Alper Ferudun · Mercury Software GmbH
Complete-result manuscripts from the Open Math Problems programme, with the reading PDF, LaTeX source, reproducibility material, and a public verification report kept together.
Let A = {n₁ < ⋯ < nᵣ} be a finite set of positive integers. Choose one residue class aᵢ mod nᵢ 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,…
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…
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…
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…
Let h ≥ 2 be fixed, let f be a natural-valued polynomial of degree d ≥ 2, and suppose that A ⊆ ℕ satisfies f(ℕ) ⊆ hA. We prove |A ∩ [0,X]| ≥ X^{4/(3hd)−o(1)} when h is even, and |A ∩ [0,X]| ≥ X^{4/((3h+1)d)−o(1)} when h is odd. Both exponents are strictly larger than the elementary exponent 1/(hd).…
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 φ on the finite branch-value set and by a uniform fiber formula. Reducing the affine map on…
Let (M,g) be a compact semi-Riemannian manifold of indefinite signature whose null geodesics are complete. We prove that no C¹ one-form η can have the property that, along every nonconstant affinely parametrized null geodesic γ: ℝ → M, the function η(γ̇) is affine with nonzero slope. This gives a…
Fix K > 1. We construct finite nonempty sets A,B ⊆ 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 all…
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 N² = 0, although neither the manifold nor any connected double cover admits a nonzero parallel vector field. This gives a negative answer,…
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…
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 ⊗_min and ⊗_max…
An explicit question of Abért asks for a free-spanning-forest proof that the first L²-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 ≤ Γ of index k, lift a spanning tree of the finite Schreier…
We characterize the functions arising as Fuglede–Kadison determinant transforms of bounded commuting tuples in a fixed II₁ 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…
For α ∈ ℝ, let ℓ²_α be the sequence space with squared norm ∑_{n≥0}|x_n|²(n+1)^α. 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…
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 ≥ 2 we construct a linearly growing UPG automorphism of…
An AIM problem asks for a characterization of the face-angle and dihedral-angle data of a triangulated polyhedral surface in ℝ³ 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,…
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 π₁(S_g) → F_g that kills a…
Let Γ be a countable subgroup of SL₂(ℚ_p) containing no noncentral element of trace 2 or −2. We prove that, outside a Haar-null set of g ∈ SL₂(ℚ_p), the generated group ⟨Γ,g⟩ has the same property. Thus adjoining one random element preserves the absence of parabolics almost surely, answering…
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 contained in D, its setwise stabilizer is a maximal proper subgroup of countably infinite index in T. If |A| = k, then Stab_T(A) is isomorphic to F^k…
Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on the operator-norm…
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…
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…
For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic and degree. A…
Let E be the standard object in the generic polynomial Hecke category over 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 tensor U): in degree n >= 2, only the one-row…
We construct a homotopy through proper rational holomorphic maps from the unit ball in C^2 to the unit ball in C^4, joining (z,w) -> (z^3,sqrt(3)zw,w^3,0) to (z,w) -> (z,w,0,0). The construction answers the explicit target-four question in the AIM list on the Cauchy-Riemann equations. A mixed-term…