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…
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…
At the 2014 Oberwolfach workshop on combinatorics, P. Heinig asked whether, for every odd n ≥ 7, every n-vertex graph with minimum degree at least ⌈n/2⌉ 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…
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…
At the 2018 Oberwolfach workshop on enumerative combinatorics, A. Sportiello compared two families of objects attached to a digitally convex shape λ (a convex polyomino): the permutations whose graph lies in λ and that avoid a 123-pattern whose two corner cells also lie in λ, counted by A_λ, and…
Rödl, Ruciński and Szemerédi stated the following conjecture, which they attribute to Katona and Kierstead: every k-uniform hypergraph on n ≥ k+1 ≥ 4 vertices in which every (k−1)-set lies in at least ⌊(n−k+3)/2⌋ edges has a tight Hamiltonian cycle. They proved it for k = 3 and all sufficiently…
Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every 1 ≤ k ≤ 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,…,n}? Second, for every B ⊆ {1,…,n} with |B| ≥…
We study computability of sections of the restriction map S_1(N) to S_1(M) for countable stable models M elementary in N. Types are represented by characteristic functions on formulas with named parameters, and the two elementary diagrams are separately decidable. Classical stable definability…
For independent indeterminates a, x_1,...,x_N, y_1,...,y_N, we prove that the sequence b_k = a e_k(X) + sum_i y_i e_k(X without x_i) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients…
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 is…
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-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…
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…
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…
For every prime p and every rank r >= 1, let H_r be the compact subgroup of GSp_{2r}(Q_p) consisting of diag(I_r,aI_r) for p-adic units a. 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…
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…
For positive-definite Hermitian matrices A,B, we study the zeros of P_m(z)=Tr((A+zB)^m). Loewner bounds lB<=A<=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…
For real symmetric zero-diagonal weighing matrices W with W^2=dI, we deduce induced-l_p epsilon-paving into at most ceil(36 epsilon^(-q)) 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…
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…
For the degree-d Fermat fourfold in complex projective 5-space, and for delta>=2 with d>10delta-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…
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…
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 ≥ 2 vertices satisfies χ(G) > ⌈⌊w(G)/2⌋/(n − w(G))⌉. We show that the conjecture is false.…
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 G(J) be its set of ground states. Bäumler proved that on every locally finite tree, for coupling laws of linear growth, |G(J)| is almost…
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…
The Bernoulli displacement model is the random Schrödinger operator h_{ω,λ} = h_0 + V_ω on ℓ²(Z) in which each cell of two neighbouring sites carries one single-site potential λ ≠ 0. The potential sits on the left or on the right site of the cell according to an independent Bernoulli variable ω_k.…
In the ε-Dense Steiner Tree problem of Karpinski and Zelikovsky, every terminal is adjacent to at least an ε-fraction of the non-terminals, and a Steiner tree with the fewest edges is sought. For every fixed ε > 0 the problem has a polynomial-time approximation scheme. At an Oberwolfach problem…
For 2k > d + 1, the centred and normalized total k-volume of a stationary Poisson process of k-planes in hyperbolic space 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}/(Var…
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 R² with orthonormal frame X₁ = ∂ₓ, X₂ = x(x^{2ℓ} + z²)∂_z essentially…
Let T_f φ = φ∘f be a dissipative composition operator on L^p(X, B, μ), 1 ≤ p < ∞, induced by a bimeasurable bijection f such that μ∘f and μ∘f⁻¹ are bounded by a multiple of μ. D'Aniello, Darji and Maiuriello proved that if f has bounded distortion, then T_f has the shadowing property if and only if…
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,…
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>=2, the…
Fix s>=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 binomial(s,3). Consequently…
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>=1 and n>=2k+2, ceil(log2(2k+2)) <= eq(C_n^k) <= 4 ceil(log2(k+1))+1. The upper bound partitions…
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…
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 II1 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…
A real polynomial is multi-affine if it has degree at most one in each variable. Basu and Perrucci proved that the real zero set of one multi-affine polynomial of degree d in R^n has at most 2^(d-1) connected components, independently of n. They asked whether the number of connected components of…
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 + ∇φ_A, where A is divergence free and φ_A solves div(σ∇φ_A) = −div(σA). To solve this equation, Arnold and Harrach assume a conductivity σ that…
Two disks of radius 1 centred at (±1, 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…
Let E_n be the Euler up/down numbers, Σ_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…
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 E N_i is asymptotic to…
Let G be a finite connected subgraph of the grid ℤ³, not necessarily induced, and let X(G) be the cube complex whose cells are the squares and 3-cubes of ℤ³ 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 v₁,…
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 HC(T) ∪ {0} contain the range of an injective operator S ∈ L(X)? Second, for such an S, is every linearly…
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…
Kahle, Kubjas, Kummer and Rosen showed that for a set E of observed entries of a d_1 × ⋯ × d_n tensor with |E| = Σ_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…
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…
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 - k a_(k-1)a_(k+1) is positive. Katkova, Shapiro, and Vishnyakova already disproved this bound in 2024. We give an alternative…
