraically closed field of characteristic zero. We…
OPG-164Complete spider reconstruction theorem
For a graph \(G\), let \(a_j(G)\) be the number of vertices in its \(j\text{th}\) iterated line graph.…
AMR-021-0006Complete separated-system theorem
Let \(k\ge 2\) be even. We determine the largest finite real solution set of a separated system \(f_i(x_i)=g_i(t)\), \(1\le i\le m\), with all degrees at most \(k\). The maximum is…
OWR-720-011Solved: the canonical normalized tight frame of a wavelet frame need not have wavelet structure
D. Han asked in 2003 (Trans. Amer. Math. Soc. 355, Question 4) whether \(S^{-1/2}\psi\) is always a wavelet frame when \(\psi\) is a wavelet frame with frame operator \(S\); M. Frank restated the question in 2004 (Oberwolfach Rep. 1): does the canonical normalized tight frame \(\{S^{-1/2}D^jT_k\psi\}\) of a wavelet frame always have…
AIM-SEVERAL_COMPLEX_VARIABLES-0026Complete sharp polynomial-kernel theorem
Let \(V\) be an \(r\)-dimensional space of complex polynomials and let \(\mathcal{P}_k\) denote positivity on every \(k\)-point evaluation matrix.…
EP-668Complete finite congruence classification
Alexeev, Mixon and Parshall proved that twenty points in the plane have at most 54 unit-distance pairs and that every maximizing set has the same abstract unit-distance graph.
AIM-PROBABILITY-0007Complete scoped counterexamples
We construct finite nonglobal local maxima of exact empirical likelihood for a binary restricted Boltzmann machine with one hidden unit and three visible units.
AIM-PROBABILITY-0103Complete scoped counterexamples
A bounded conjugate variable does not imply a coarse relative free Poincare inequality, even for an irreducible inclusion of \(\mathrm{II}_1\) factors. We first give a commuting-extension construction: for a diffuse coefficient algebra \(B\), a bounded self-adjoint \(Y\) commuting with \(B\), and \(A=W^*(B,Y)\), the variable…
OWR-9790365-007Complete proof of question 1
Let \(X\) have an arbitrary Borel probability law \(\mu\) on \(\mathbb{R}^n\), and let \(N\) be an independent standard Gaussian. We…
AIM-SEVERAL_COMPLEX_VARIABLES-0020Boundary counterexample and jet theorem
For every \(n\ge 2\), we construct one bounded domain \(D\) in \(\mathbb{C}^{n+1}\) with smooth strongly convex boundary and a circle of distinct proper holomorphic embeddings of \(\mathbb{B}^n\) into \(D\). All embeddings…
OWR-15585-006Complete affirmative proof
Every finite abelian action on the C*-algebra of a finite strongly connected noncycle graph that fixes vertices and scales edges by characters is strongly approximately inner.
AIM-COMPUTATION-0040Explicit theorem and sharp bound
We give a connected compact rational polygonal model on four outcomes, all of whose probabilities are at least 1/32, with a nonsemialgebraic logarithmic Voronoi cell at a smooth rational point.
AIM-COMBINATORICS-0181Complete counterexample
For every integer \(k\ge 3\) and \(n\gt k(k+1)\), the oriented cyclic Cayley graph with increments \(1,k,k+1\) has directed girth greater than \(k\), yet every vertex has \(2k-1\) vertices at shortest directed distance \(k-1\) and only \(k+1\) at distance \(k\). This…
AMR-122-0012Complete proof for every n
Let \(K_n\) be the Kasteleyn–Percus matrix of the Aztec diamond of order \(n\): the adjacency matrix between its black and its white squares in which the entry of every vertical domino whose white square is on top is replaced by \(-1\). In Problem 12 of his list of open problems in the enumeration of matchings (1996,…
KOU-16.60Complete affirmative answer
For a finite group \(G\) let \(T(G)\) be the sum of the degrees of the irreducible complex characters of \(G\), and for an automorphism \(\alpha\) of \(G\) let \(S_\alpha\)\({}=\{g\)\({}\in G\)\({}:\alpha(g)\)\({}=g^{-1}\}\). D. MacHale asked in the Kourovka Notebook (Problem 16.60) whether \(|S_\alpha|\le T(G)\) for every \(\alpha\)\({}\in\operatorname{Aut} G\). The inequality is stated as Theorem 3.1 of a 2026…
AMR-126-0025Negative answer (sequence reading)
Benedetto, Ingram, Jones, Manes, Silverman and Tucker asked whether, for every dominant rational map \(f\) of \(\mathbb{P}^N\), there are finitely many linear recursions with constant coefficients \(L^{(1)},\)\(\dots,L^{(r)}\) and an index function \(\iota\colon\mathbb{N}\)\({}\to\{1,\)\(\dots,r\}\) such that \(\deg(f^n)\)\({}=L^{(\iota(n))}_n\) for all \(n\ge1\) (Bull. Amer. Math. Soc. 56 (2019); Question…
AMR-152-0049Conjecture false for every rational Jacobian
In a list of open problems on Hénon maps (Arnold Math. J. 10 (2024)), P. Ingram conjectured that for all but finitely many \(\delta\in\mathbb{Q}\) every \(\mathbb{Q}\)-rational periodic point of every map \(f(x,y)\)\({}=(y,\,y^2\)\({}+c\)\({}-\delta x)\) with \(c\in\mathbb{Q}\) has period dividing \(2\). We show that the statement, as printed, is false for every \(\delta\in\mathbb{Q}\), in each…
KOU-13.19Complete answer: regular iff n ≤ p
Let \(Q\) be a subgroup of a direct product \(G_1\)\({}\times\)\(\dots\)\({}\times G_n\) of groups and \(H\) a normal subgroup of \(Q\) such that \(Q\) and \(H\) both project onto every factor. Yu. M. Gorchakov asked in the Kourovka Notebook (Problem 13.19) whether \(Q/H\) is a regular \(p\)-group whenever it is a \(p\)-group; in 2009 he stated this as a conjecture…
AMR-150-0035Complete answer: exactly the left-Σ2 reals
In a 2025 list of open problems in computability theory, A. Nies notes that every left-\(\Sigma_2\) real \(r\in[0,1]\) is the Hausdorff dimension of a computable subgroup of a computable profinite group, and asks which values occur when the group is topologically finitely generated. Reading “computable subgroup”…
KOU-7.27Affirmative answer for all large q
V. M. Levchuk asked in 1980 (Kourovka Notebook, Problem 7.27) whether, for all sufficiently large \(q\), the group \(\mathrm{SL}_n(q)\) contains a diagonal matrix lying in no proper irreducible subgroup except block-monomial ones. We show that the answer is yes for every \(n\), as a short corollary of density theorems…
AIM-GEOMETRIC_GROUP_THEORY-0024Complete proof (frozen ab formulation)
Let \(\phi\) be the automorphism of \(F(a,b,c)\) given by \(\phi(a)=a\), \(\phi(b)=ab\) and \(\phi(c)=bcb\). We construct a compact nonpositively curved cube complex with fundamental group \(F(a,b,c)\rtimes_\phi\mathbb{Z}\). It has one vertex, seven…
AIM-OTHER-0095Complete normalization obstruction
For a nonzero real binary form of degree \(d\), we prove that all norm sublevel sets in \(\operatorname{SL}(2,\mathbb{R})\) have finite Haar volume if and only if every real projective root has multiplicity strictly less than \(d/2\).…
AIM-ALGEBRAIC_GEOMETRY-0162Complete scoped counterexample
Let \(k\) be any nonzero commutative ring of characteristic two. The ghost map from the constant \(C_2\)-Tambara functor on \(k\) is finitely presented and has unique lifting against square-zero extensions, but is not flat for the relative Mackey box product.…
AIM-CONVEX_GEOMETRY-0026Complete scoped counterexample
Every centrally symmetric convex body is an orthogonal projection of a centrally symmetric convex body whose linear functionals have \(\psi_2/L_2\) ratio at most \(2\sqrt2\), independently of dimension. The lift is…
AIM-GEOMETRIC_GROUP_THEORY-0023Complete family classification
Let \(F\) be freely generated by \(a,b_1,\ldots,b_m\) and let \(\phi(a)\)\({}=a^\varepsilon\) and \(\phi(b_i)\)\({}=a^{p_i}b_{\sigma(i)}^{\delta_i}a^{q_i}\), where \(\varepsilon\) and \(\delta_i\) are…
AIM-GEOMETRY-0093Complete lattice theorem
For each perfect strictly eutactic lattice, we determine the large-parameter Gaussian energy minimum in a fixed neighborhood of its shape.
