Alper Ferudun

Property tau Need Not Survive Intersections of Normal Subgroups

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...

A Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class

We give four explicit matrices defining a genus-two surface-group representation over R[s,t,1/(st(1+t^2))], where R denotes the real numbers. Every real specialization has zero Witt class and zero Toledo number. Over R(s,t), however, the Witt class is the anisotropic Pfister form...

Open Sets of Linear Systems with n = mp That Are Not Stabilizable by Static Output Feedback: New Cases of a Problem of Eremenko

A real linear system 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 < mp, and when n = mp and the degree d...

Fourier Shadows and Tracial Free Infinite Divisibility

We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to the scalar Fourier series tau(exp(i sum u_j X_j)). The Fourier shadows obtainable from free sums form a proper subset of products of such...

Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

In his list of questions on scalar curvature, Gromov states that a complete Riemannian manifold X with Sc(X) ≥ 0 which is isometric at infinity to a complete flat manifold X_fl is flat, provided there is a homomorphism π₁(X) → π₁(X_fl) compatible with the isometry at infinity. He...

Minimal Geodesics under Finite Covers

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 or...

Scales of Convergent Series and the Null Ideal: An Answer to Hinkkanen's Problem 7.62

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 α a convergent series Σ_n x_{n,α} with non-negative non-increasing terms such...

Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written Proof for a Problem of the K3 List

Problem 1.19 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026) asks: if Y = Y₁ # Y₂ is a connected sum of 3-manifolds, neither of them S³, and the twist Φ about the connected-sum sphere is not isotopic to the identity, is ther...

Equilibria of Inverse-Square Repulsion on the Line Are Arithmetic Progressions

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 le...

The Equation J_0(z) = 1 Has at Most One Solution on Each Ray: An Answer to a Question of Zalcman

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...

Open Sets of Linear Systems with n = mp That Are Not Stabilizable by Static Output Feedback: New Cases of a Problem of Eremenko

A real linear system 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 < mp, and when n = mp and the degree d...

The Equation J_0(z) = 1 Has at Most One Solution on Each Ray: An Answer to a Question of Zalcman

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...

Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

In his list of questions on scalar curvature, Gromov states that a complete Riemannian manifold X with Sc(X) ≥ 0 which is isometric at infinity to a complete flat manifold X_fl is flat, provided there is a homomorphism π₁(X) → π₁(X_fl) compatible with the isometry at infinity. He...

Fourier Shadows and Tracial Free Infinite Divisibility

We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to the scalar Fourier series tau(exp(i sum u_j X_j)). The Fourier shadows obtainable from free sums form a proper subset of products of such...

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...

The Sharp Hexagonal Constant in Steinerberger's Geometric Uncertainty Principle on the Square

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 confi...

Knots with Signature 4 and Determinant 4k+1: An Answer to a Question of Shinohara

By a congruence of Murasugi, the determinant and the signature of a knot K satisfy det K ≡ (−1)^(σ(K)/2) (mod 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...

Focal Inversion Area Products in Elliptic Billiards

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...

A Negative Answer to Wolansky's Question on Mutually Dominating Multiphase Measure Spaces

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 ad...

Focal Distance Products of Outer Elliptic Billiard Polygons

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 per...