Alper Ferudun

Products of the Two Focal Inversion Areas in Elliptic Billiards

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 res...

Exact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion

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)), wher...

Original Antipedal Centroids in Elliptic Billiards

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 inter...

A Six-Periodic Zero of Focal Outer-Antipedal Area

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 there...

A Six-Periodic Zero of Focal Outer-Antipedal Area

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 there...

On a Question of Freslon, Gerontogiannis and Skalski about Quantum Isometries of Cuntz–Krieger Algebras

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 gro...

Components and Cohomology of Fock's Baby Teichmüller Space

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...

A Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary

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 ope...

The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism

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...

On a Question of Freslon, Gerontogiannis and Skalski about Quantum Isometries of Cuntz–Krieger Algebras

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 gro...

Extremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami

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 mo...

Components and Cohomology of Fock's Baby Teichmüller Space

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...

All Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable

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...

Extremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami

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 mo...

A Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary

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 ope...

All Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable

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...

The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism

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...

Finite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs

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...

Finite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs

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...

A Negative Answer to Melnikov's Valency-Variety Problem

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 c...