OWR-15951-003Partial answer: yes for constant bumps and all couplings; conditional result for bumps of bounded variation
J. Fillman asked (Oberwolfach Report 46/2017) whether, for the continuum Fibonacci potential \(V_\omega\) built from two bump functions \(f_0,f_1\in L^2[0,1)\) on unit cells, the spectrum of the separable operator \(-\Delta\)\({}+\lambda_1V_\omega(x)\)\({}+\lambda_2V_\omega(y)\) on \(L^2(\mathbb{R}^2)\), which is the sum \(\sigma(L_{\lambda_1V_\omega})\)\({}+\sigma(L_{\lambda_2V_\omega})\) of two one-dimensional spectra, contains…
AMR-155-0008Affirmative answer for every b in (1/8, 1/6]
For \(0<b<1\) let \(\psi_b\)\({}\in L^2(\mathbb{R})\) be given by \(\widehat{\psi_b}\)\({}=\mathbf{1}_{(-1,-b)\cup(b,1)}\). Bownik and Weber (Studia Math. 159 (2003)) determined for which \(b\) the dyadic affine system of \(\psi_b\) is a frame, except for \(1/8<b\le 1/6\), where they believed it to be a frame and called \(b=1/6\) the hardest case; Bownik and Rzeszotnik list…
AMR-154-0006Explicit regular conservation laws for every normal form; all real-analytic laws described
Bolsinov, Konyaev and Matveev asked in 2024 to describe all conservation laws of the normal forms of two-dimensional gl-regular Nijenhuis operators, or at least to find one regular conservation law for each of them. A conservation law of L is a function F with d(L*dF) = 0, regular if dF and L*dF…
AMR-129-0019Partial answer: dense in balls (source not the centre) and in ellipses (source not a focus)
Tabachnikov asked in 2026 whether, for a domain with smooth boundary in Euclidean space and a generic point \(P\) of it, the time-\(t\) wave front (the positions at time \(t\) of the unit-speed photons that leave \(P\) in all directions and are reflected optically at the boundary) becomes dense as \(t\to\infty\); he added that…
AIM-ALGEBRAIC_GEOMETRY-0318Complete projection counterexample
We construct a degree-ten hypersurface in complex projective four-space with exactly sixty ordinary double points whose general plane projection lies on a sextic.
AMR-074-0117Answer depends on the reading: no under the null-cover reading (RCA0 proves LDT)
Question 17 of Montalbán's survey of open questions in reverse mathematics (Bull. Symbolic Logic 17 (2011)) asks whether the Lebesgue differentiation theorem (LDT) implies WWKL0 over RCA0; Simpson had observed, using Pathak's work, that LDT is provable in WWKL0. The question does not fix how "for…
OWR-12339-005Unit coefficients under weak nondegeneracy
Let \(P\) be a finite union of compact convex polytopes in real \(d\)-space, and let \(S\) contain its algebraic vertices.…
OWR-17464-019Complete degree-two bound for all uniformities
For every integer \(r\ge 2\), we prove the Balogh-Bollobas-Narayanan independent-set bound for finite simple \(r\)-uniform, 2-regular hypergraphs. If…
OWR-2090-028Complete measure-zero theorem
Let \(A\), \(B\), \(C\) be pairwise disjoint convex subsets of \(\mathbb{R}^3\).…
OWR-13941-002Complete convexity counterexample
The level-one Artin cone consists of homogeneous forms that become sums of squares after multiplication by some nonzero quadratic sum of squares.
AIM-COMPUTATION-0052Explicit polynomial certificate
The nonexistence of a Hadamard matrix of order 28 with one circulant core is classical.
OWR-2090-019Complete negative answer
For every integer \(m\ge 3\) we construct \(m+4\) pairwise disjoint compact convex bodies with nonempty interiors in \(\mathbb{R}^3\) and a common transversal line such that every subfamily pinning that line contains at least \(m\) members. Thus…
OWR-16412-006Complete counterexample to Question 5
We construct a bounded convex balanced domain in the complex Hilbert space \(\mathbb{C}\)\({}\oplus\ell^2\) that is not holomorphic homogeneous regular. The domain is…
AIM-PHYSICS-0025Complete connected counterexample
Consider the discrete-time coloring chain that chooses a uniform vertex and a uniform color from all \(q\) colors, accepting the proposed recoloring when it is proper and otherwise staying in place.…
OWR-16930-001Complete canonical percolation proof
We derive high-density percolation for uniform canonical hard-disk packings from Georgii's equivalence of ensembles and Magazinov's high-activity percolation theorem.
AMR-029-0087Complete banana-tree algorithm
A banana tree is a connected loopless multigraph whose underlying simple graph is a tree. We give an exact algorithm that computes its divisorial gonality and an effective divisor attaining it in \(O(n^2)\) word operations and \(O(n^2)\) space after reading the input, where \(n\) is the number of vertices. Multiplicities…
AIM-COMBINATORICS-0131Complete kernel theorem
For every finite connected simple graph, we prove that the natural actions of its automorphism group on the finite critical group and on real first homology have the same kernel.
OWR-13490-013Counterexample and repaired theorem
For a finite nonempty simplicial complex of dimension at most two, we prove that tightness over a field of characteristic two is equivalent to a complete one-skeleton together with symmetry of the link-component counts in every induced subcomplex.
OWR-1787-006No: polynomial multiples of MacLane universal functions need not be universal
An entire function \(f\) is universal in the sense of MacLane if its derivatives \(f,f',f'',\dots\) are dense in the space \(H(\mathbb{C})\) of entire functions with the topology of locally uniform convergence; \(\mathrm{HC}(D)\) denotes the set of these functions. G. Costakis proved (Oberwolfach Report 6/2008, Theorem 6) that there…
OWR-1315-001Complete proof
Let \(G\) be a finite group and let \(k\) be any field of odd positive characteristic. We prove that the symmetric and skew-symmetric Cartan homomorphisms from the Witt groups of projective \(kG\)-modules to those of all finite-dimensional \(kG\)-modules are isomorphisms.…
OWR-12868-001Complete calculation for explicit families
We compute every cohomology-annihilator ideal and its least nonnegative stabilization index for \(S[x]/(x^n)\), where \(n\ge 2\) and \(S\) is a polynomial or formal power-series ring over any field.
