Lenny Fukshansky

Virtual links can be represented as equivalence classes of Gauss diagrams under Reidemeister moves. The Forbidden Moves are moves which look plausible but change the virtual isotopy class of the […]

We will examine the multiplicative structure of two skein algebras--- the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang […]

This is a talk in two parts covering two projects that the speaker mentored over the summer of 2025. The first project deals with the study of polytopes that arise […]

The classical Siegel's lemma (1929) asserts the existence of a nontrivial integer solution to an underdetermined integer homogeneous linear system, whose "size" is small as compared to the size of the coefficients of the system. Far-reaching generalizations of this theorem, producing a full basis for the solution space, were obtained over number fields by Bombieri […]

Hecke algebras play a central role in both number theory and representation theory. While some Hecke algebras have explicit descriptions in terms of generators and relations, others are understood through structure constants that encode multiplicities in tensor products of representations. In this talk, I will discuss several projects with Thibaud van den Hove and Jakob […]

A key problem in computer proofs based on solutions from copositive optimization, is checking whether or not a given quadratic form is completely positive or not. In this talk we describe […]

A low autocorrelation binary sequence of length $\ell$ is an $\ell$-tuple of $+1$s and $-1$s that does not strongly resemble any translate of itself.  Such sequences are used in communications and remote sensing for synchronization and ranging, where translation represents time delay.  A single number that indicates how good a sequence is for such purposes, […]

In this talk, I'll attempt to give a friendly introduction to tropical linear series and explore their relationship to matroid theory. Along the way, we'll stop to admire the beautiful view from enumerative geometry and combinatorics. This is joint work with Chih-Wei Chang, Matthew Dupraz, Hernan Iriarte, David Jensen, Sam Payne, and Jidong Wang, and also with […]

Inspired by the categorification program for a numerical invariant of three-manifolds, series invariants for closed manifolds and for knot complements were introduced. This in turn motivated an extension of the series invariant of the former case to Lie superalgebras. It was recently generalized to knot complements. In this talk, we review the original series invariants […]

Diophantine avoidance has been studied by several authors in recent years. This refers to effective results on existence of points of bounded size in a given algebraic set avoiding some […]

One of the key objects used in Ngo's proof of the fundamental lemma is the group scheme of universal centralizers associated to a split reductive group G. In this talk, we'll discuss forthcoming work, […]

Many aggregation problems ask us to turn individual judgments into a single collective outcome. In this talk, we model each voter’s input as a relation on a set of alternatives, […]