Current students

Abstract: The totally asymmetric simple exclusion process (TASEP) is a finite Markov chain of particles hopping between adjacent sites on a one-dimensional lattice. The multispecies TASEP is a generalization in which particles have different types. These processes have interesting connections to algebraic combinatorics: the stationary distribution of the TASEP on a circle is connected to Macdonald polynomials at t=0, whereas the stationary distribution of the open-boundary TASEP is connected to Koorwinder polynomials at t=0.

Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has been a long-standing open problem to find a combinatorial formula for the stationary distribution of the multispecies TASEP with open boundaries. Recently, we studied the combinatorics of Ferrari–Martin multiline queues using type A crystals. In this talk, we use crystals of type C to construct an analog of multiline queues and give a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for a certain specialization of the boundary parameters. This is joint work with Olya Mandelshtam.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: A biased graph is a pair (G,B) consisting of a graph G and a collection B of “balanced" cycles in G. Biased graphs arise naturally as the cycles that have identity weight in a group-labelling of arcs where opposite arcs have inverse weight. I present an excluded minor characterization for biased graphs that have such a group-labelling over Z₃.

Abstract:  When you encounter an algebraic variety in the wild, you might ask: What do its components look like? Which components are smooth? How do the components intersect? For Springer fibres, answers to these questions are only known in some very special cases. This is particularly surprising because other aspects Springer fibres have been studied for the past 50 years and they appear throughout combinatorics and adjacent areas. For just one example: Hall–Littlewood polynomials can be obtained from the cohomology of Springer fibres by taking graded Frobenius characteristic.

One classical theorem says that the components of Springer fibers are indexed by standard Young tableaux. In this talk, we will discuss the benefits of instead using webs to index the components in two cases: the "two row" case, and our recent contributions in the "two column" case. We will see that in these cases, webs both characterize and describe the smooth components of Springer fibres, and give a geometric interpretation of rotation of webs.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: The sign coherence of c-vectors is one of the fundamental theorems of cluster algebras with principal coefficients.  Gekhtman and Nakanishi posed the Asymptotic Sign Coherence Conjecture for cluster algebras with arbitrary coefficients, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence.  We prove that for cluster algebras from quivers of arbitrary rank, their conjecture holds with probability 1 for a random mutation sequence.  Our results also establish the conjecture in full generality for many families of quivers.  This is joint work with Amanda Burcroff.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: The Fujisaki-Okamoto (FO) transform is a fundamental building block in new post-quantum cryptography standards like NIST's ML-KEM, where it is used to convert a weakly secure public key encryption scheme into a key encapsulation mechanism (KEM) secure against active attackers. In this talk, we'll explore two approaches to add extra security and functionality to post-quantum KEMs by enhancing the FO transform. First, we see how a birthday-style collision argument lets an attacker who collects many ciphertexts halve the security of the FrodoKEM and HQC standards, and how extending the FO transform with public salts thwarts this multi-target attack. Second, we turn to implementation flaws: for 19 months, HQC's reference implementation effectively skipped a security-critical verification step, yet basic correctness tests still passed. We show how the principle of "verifiable verification", via an extension of the FO transform, ties security to functionality, so that an implementation which that skips it visibly breaks.