AMR-103-0230Partial answer: new infinite families and all open determinants up to 10^6
Stoimenow asked (Problem 12.25 of the problem list edited by Ohtsuki, 2002; Conjecture 4.2 of his paper in Comm. Anal. Geom. 13 (2005)) whether every odd sum of two squares \(n\notin\{1,9,49\}\) is the determinant of a prime alternating achiral knot. He proved this for squares and, with rational knots, for…
OPG-60013Partial answer: exact values for n up to 13 and linear bounds
For graphs \(G\) and \(H\), \(\operatorname{wsat}(G,H)\) is the least number of edges of a spanning subgraph \(F\) of \(G\) such that the edges of \(G\) missing from \(F\) can be added one at a time, each completing a new copy of \(H\). Morrison and Noel asked (Open Problem Garden, 2016) to determine \(\operatorname{wsat}(K_n,Q_3)\), where \(Q_3\) is the cube. We give…
OWR-11129-006Yes: order 1/log n, with a log-periodic asymptotic
Let \(G\) be a finite \(n\)-dimensional classical group over a field of odd order \(q\), and let \(Q\) be the set of elements \(g\) of even order whose involution \(g^{|g|/2}\) has a fixed space of dimension in \([n/3,2n/3)\). Lübeck, Niemeyer and Praeger proved \(|Q|/|G|\ge c/\log n\), and Praeger, with Lübeck and Niemeyer, asked at Oberwolfach…
OWR-1061-006Partial answer: non-regenerative for three classes and off one mutation rate
M. Möhle (Oberwolfach Report 40/2005) conjectured that the sampling distributions of the infinitely-many-alleles model along a coalescent with simultaneous multiple collisions (a \(\Xi\)-coalescent), with mutation rate \(\rho>0\), are never regenerative in the sense of Gnedin and Pitman, except for Kingman's…
AMR-123-0003Partial answer: an infinite family and all open cases up to max(a,b) = 60
Propp asked whether every \((a,b)\)-benzel with nonnegative Conway–Lagarias invariant has a tiling by right-pointing stones and bones of all three orientations, a tiling of type 103 (Problem 7 of his list of twenty problems). By published theorems of Defant, Li, Propp and Young, of Defant, Foster, Li,…
OWR-14299288-010Complete scoped sampling theorem
Let \(A\) be a finite spanning collection of primitive integral normals in dimension \(n\), and let \(E_A\) be the least common multiple of the largest Smith invariant factors of its nonsingular full-rank submatrices.
OWR-3392-009Complete ambient-invariance counterexample
A proposed independence relation declares two subalgebras independent when their joint state is realized in two different copies of an extreme exchangeable state on the infinite unital free product of the ambient algebra.
AMR-119-0003CR or anti-CR: no as literally posed, yes for coorientation-preserving maps
Puqi Tang asked in the 1995 Several Complex Variables Problem List whether a homeomorphism between strongly pseudoconvex CR 3-manifolds that maps infinitesimal spheres of a Carnot–Carathéodory metric to infinitesimal spheres is CR; for this metric reading the answer is no as literally posed,…
OWR-4530-006Yes: the two-chunk DFC system is stable although its large-system limit is not
In Oberwolfach Report 48/2010, I. Norros (joint work with H. Reittu) considered the two-chunk Deterministic First Chunk (DFC) file-sharing system of Norros, Reittu and Eirola, in which one permanent seed holds both chunks and arriving peers choose by a fair coin the chunk they download first. The…
OWR-17469-002The limit exists and depends on the domain only through its volume
For a bounded open set \(D\subset\mathbb{R}^d\) let \(e_n(C^k(D),\mathrm{INT})\) be the minimal worst-case error of algorithms that use \(n\) function values to integrate, over \(D\), the functions all of whose partial derivatives up to order \(k\) are bounded by \(1\). In Oberwolfach Report 6/2020, Novak asked whether the asymptotic constant \(\limsup_{n}n^{k/d}e_n(C^k(D),\mathrm{INT})\)…
OWR-4425-012No in the universal reading; fixed alphabets classified up to one small window
Shallit asked (Problem 38 in Oberwolfach Reports 7 (2010)) whether, for a real number \(\alpha>1\) that is not an integer, there is a set \(P\) of patterns, finite or infinite, such that a word contains an \(\alpha\)-power, that is, a factor which is an exact \(p/q\)-power with \(p/q\ge\alpha\), if and only if some factor of the word…
OWR-12339-003Yes: the Gaussian Gabor system over Lambda_3 is a frame (computer-assisted)
Gröchenig and Lyubarskii (Oberwolfach Report 11/2013) considered the Gabor system \(\mathcal{G}(g,\Lambda_3)\) of time-frequency shifts of the Gaussian \(g(s)\)\({}=2^{1/2}e^{-\pi|s|^2}\) on \(\mathbb{R}^2\) along the complex lattice \(\Lambda_3\)\({}=A_3(\mathbb{Z}\)\({}+i\mathbb{Z})^2\), \(A_3\)\({}=\bigl(\begin{smallmatrix}1&2/5\\0&\sqrt{21}/5\end{smallmatrix}\bigr)\). They conjectured that it is a frame for \(L^2(\mathbb{R}^2)\)…
OWR-13498-009No for nonconvex polyhedra: an explicit pair of triangulated bipyramids
At the 2015 Oberwolfach workshop on discrete differential geometry N. Amenta asked whether Stoker's conjecture extends to nonconvex polytopes: for a polytope with given combinatorics, do the dihedral angles, all different from \(0\) and \(\pi\), determine the face angles? One may assume that the polytope is…
AIM-PROBABILITY-0016Complete scoped heat-flow theorem
Independent bit-flip heat flow preserves binary restricted Boltzmann machines with one hidden unit, by their classical representation as mixtures of two product distributions.
AMR-109-0231Yes, as a promise problem: an algorithm that outputs Mosher's exponent M
Let \(S\) be a closed oriented surface of genus at least \(2\). By a theorem of Mosher (1997), for pairwise independent pseudo-Anosov mapping classes \(\varphi_1,\)\(\dots,\varphi_n\) of \(S\) there is an integer \(M\) such that for all integers \(\alpha_1,\)\(\dots,\alpha_n\)\({}\ge M\) the classes \(\varphi_i^{\alpha_i}\) freely generate a subgroup \(H\) whose surface group extension \(\Gamma_H\) is word…
OWR-12872-018Proved for every corner weight a >= 0; the range -1 to 0 stays open
For a real number \(a\) let \(Q_n(a)\) be the sum of \(a^{c(w)}\) over the walks \(w\) of length \(2n\) with steps N, E, S, W that start and end at the origin and stay in the quadrant \(\{x\ge0,\ y\ge0\}\), where \(c(w)\) is the number of factors NW and ES of \(w\), and let \(L_n(a)\) be the same sum without the quadrant condition. Albert and Bousquet-Mélou…
AMR-021-0002Complete rational-exponent theorem
We extend the compact-curve method of Enciso and Peralta-Salas for Coulomb equilibria in three dimensions.
