Future students

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.

Abstract: Around 1900 Young and Frobenius (independently, and through very different techniques) obtained a formula for the dimensions of the irreducible representations of the symmetric group. Some 53 years later, Frame, Robinson and Thrall noticed that the Young-Frobenius formula simplified into the now famous hook length formula. Nowadays there are many proofs, but the hook length formula remains something of a mystery, as if some deeper understanding lies just out of reach. One aspect of this mystery is that none of the proofs seem to indicate how one might come up with the formula in the first place, other than just guessing.

I will attempt to answer that question. It is an improbable tale that meanders through scenes of Young symmetrizers, Schur-Weyl duality, Weyl algebras, elementary combinatorics, and Plücker relations. All because Google's AI gave me a very obviously wrong answer when I was trying to find out the square of a Young symmetrizer.

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