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Maryam Mohammadi Yekta

Publications and source records attributed to Maryam Mohammadi Yekta.

4 recordsLinked to original sources
arXiv2026

Log-concavity and Approximate Counting for Totally Unimodular Polytopes

We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm for approximate counting of the lattice points up to an explicit exponential factor. We achieve our bounds by showing that the associated generating polynomials fit into a new general class of log-concave polynomials called VLC ("variable-wise log-concavity''). This also implies a conjecture of Ferroni and Higashitani on the evaluations of the Ehrhart polynomials of unimodular polytopes. The essential ingredient for these results is the resolution of Barvinok's log-concavity conjecture for contingency tables on lines, which was proven using ChatGPT 6 Astra. We conjecture a generalization of Barvinok's conjecture, which we believe will lead to stronger and more general bounds.

math.CO↗
arXiv2026

New Bounds for Integer Flows and Verma Modules, via Denormalized Lorentzian Laurent Series

The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers. In this paper, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic $\mathfrak{sl}_{n+1}(\mathbb{C})$ Verma modules.

math.CO↗
arXiv2024

Optimal Linear Sofic Approximations Of Countable Groups

A countable group G is called k-linear sofic (for some 0 <k \le 1) if finite subsets of G admit "approximate representations" by complex invertible matrices in the normalized rank metric, so that non-identity elements are k-away from the identity. This class of groups was systematically studied by Arzhantseva and Paunescu [AP17], where it is shown that such a group is always 1/4-linear sofic. In this paper, we will study the optimality of this result for general countable groups and show that every linear sofic group is 1/2-linear sofic, and 1/2 cannot be improved. However, if G is assumed to be torsion-free, then it is 1-linear sofic. These results answer a question posed by G. Arzhantseva in her talk in the IAS Stability and Testability lecture series. We also study the optimal linear sofic constant of finite groups over C and fields of positive characteristic. For the proof, we establish effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest. Another ingredient of the proof involves bounding integrals of certain trigonometric sums.

math.GR↗