OWR-12978-004No for group algebras: odd-order Brauer pairs
In the open questions of the Oberwolfach mini-workshop on infinite dimensional Hopf algebras (Report 20/2014), Montgomery asks whether two semisimple Hopf algebras over \(\mathbb{C}\) with the same set of Frobenius–Schur indicator values have gauge equivalent, that is tensor equivalent, representation…
OWR-13498-012Yes at generic positions, for every graph with 2n-2 edges
In Problem 8 of Oberwolfach Report 13/2015, G. Rote gives a self-stress \(\omega\) of a wheel in general position in the plane with \(\sum_{ij\in E}\omega_{ij}[a\,p_i\,p_j][b\,p_i\,p_j]\)\({}=1\) for all points \(a,b\), and asks whether other graphs with \(n\) vertices and \(2n-2\) edges carry such a self-stress (the triangular prism and \(K_{3,3}\), each…
OWR-4990373-005False as stated: explicit finite graphs with exact certificates
Amir, Baldasso and Beilin conjectured (Oberwolfach Report 4/2021, Conjecture 1; Stoch. Proc. Appl. 155 (2023), Conjecture 3.1) that in median dynamics on any locally finite graph (i.i.d. uniform opinions in \([0,1]\), independent rate-one Poisson clocks, a ringing vertex takes the median of its…
OWR-14213-008False for n = 8, 9 and 11
Let \(\varphi\) be a non-degenerate quadratic form of dimension \(2n+1\) over a field \(F\) whose even Clifford algebra \(C_0(\varphi)\) is a division algebra, let \(X\) be its maximal orthogonal Grassmannian and \(S\) the Severi–Brauer variety of \(C_0(\varphi)\). N. Karpenko conjectured (Math. Z. 285 (2017), Conjecture 4.7, also Conjecture 3 of…
OWR-17294-007Complete orbit-lattice calculation
Let \(Q\) be a finite quandle whose right translations commute, with \(r\) orbits and stabilizer lattices \(H_i\) in \(L=\mathbb Z^r\).
OWR-15219-013No: one symmetric cubic refutes Conjectures 2 and 5
In an extended abstract in Oberwolfach Rep. 14 (2017), no. 1, 771–862 (pp. 825–826), Brändén considered the set \(\mathcal{D}_{m,d}\) of stable homogeneous polynomials \(P\) of degree \(d\) in \(m\) variables with \(\partial P\)\(/\partial x_i(\mathbf{1})\)\({}=d\)\(/m\) for all \(i\), and the zero vector \(\lambda(P)\) of the mixed characteristic polynomial \(\chi_P(t)\)\({}=\prod_i(1\)\({}-\partial\)\(/\partial x_i)P|_{x=t\mathbf{1}}\).…
AIM-PROBABILITY-0153Complete H3 calculation
We compute the homotopy type of the absolute order of the finite Coxeter group \(H_3\) after deleting its identity. The resulting complex is homotopy equivalent to a wedge of four circles and 872 two-spheres.
OWR-14299097-014Complete unitary-family theorem
Let \(q=p^e\) be a prime power and \(n\geq2\). We construct a nontrivial singular mixed identity for \(\operatorname{PSU}_n(q)\) of variable length at most \(16q+8p+12\).
AMR-022-7023No in the range asked: sufficient exactly when q is below 1/3
For \(f\) analytic in the unit disc \(\mathbb{D}\) let \(u(z,\zeta)\)\({}=f'(z)f'(\zeta)\)\(/(f(z)\)\({}-f(\zeta))^2\)\({}-(z\)\({}-\zeta)^{-2}\). Kühnau showed that a quasiconformal extension of \(f\) to the whole plane with \(|\mu|\le q<1\) implies \(|u(z,\zeta)|\)\({}\le q(1\)\({}-|z|^2)^{-1}(1\)\({}-|\zeta|^2)^{-1}\). J. G. Krzyż asked (Hayman–Lingham, Research Problems in Function Theory, Problem 7.23) to show…
OWR-14299580-006No: Aff(Z/5,2) and Aff(Z/5,3) are twist-related but not isomorphic
In an extended abstract in Oberwolfach Report 49/2025 (Oberwolfach Rep. 22 (2025), no. 4, 2633–2672; pp. 2645–2649), D. Ferri proposes to classify set-theoretic solutions of the Yang–Baxter equation up to Drinfeld twists, and asks whether two rack solutions on the same set that are related by a…
OWR-17471-010Complete counterexample
A question in Oberwolfach Report 8/2020 asks whether every hyperbolic knot without finite surgery has a rank-two free subgroup that injects under every rational Dehn filling. The figure-eight knot gives a negative answer.
OWR-17288-004No in the same-R^N reading, for every N; flat tori fail the hypothesis
A loose embedding of a metric space \((X,d)\) in \(\mathbb{R}^N\), in the sense of Chirvasitu, is a continuous map \(\varphi\) such that \(|\varphi x\)\({}-\varphi y|\)\({}=|\varphi x'\)\({}-\varphi y'|\) if and only if \(d(x,y)=d(x',y')\). In his abstract in an Oberwolfach report of 2019, Chirvasitu asked whether a compact metric space all of whose finite subspaces are loosely…
OWR-14299292-011Partial answer: yes for p = 5; no for p = 7, 13, 19; open from p = 31 on
M. Lee asked at the Oberwolfach workshop “Computational Group Theory” (June 2025) whether, for a prime \(p\) such that \(p-2\) is also prime, the alternating group \(A_p\) has a faithful irreducible representation over a field of some characteristic \(r>0\) in which all elements of orders \(p\) and \(p-2\) act without…
KOU-19.93No for p = 2, 3, 5: part b) of Spiga's conjecture is false
Problem 19.93 of the Kourovka Notebook, posed by P. Spiga in 2018, conjectures that there is a function \(f\colon\mathbb{N}\)\({}\times\mathbb{N}\)\({}\to\mathbb{N}\) such that every factor of the lower \(p\)-central series of a finite \(p\)-group with \(d\) generators and without an epimorphic image isomorphic to \(C_p\wr C_p\) has order at most \(f(p,d)\); part a) concerns…
KOU-15.98No: a counterexample of order 2304, the least possible
In Problem 15.98 of the Kourovka Notebook (2002), L. A. Shemetkov asked whether, for a saturated formation \(\mathfrak{F}\) and a finite soluble minimal non-\(\mathfrak{F}\)-group \(G\) whose \(\mathfrak{F}\)-residual is a Sylow \(p\)-subgroup, this residual is always isomorphic to a factor group of the Sylow \(p\)-subgroup of \(\mathrm{PSU}(3,p^{2n})\). For the…
KOU-2.78Partial answer: both sets are finite; the second question stays open
P. I. Trofimov (Kourovka Notebook, Problem 2.78, 1967) calls the set of all subgroups of one order of a finite group \(G\) an \(\mathit{IE}\)-system if it contains a non-normal subgroup, so that the number \(k(G)\) of \(\mathit{IE}\)-systems is the number of orders of non-normal subgroups of \(G\). A positive integer \(k\) is a soluble…
KOU-19.82Partial answer: yes for chief factors of type PSL(2,q) and Sz(q), with one restriction
Let \(h^*(G)\) be the generalized Fitting height of a finite group \(G\), and \(h(K)\) the Fitting height of a soluble group \(K\). P. Shumyatsky asked (Kourovka Notebook, Problem 19.82) whether every finite group \(G\) has a soluble subgroup \(K\) with \(h^*(G)=h(K)\), that is, whether \(h^*(G)\) never exceeds the largest Fitting height…
KOU-11.16Partial answer to part b): a reduction to the Suzuki groups
For \(r\ge1\) let \(V_r\) be the class of finite groups \(G\) that satisfy a law \([x,{}_r\,y]\)\({}=[x,{}_s\,y]\) for some \(s=s(G)>r\). R. Brandl asked (Kourovka Notebook, Problem 11.16 b)) whether \(V_r\) contains only finitely many nonabelian simple groups. Brandl proved that \(V_2\) contains none (1983), and Brandl and Nikolova proved…
KOU-17.16Partial answer: linear except possibly for six exceptional types
Problem 17.16 of the Kourovka Notebook, posed by V. G. Bardakov, asks whether the automorphism group \(\operatorname{Aut}(A)\) of an Artin group \(A\) of finite type is linear, as \(A\) is by the theorems of Bigelow, Krammer, Digne and Cohen–Wales; the Notebook records the affirmative answer of Bardakov and Bryukhanov for…