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…
A bipyramid cell is a tetrahedral partition with one interior vertex v₀, n boundary vertices that are coplanar with v₀ 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 S¹_d(Δ) of C¹ splines…
Let M be a loopless matroid on a finite set E with rank function r, and let Σ(M) ⊆ ℝ^E be the support of its matroid fan. Draisma, Eggleston, Pendavingh, Rau and Yuen defined adim(M) as the minimum of 2 dim(Σ(M) + R) − dim R over rational subspaces R ⊆ ℝ^E; for the matroid of a complex linear space…
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…
For a bipartite graph G = (U ∪ V, E), a set I ⊆ U ∪ V and rationals b_ij, let S(G,I) = {x ∈ R^(U∪V) : x_i + x_j ≥ b_ij (ij ∈ E), x_i ∈ Z (i ∈ I)}, and let k be the least positive integer with kb integral. In the 2008 Oberwolfach report on combinatorial optimization, Conforti conjectured that conv…
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…
The star transform integrates a function on the plane along m rays with a common vertex, with directions γ_1, …, γ_m and nonzero weights c_1, …, c_m. In the inversion formula of Ambartsoumian and Latifi, the directions ψ at which Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ vanishes, called singular directions of Type…
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 C(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…
Let M_0 = ℂ and M_1 = S², with complete metrics g_κ of constant curvature κ ∈ {0, 1}. Every complete smooth metric of nonnegative curvature on M_κ can be written uniquely as φ*(e^{−2u} g_κ), where φ is an orientation-preserving diffeomorphism fixing 0 and 1 (and ∞ if κ = 1) and u is a smooth…
Alfieri and Binns say that an F[U]-module M satisfies the strong geography restriction if it has a direct summand F[U]/U^ℓ ⊕ F[U]/U^(ℓ−1) ⊕ ⋯ ⊕ F[U]/U, where ℓ is the least integer with U^ℓ M_red = 0. They showed that HF⁻(Y) satisfies it when Y is surgery on a knot in S³ or large surgery on a link,…
For 0 ≤ i ≤ 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 × B^(d−i−1). We show that it is, for all 0 ≤ i ≤ d…
Let E_1 ≤ E_2 ≤ ⋯ be the eigenvalues of the Dirichlet Laplacian on a bounded open set Ω ⊂ R^n, and let M_p(J) = ((n+2p)/n · (1/J) Σ_{j≤J} E_j^p)^{1/p}. Harrell and Stubbe proved that M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2, and in a problem list of the 2009 Oberwolfach workshop on low eigenvalues…
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,…
For points A_1, …, A_n, A'_1, …, A'_n in K², where K is a field, write [XY] for the 2 × 2 determinant and let α(π) be the cyclic quotient ∏_k [A_{π_k} A_{π_{k+1}}] / ∏_k [A'_{π_k} A'_{π_{k+1}}], with indices read cyclically. Below, Krummeck and Richter-Gebert conjectured in 2003 that Σ sgn(π) α(π)…
Mean value coordinates of a planar polygon P with n vertices define a map from P to 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…
Let f ∈ C[x] have at most k monomials and put M_n = ∫_0^1 f(x)^n dx. A. Gasull asked whether there is a number N(k) such that M_1 = ⋯ = 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,…
F. Santos asked in an Oberwolfach problem session (2019) whether every combinatorial d-sphere S with n ≥ 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 lk(v) ∩ lk(w) = lk(vw), equivalently an edge that lies in…
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…
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…
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…
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…
For a primitive 0–1 matrix A, Freslon, Gerontogiannis and Skalski computed the quantum isometry group G_A^∞ of the log-Laplacian spectral triple on the Cuntz–Krieger algebra O_A. They showed that the quantum automorphism group QAut(A) is a quantum subgroup of a larger quantum group G_A^1 and asked…
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: ℤ/n → ℝP¹ with f(i) ≠ f(i+1) and at least three values, modulo SL(2,ℝ). We answer both questions for all n ≥ 3. The space B_n is a real-analytic (n−3)-manifold with…
For even k ≥ 4 let f_k = 1 + O(q^{dim M_k}) be the extremal modular form of weight k for SL_2(ℤ). Bannai, Koike, Shinohara and Tagami conjectured that if k = 12μ and f_k mod p is a nonconstant power series in q^p (their Case (2)), then f_k(τ) ≡ g(p^r τ) (mod p) for an extremal modular form g of…
For a set-theoretic solution (X, σ) 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, σ) can be chosen to be the quantum symmetriser of −σ, and that it factors through a…
Four agents in the plane move by the standard distance-based gradient law ṗᵢ = Σ_{j≠i} e_ij (p_j − pᵢ), e_ij = ‖pᵢ − p_j‖² − d̄_ij², 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…
Let f̄(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 f̄(2,n) ≤ n − 3. We show that the answer is no. A triangulation M₇ of the real projective plane minus two disjoint open discs has 7…
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…
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…
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…
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…
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…
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…
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…
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…
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…
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…
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…
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…
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…
For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 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,…
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…
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…
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…
In an Oberwolfach report of 2017, Wolansky introduced a domination order between vector-valued measure spaces (X, σ) and (Y, η) with k components, defined by one Markov kernel that transports every σ_i to η_i, and asked whether two atomless spaces that dominate each other must admit deterministic…
For a partition of [0,1]² into N measurable pieces Ω_i, let F = Σ_i |Ω_i| A(Ω_i) + Σ_i (|Ω_i| − min_j |Ω_j|), where A is the Fraenkel asymmetry with respect to discs. Steinerberger proved F ≥ 1/60000 for large N and asked, in Oberwolfach Report 49/2013, whether the extremal configuration is a…