arXiv subjects

Kyle Binder

Publications and source records attributed to Kyle Binder.

4 recordsLinked to original sources

arXiv2026

The singular cohomology ring of a uniform matroid: combinatorics and Lefschetz properties

The singular cohomology ring of a matroid is an algebraic invariant which generalizes the Chow ring of a matroid. We study combinatorial and Lefschetz properties of the singular cohomology ring of a uniform matroid. Combinatorially, we construct an explicit basis for the singular cohomology ring in terms of Koszul homology. From this basis we derive multiple formulas for the Hodge numbers of the cohomology ring that recover and extend known formulas for the Chow polynomial of a uniform matroid. We also use this basis to show that the singular cohomology ring of a uniform matroid satisfies the "quasi-projective strong Lefschetz property" -- a slight weakening of the Hard Lefschetz property found in the Chow ring of a matroid.

math.CO↗

arXiv2024

The Singular Cohomology Ring of a Matroid

We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal flips of Adiprasito, Huh, and Katz, we prove sharp vanishing results for the cohomology ring and compute the dimension of the top-weight cohomology in terms of the M\"{o}bius invariant of the matroid. In the case of uniform matroids, these techniques give a recursive formula for the Hodge numbers. Finally, we generalize the singular cohomology ring to arbitrary building sets on the lattice of flats, and we show how the cohomology depends on the building set.

math.CO↗

arXiv2024

The Unipotent Tropical Fundamental Group

We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable in that it satisfies a Seifert--Van Kampen theorem and has a description for fans in terms of a bar complex. We then review an analogous classical object, the unipotent de Rham fundamental group of a sch\"{o}n subvariety of a toric variety. Our main result is a correspondence theorem between classical and tropical unipotent fundamental groups: there is an isomorphism between the unipotent completion of the fundamental group of a generic fiber of a tropically smooth family over a disc and the tropical unipotent fundamental group of the family's tropicalization. This theorem is established using Kato--Nakayama spaces and a descent argument. It requires a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.

math.AG↗

arXiv2019

Limit Sets and Internal Transitivity in Free Group Actions

It has been recently shown that, under appropriate hypotheses, the $\omega$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the action of a finitely generated free group or monoid. We give general definitions for several types of limit sets and analogous notions of internal transitivity. We then demonstrate that these limit sets are completely characterized by internal transitivity properties in shifts of finite type and general dynamical systems exhibiting a form of the shadowing property.

math.DS↗