KOU-15.75Partial answer: two infinite families of PSL(2,p) and finite certificates
B. I. Plotkin asked in the Kourovka Notebook (Problem 15.75 b), 2002) whether a finite group is soluble if and only if it satisfies one of the identities \(u_n=1\), where \(u_1=[x,y]\) and \(u_{n+1}\)\({}=[[u_n,x],[u_n,y]]\). We give a partial answer: the question reduces to the minimal simple groups; \(\mathrm{PSL}(2,p)\) is settled for…
KOU-11.18Partial answers: seven infinite families of G(a,b); SL(2,K) images reduce to G(d,d)
R. Brandl asked in the Kourovka Notebook (Problem 11.18, 1990) whether the group \(G(a,b)\)\({}=\langle x,y\mid x\)\({}=[x,{}_{a}\,y],\ y\)\({}=[y,{}_{b}\,x]\rangle\) is finite; he noted that \(G(1,b)=1\) and \(G(2,2)=1\) and that nothing is known about \(G(2,3)\). O. Macedo\'nska asked (Problem 17.80, 2010) whether \(M_n=G(n,n)\) is infinite for every \(n>2\) (UnsolvedMath…
OWR-14224-004Counterexample to the star endpoint for full-cone least squares
For Gaussian denoising of nonnegative leaking flows, consider least squares over the full flow cone, with a fixed bound on the true root flow. We construct an \(N\)-vertex broom whose worst-case normalized risk is \(O(\sqrt{\log N}/N)\), while the \(N\)-vertex rooted star has risk \(\Omega(\log N/N)\) already at the zero signal. The…
OWR-14299097-016Counterexample to the all-character formulation; exact repair
A workshop problem asks for an exponent \(0\le\alpha(g)\lt 1\) such that \(\lvert\chi(g)\rvert\)\({}\le\chi(1)^{\alpha(g)}\) for every irreducible complex character of a finite almost quasisimple group and every noncentral element \(g\). The unrestricted formulation has an elementary quotient obstruction. The almost simple group \(\operatorname{P\Gamma L}_3(4)\) has quotient…
AMR-059-0016Counterexample to openness of Hessian-metric existence
The property of admitting a positive definite Hessian metric is not open under small affine deformations of a compact manifold. On the fixed torus \(\mathbb{T}^2\), the classical shear family with sole nonzero Christoffel coefficient \(\Gamma^1_{22}=t\) consists of complete, volume-preserving affine connections and converges…
AMR-098-0008Counterexample: allocation invariance without mass stationarity
Invariance under every measurable equivariant measure-preserving allocation does not characterize mass-stationarity of a diffuse random measure without an independent stationary background. For every dimension \(d\ge 2\), a periodic measure supported on parallel hyperplanes with distinct positive weights…
OWR-9790358-009Scoped counterexample and a corrected hereditary criterion
A finite family of affine algebraic sets may allow polynomial interpolation of every pointwise-compatible tuple although a subfamily does not. For every integer \(m\ge 2\), this note constructs an irredundant family of three proper, reduced, pure curves in \(\mathbb{C}^3\) with this property, while the sum of two vanishing…
OWR-5152-009Linear corner growth and a non-geometric counterexample
For a uniformly chosen nonnegative integer \(n\)-by-\(n\) matrix with both margin vectors \((3n,n,\ldots,n)\), the corner expectation has order \(n\). An explicit finite weighted-switching bound gives \(\liminf {\mathbb{E}[X_{11}]/n}\)\({}\ge 2\)\({}-\sqrt{2}\). The same corner law has total variation distance at least \(1/4096\) from every geometric distribution, including…
OWR-13494-006Counterexample to additivity of linear computability
Over an algebraically closed field of characteristic zero, the sum \(F_m\) of \(m\) disjoint binary cubics \(x_i y_i^2\) has Waring rank \(3m\), but the largest linear-computability colon bound is \(2m+1\). An exact inverse-system calculation covers every linear ideal and identifies the unique maximizer. In particular,…
OWR-1535-001Complete proof in the stated sets-of-revolution scope
For every \(0\lt s\lt 1\), we construct a connected compact set of revolution whose Riesz \(s\)-equilibrium measure omits a positive-capacity circle of its outer boundary. The generating set can also be chosen connected with nonempty interior. This gives an affirmative answer to the existential support question…
OWR-1587-003Complete attainability theorem for graph algebraic immunity
Let the graph algebraic immunity of a Boolean map \(f\)\({}:\mathbb{F}_2^n\)\({}\to\mathbb{F}_2^m\) be the least degree of a nonzero reduced Boolean polynomial vanishing on its graph. We prove that the dimension bound, the least \(d\) with \(\sum_{j=0}^d\binom{n+m}{j}\)\({}\gt 2^n\), is attained for every \(n,m\ge 1\). Translation invariance gives a rank-density inequality…
OWR-4086-008Counterexample to an adjacent jet-closure conjecture
Chen and Zuo conjectured that the \(r\text{th}\) jet closure of the Jacobian ideal of a weighted homogeneous polynomial equals that ideal plus the \((r+1)\)st power of the maximal ideal. For \(f=z(x^2+y^3)\), we exhibit \(xy^3\) in the fourth jet closure but outside \(J(f)+(x,y,z)^5\). Membership follows from an explicit identity…
OWR-11136-022Exact suspension counts and a three-arc gap; general problem open
Assign a positive radius to each point of a finite set in Euclidean three-space and count directed pairs at the assigned distance. Put \(B(n)\)\({}=\lceil{n^2/4+5n/2}\rceil\)\({}+1\). We prove that the exact maximum among \(n\)-point suspensions is \(B(n)\) for every \(n\ge 13\), using Swanepoel's construction for sharpness. A maximum principle…
OWR-3392-002Scoped counterexample to a coordinate-intersection implication
For every prime \(p\ge5\), we construct a minimal exchangeable sequence of trace-preserving embeddings of \(L(C_p)\) into a hyperfinite finite von Neumann algebra whose tail algebra is scalar but whose canonical filtration fails the intersection axiom for noncommutative factorizations. The intersection of the…
OWR-14299093-005Signed-weight counterexample; positive-weight scope not resolved
We construct a smooth signed radial weight for which a Grushin Hardy--Rellich inequality holds for every compactly supported smooth test function on the punctured unit gauge ball, but the associated Bessel equation has no positive solution. The principal weight is the constant one and the…
OWR-3392-012Scoped counterexample to a GNS-descent implication
We construct a general dilation of the Ornstein-Uhlenbeck Markov semigroup whose ambient commutative von Neumann algebra is generated by the initial algebra and its nonnegative time shifts, but whose GNS representation kernel is not invariant under those shifts. The dilation endomorphisms are…
OWR-1740-002Counterexamples for arbitrary homogeneous monomial maps
The optimized affine winner-cell bound gives a combinatorial lower bound for the dimension of a secant cone of a monomial image. We show that it need not equal the actual dimension, answering negatively the unrestricted monomial formulation of a question in Draisma's Oberwolfach report. For the six…
KOU-15.67Partial answer: F4(Z) is generated by three involutions, two of which commute
