Alper Ferudun

A Finite Gap Bound for Local Progression-Free Density

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(...

Dirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig

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 unti...

Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation

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 exact...

A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces

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 conne...

Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One

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 complet...

Dirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig

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 unti...

Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One

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 complet...

A Semi-Abelian Group of Order 768 That Is Not an M-Group

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...

A Semi-Abelian Group of Order 768 That Is Not an M-Group

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...

An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

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 equat...

An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

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 equat...

Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

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 an...

Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

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 an...

Maximal Finite-Set Stabilizers in Thompson's Group T

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 is...

Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

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...

A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

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 an...

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

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...

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

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...

Maximal Finite-Set Stabilizers in Thompson's Group T

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 is...

A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

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 an...