In Problem 15.67 of the Kourovka Notebook (2002), Ya. N. Nuzhin asked which adjoint Chevalley groups of normal type over the integers are generated by three involutions, two of which commute. By work of Tamburini and Zucca, Nuzhin, Timofeenko, and Kazakova and Nuzhin, the answer was known for the…
KOU-8.3Partial answer: new regular and irregular cases; the case m = 3 is settled for every prime p ≤ 29
In Problem 8.3 of the Kourovka Notebook (1982), B. A. F. Wehrfritz asked for which \(m\) and \(n\) the group \(P_n(\mathbb{Z}_{p^m})\) of all \(n\times n\) matrices over \(\mathbb{Z}/p^m\) with diagonal entries \(\equiv1\) and entries below the diagonal \(\equiv0\pmod p\), a Sylow \(p\)-subgroup of \(GL_n(\mathbb{Z}/p^m)\), is regular. It is known to be regular when \(nm-1<p\) (by Merzlyakov's…
OWR-14299292-010Partial answer: A_n is conjugacy-expansive for 5 ≤ n ≤ 24, and two infinite families of classes satisfy the condition for every n
A finite group \(G\) is conjugacy-expansive if for every normal subset \(S\) and every conjugacy class \(C\) of \(G\) the normal set \(SC\) contains at least as many conjugacy classes of \(G\) as \(S\) does. The notion is due to Halasi, Maróti, Sidki and Bezerra, who asked whether every finite simple group is…
OWR-1274-007Negative answer: the conjecture is false for both groups
In an Oberwolfach report of 2006, L. Bartholdi conjectured that the closures of the Aleshin automaton group \(G(\mathcal{A})\) and of the Bellaterra automaton group \(G(\mathcal{B})\) in the automorphism group of the binary rooted tree are \(\mathbb{Z}_2\)-analytic, and that the closure of \(G(\mathcal{B})\) is a pro-\(2\) Sylow subgroup of \(\mathrm{SL}_2(\mathbb{Z}_2)\).…
OWR-13353-012Complete proof
For a 2-vertex-connected graph G, let beta(G) be the maximum number of initial odd ears in an open ear decomposition, counting its initial cycle. We prove that deciding beta(G) >= k, with k part of the input, is NP-complete. This gives a complexity classification for a question of Benchetrit and…
AIM-COMBINATORICS-0025Complete scoped theorem
For integers b >= 2, n >= 1 and 0 <= r <= b-2, let D consist of the nonnegative integers with at most n base-b digits, all in {0,...,r}. We prove that the longest nonconstant arithmetic progression in D has exactly (r+1) times the largest divisor a of b^(n-1) satisfying (b-r-1)a < b terms, with the…
AIM-LINEAR_ALGEBRA-0010Quantitative refinement
Let H(z) be a finite product of positive linear numerator factors divided by positive geometric denominator factors, and let T_t be the lower triangular Toeplitz matrix generated by exp(tz)H(z). For every structurally admissible minor, we give an explicit index-sum formula for its order of…
AIM-COMBINATORICS-0014Complete scoped theorem
Consider a family of real two-dimensional subspaces with common isoclinic projection length 1/2. We prove that if one triangle of normalized cross-Gram matrices has negative determinant, then the entire family spans at most six dimensions and contains at most nine planes, independently of the…
AIM-ALGEBRAIC_GEOMETRY-0185Complete scoped theorem
Let two fixed projective cameras with distinct centers have integral plane image curves of degrees m and n, neither an epipolar line, over an algebraically closed field of characteristic zero. We describe all integral reconstructions using the normalized components of the fiber product of the…
AIM-COMPUTATION-0090Complete scoped theorem
Let p be prime, let S be a nonempty subset of F_p with d elements, and let h be a nonconstant polynomial with no zero in S. Let G_h be the subgroup of (F_p[X]/(h))^* generated by the classes of X-s for s in S. We determine the order, invariant factors, and integer relation lattice of the kernel K_h…
OWR-3470-005Complete scoped theorem
Let T=(T_1,...,T_d) be a tuple of contractions on a complex Hilbert space with T_i T_j=0 for all i,j. Relative to the common kernel F and its orthogonal complement E, write T_i=[0 0; W_i 0]. We prove that T is extremal among all commuting contractive extensions if and only if every W_i is a…
AMR-029-0066Complete scoped theorem
Let F be a finite forest. At each vertex v, choose a subset S_v of its neighbors and an integer interval [l_v,u_v]. We consider independent sets T for which l_v <= |T intersect S_v| <= u_v whenever v is not in T. Their convex hull is described by vertex bounds, edge inequalities and the perspective…
OWR-13346-004Complete scoped obstruction
Ambient Gaussian Landmarks and a Unit-Circle Obstruction to Nyström Approximation Alper Ferudun, Mercury Software GmbH. English unrefereed preprint, version 1.0, 24 October 2026. For k(x,y)=1+x^T y and m ambient standard Gaussian landmarks in R^n, the manuscript derives an exact chi-square…
OWR-720-006Complete proof
We prove that a Bessel wave packet system generated by a nonzero L2 function, with an arbitrary discrete parameter set in the affine Weyl-Heisenberg group, has a parameter set missing a left translate of every compact set. Hence all lower densities in the Czaja-Kutyniok-Speegle convention vanish…
OWR-720-010Complete proof
For a closed-range synthesis operator F, let P_F be the projection onto the orthogonal complement of its kernel. We study the metrics d_(p,kappa)(F,G) = (||F-G||^p + kappa^p ||P_F-P_G||^p)^(1/p), for finite p >= 1. The polar factor is universally a nearest Parseval synthesis operator among partial…
OWR-13948-006Complete counterexample
We construct a smooth polynomial, uniformly gradient-convex energy on two-valued maps of the unit disk whose minimizers have completely collapsed zero boundary data but admit no global labeling by two Sobolev functions. The potential is lambda times the squared modulus of y^2-(1-|z|^2)^2z. An…
OWR-1538-001Explicit counterexamples
We compute two small examples separating maximality, maximal transcendence degree and flatness in symmetric Poisson algebras. For the four-dimensional diamond Lie algebra we classify the regular argument shifts whose natural morphisms are flat. The shift at x* is not flat despite satisfying the…
OWR-13346-003Complete scoped theorem
Let H be a connected bipartite graph specifying interior observations of a generic rank-two matrix. Suppose every entry in a border row and a border column is also observed, except their intersection. We determine the number of completions from the cycle structure of H. A tree gives a…
OWR-15178-020Complete realization theorem
For every field k and every integer i >= 0, we construct a fixed complete regular local ring (R,m) of dimension 2i+2 and a fixed prime complete-intersection ideal I with the following property. Every sequence (a_n) of nonnegative integers is realized by ideals J_n satisfying I^(n+1) contained in…
OWR-4425-011Complete proof using the four-color theorem
Cooper, Rowland and Zeilberger conjectured that two ordered full binary trees sharing a pair of consecutive uncle-nephew leaves have a common parse word with equal letters at that prescribed pair. We prove this existence assertion using the four-color theorem. Gluing the two corresponding polygon…
OWR-14298803-004Complete scoped counterexample
We construct noiseless data for which extended dynamic mode decomposition (EDMD) converges uniformly to the true generator on a fixed polynomial auxiliary space, while the associated unregularized auxiliary optimum is minus infinity instead of 1. For dx/dt = -x^5 on [-1,1], we use quadratic…
OWR-15962-003Complete scoped theorem
We give an exact reduction from competing first-passage percolation with initially infected hubs to the same process with many seeds on a regular configuration model. Merging vertices already occupied by the same type preserves the entire infection process outside those vertices. Combining this…
OWR-12975-010Complete scoped theorem
Let G have a finite-index subgroup isomorphic to Z^d. We give an explicit finite-index compression formula and use it to show that positivity in all finite-dimensional unitary representations is equivalent to a sum of Hermitian squares at every matrix level exactly when d<=2. In the positive cases,…
OWR-1194-009Complete proof
For a finite simplicial complex K, let e_PL(K) and e_L(K) denote its minimum piecewise-linear and simplex-wise linear embedding dimensions. We prove that, for every d>=3, a finite d-complex satisfies e_PL(K)=d and e_L(K)=2d+1; this is the largest possible gap in that dimension. In dimension two we…
OPG-60024Complete proof
For a monotone map between complete lattices, let its two adjoint transforms be the supremum of sublevel sets and the infimum of superlevel sets. Porton's diagram problem asks for the missing transform of the outward reloid map and whether its iterated transforms form a finite set. Working in ZFC,…
OWR-12485-006Complete counterexample to a subquestion
We give an explicit embedded planar loop of length one with finite integral Menger curvature M_3 whose arclength parametrization is not differentiable at one point. The curve is smooth everywhere else. Its local graph has slope a sin(log log(1/|x|)) plus a smooth term. A direct three-point estimate…
KOU-16.46Partial answer: no non-trivial finite quotient of order below 10^8 (below 10^9 with Jambor 2015)
Problem 16.46 of the Kourovka Notebook, posed by M. Conder and going back to his paper of 1992, concerns the free product \(G=G_3*_{G_1}G_4\) of two of the Djoković–Miller groups that act arc-transitively on the cubic tree, amalgamated along the modular group \(G_1\), and the normal subgroup \(K\) of index \(8\) of \(G\), which…
AIM-PROBABILITY-0167Complete counterexample to a subquestion
For every finite root system Phi of rank n>=3, we prove that the generalized cluster complex Delta^2(Phi) is not the full (n-1)-skeleton of a finite polytopal PL manifold, with or without boundary, in any dimension. A face consisting of negative simple roots has link K3,3. A connected cubic…
OPG-37163Quantitative refinement
Lin and Zeng proved that every 5-regular or 8-regular graph with at least 10000000 vertices has an internal partition. We refine their argument to the explicit threshold 500000. A third round of degree-p deletion in their weighted local-density argument bounds the number of degree-p vertices by…
OPG-37413Complete computer-assisted proof
For every integer n>=8, put K=floor(n^(1/3)/2) and take r_i to be the least nonnegative residue of (2i+1)^(n-1) modulo n. We prove Alexa's corrected criterion: n is prime if and only if the ordinary integer sum of r_i for i=1,...,K equals K. Residue counting excludes cancellation by zero terms. A…
AMR-010-0302Complete counterexample (universal bound)
There is no universal constant K such that V(A intersection B) <= K V(A)V(B) for torsion-free convex-cocompact Kleinian groups, where V is three-dimensional convex-core volume. An explicit rank-two classical Schottky deformation has positive core volume tending to zero and constant systole…
OWR-15587-015Complete counterexample family (dimension refinement)
Ferroni constructed smooth 40-dimensional lattice polytopes with the integer decomposition property whose h-star polynomials are not unimodal. This paper refines the parameters in his Cayley-box construction to obtain infinitely many pairwise lattice-inequivalent examples in dimension 34. For every…
OWR-2090-008Complete proof (optimal constants)
For every dimension d and integer dilation n >= d+1, the ratios of the coefficients of a dilated Ehrhart h-star polynomial to its last coefficient are maximized simultaneously by the unimodular simplex. The same sharp inequality holds for every nonnegative numerator of degree at most d under the…
OWR-13674-019Complete scoped theorem
Let a finite family of monic real-rooted polynomials have a common interlacer and the same sum of roots. If a positive weighted average has all its roots in an interval with center c and radius R, then one member has all its roots in the concentric interval of radius 2R/sqrt(3). The factor…
AMR-030-0046Complete scoped theorem
We give an exact polynomial-time algorithm for counting successful pressing sequences of every labeled bicolored graph whose underlying graph has maximum degree at most two. Paths, cycles, arbitrary colorings, singular matrices, full-length sequences and disconnected unions are included. A…
OWR-2041-002Complete scoped theorem
We classify the squared geodesic loss class of smooth totally geodesic maps from a connected Riemannian manifold without boundary into the unit circle. Its pseudodimension is two exactly when there is no nonzero integral-period parallel one-form; otherwise its fat-shattering dimension at margin…
AMR-030-0030Complete scoped theorem
For the Cooper-Dutle normalized adjacency tensor of the Fano plane, we explicitly identify the entire projective eigenscheme at each root of lambda^2+lambda+2 with the bicanonical Klein quartic. A Fourier transformation turns the seven incidence equations into an adjugate equation and a quadric. A…
AMR-029-0040Complete scoped theorem
For each fixed finite connected loopless multigraph G and degree d, we give an effective finite union of finitely generated additive submonoids describing all positive k for which the regular k-subdivision has divisorial gonality at most d. Consequently the gonality sequence eventually depends only…
AMR-021-0017Complete auxiliary theorem
Let b_n count the nondegenerate orders of all derivative zeros satisfying Rolle interlacing, and let a_n count those realizable by ordinary degree-n real polynomials. We prove a_n <= (D_n+1)(2D_n+1)^n for D_n=n(n^2-1)/2. The classical enumeration of b_n then yields log(a_n/b_n) = -n^2 log(n)/2 +…
OWR-3389-023Complete counterexample to one source question
For the Dirichlet Schrodinger operator -d^2/dx^2 + V on (0,1), define M_j(V) as the expectation of V in the j-th L2-normalized eigenfunction. We prove M_2(t x)-M_3(t x)>t^2/20000 for every 0<t<=1. Every real bounded potential with ||V-x||_infinity<=10^-5 also has M_2(V)-M_3(V)>1/40000. In…
AMR-022-6050Complete counterexample
A question of D. K. Thomas, recorded as Problem 6.50 in Hayman and Lingham's Research Problems in Function Theory, asks whether geometric means of normalized exterior univalent functions satisfy the area-theorem coefficient inequality. We give an explicit counterexample at exponent 1/2. The two…
AMR-022-7077Complete scoped theorem
Rubel asked for functions whose derivative has the same value distribution as the function (Hayman-Lingham Problem 7.77). We parametrize all strictly positive C1 solutions on a compact interval by continuous Lebesgue-measure-preserving maps satisfying a spectral-radius condition.…
AMR-022-7069Counterexample to part (a)
We give an explicit counterexample to the fixed-coefficient necessity in the existence formulation of Hayman-Lingham Problem 7.69(a). In the commutative complex algebra C direct-product C, the coefficients e1=(1,1) and e2=(i,2i) define two differently scaled Cauchy-Riemann operators. The ordinary…
AMR-022-7042Complete counterexample
Arsove and Johnson asked whether every normalized Martin kernel on a multiply connected Green domain agrees with the Harnack envelope along a Green line issuing from the normalization point. We give a negative answer on the circular annulus exp(-pi/2)<|z|<exp(pi/2), based at1. For every outer…
AMR-022-7058Complete proof
Let E be a compact subset of [0,1], regarded as a subset of the real axis L in the plane. We prove that the 2-modulus of all rectifiable curves joining E to L minus E is infinite whenever E has positive conformal capacity. This gives an affirmative answer to A. A. Gonchar's question recorded as…
AMR-022-6084Complete counterexample
We construct a bounded normalized univalent function whose logarithmic coefficients are not O(1/n), although every pth root transform satisfies the coefficient estimate O(n^(2/p-1)). This gives a negative answer to the converse question recorded as Hayman-Lingham Problem 6.84 (UnsolvedMath…
AMR-022-5057Complete strengthening
We extend Oleg Ivrii’s Gaussian lacunary construction by slowing its variance clock. If deterministic positive variances satisfy v_n <= C/n, sum v_n = infinity, and sum v_n/log(n+1) < infinity, then almost surely the holomorphic series sum sqrt(v_n) xi_n z^(2^n) has almost every boundary point…
AMR-022-5074Complete dual criterion
For a nonnegative measurable function f on the unit circle, we give an attained variational test for almost-everywhere domination by the boundary modulus of a holomorphic function with positive real part. With h = log max(1,f), the logarithm of the least center modulus for a majorant of max(1,f)…
OWR-12697689-012Complete normal-surface criterion
For a normal complex algebraic surface germ, we prove that its focused scheme-theoretic arc Hilbert-Poincare series equals (1-q)^(-3) times the product of (1-q^j)^(-2) for j >= 2 if and only if it is an ADE rational double point. The known forward formula is complemented by a converse: low-weight…
OWR-13750335-005Complete counterexample
A conjecture in the 2023 Oberwolfach report on hypoelliptic operators proposes replacing the homogeneous weight in a fractional Hardy inequality by the power of the standard Heisenberg homogeneous norm while retaining the displayed constant. We give a complete counterexample to this replacement. On…
OWR-4136-004Complete counterexample
A conjecture printed in the 2009 Oberwolfach report on convex geometry proposes quantitative Prekopa-Leindler stability in the unshifted L1 distance of probability densities with zero mean. We give an explicit counterexample in every dimension. Strictly positive real-analytic densities with mean…
OWR-14297736-026Complete proof of Robin question
Fix a positive Robin parameter alpha and a volume M. We prove that the kth Robin eigenvalue of every bounded convex domain in R^n with volume M is at least a k^(2/(2n-1)), where a>0 depends only on n,alpha,M. The proof combines a one-dimensional trace estimate, a John ellipsoid and convex-domain…
OWR-14218-004Positive-characteristic counterexample
We compute the higher Koszul homology, defined by the Euler cap differential, of A=k plus V with V squared zero and finite-dimensional V. In each total weight, the calculation is the natural map from invariants to coinvariants of a signed cyclic rotation. A word orbit contributes higher classes…
OWR-12697690-008Complete Nakayama classification
For a connected nonsemisimple split basic Nakayama algebra B, let c_i and d_i be the lengths of the projective cover and injective envelope of the same simple module. We prove that the Koszul dual E of its Auslander algebra has global dimension max_i(c_i+d_i-2) and dominant dimension…
OWR-12732-005Complete auxiliary counterexample
Does every comparison in the facial semistable order admit coordinatewise ordered representatives in the corresponding TF-equivalence classes? We give a complete counterexample for the path algebra of 3 -> 2 -> 1. The classes C = {(-a,-b,a+b): a,b > 0} and D = {(-c,0,c): c > 0} satisfy a cover…
OWR-1058-002Complete counterexample
We give a smooth counterexample to the mesh-independent second-order difference-quotient estimate in Becker and Rannacher's 2005 Oberwolfach report, equation (5), represented by UnsolvedMath record OWR-1058-002. The fixed polynomial u(x,y)=xy(1-x^2)(1-y^2) on the unit square has homogeneous…
OWR-3389-011Complete scoped counterexample
We construct smooth positive unit-area conformal metrics on the two-sphere with first nonzero Laplace eigenvalue exactly 6*pi, while the intrinsic H^-1 norm of the conformal factor diverges uniformly over all round conformal parametrizations. The norm uses the scalar-function pairing with the…
OWR-14299904-009Complete counterexample
We disprove the uniform linear threshold for adjacent-hook monotonicity in self-conjugate and doubled distinct partitions proposed in Conjecture 3 of Ae Ja Yee's 2026 Oberwolfach report. For every fixed integer K>=3, the self-conjugate ratio H_(t+1)(Kt)/H_t(Kt) tends to infinity along even t; for…
OWR-14299904-011Exact enumeration: singleton-letter words
We give exact counts of word trees whenever the defining word contains a letter exactly once. A word tree is an assignment of independent uniform transition maps whose composite has exactly one cyclic point. Conditioning on the singleton-letter map reduces the count to a two-point collision…
OWR-734-003Complete theorem: cyclic symmetry obstruction
Over a field of characteristic zero, we prove that the Pluecker ideal of Gr(k,n) has no monomial initial ideal invariant under cyclic rotation whenever gcd(k,n) has an odd divisor greater than one. More generally, for n=mq and k=rq with q>1 odd and 1<=r<m, a permutation of cycle type q^m has trace…
OWR-12012-001Complete proof
For every Fibonacci number N=F_n>=2, we prove that the unshifted rational spherical Fibonacci lattice has least Euclidean distance2/sqrt(N), uniquely attained by the north pole and the next point. This answers the separation question in Brauchart's contribution to Oberwolfach Report40/2012,p2438…
OWR-14740-003Complete scoped counterexample
We give a smooth planar counterexample to simultaneous bandwidth-independent variance and bias limits for the mean symmetric-difference risk of kernel density level sets. For a standard bivariate normal density, a compactly supported radial C2 polynomial kernel, and critical bandwidths h_n(t)…