Seminar

Title: Fast quantum state transfer on paths via localized eigenvectors

Abstract: 

Title:The Heron-Rota-Welsh conjecture via Lorentzian polynomials

Abstract:The Heron-Rota-Welsh conjecture asserts that the characteristic polynomial of a matroid has log-concave coefficients. This conjecture was open since the 1970s until it was proven by Adiprasito, Huh, and Katz in 2018 using their newly developed combinatorial Hodge theory. Their proof was groundbreaking, but rather complicated. In this talk, we will give a proof of this fact using Lorentzian polynomials, which otherwise will use nothing more than basic theory of matroids, linear algebra, and convexity.

Title:Almost Regular Matroids

Abstract: Regular matroids form an important and extensively studied class of matroids and have numerous known descriptions and characterizations. In the 1980s, Truemper gave a constructive description of the related class of almost regular matroids: non-regular matroids $M$ such that for every element $e$ of $M$ either $M/e$ or $M\backslash e$ is regular. In this talk, we present a description of almost regular matroids in terms of grafts. A consequence of this description is that almost regular matroids have bounded complexity. We outline some of the proof ideas after introducing the necessary background.

Title: Generalized Abelian Complexities for Pisot-Type Substitutive Sequences

Abstract: Two finite words are said to be abelian equivalent if one is a permutation of the letters of the other. For an infinite word, one can investigate the associated complexity function, called Abelian complexity, which is a classical object of study in combinatorics on words. In particular, many works study the abelian complexity of automatic sequences, where a longstanding conjecture states that the abelian complexity of an automatic sequence is a regular sequence. We have studied when the abelian complexity can be computed efficiently, in particular using the theorem prover Walnut. To this end, we study words that are fixed points of Pisot-type substitution and prove that these words satisfy the conjecture. If time permits, I will present k-abelian complexities, which are intermediate complexities between the abelian complexity and the factor complexity. I will also explain how our results can be extended to these
complexities and how we can obtain a two-dimensional linear representation of some examples. This talk is based on joint work with J-M Couvreur, M. Delacourt, N. Ollinger, J. Shallit, and M. Stipulanti (arXiv: 2504.13584).

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

Title:Seems Legit: Automated Analysis of  Subtle Attacks on Protocols that Use Signatures

Abstract: The standard definition of security for digital signatures—existential unforgeability—does not ensure certain properties that protocol designers might expect. For example, in many modern signature schemes, one signature may verify against multiple distinct public keys. It is left to protocol designers to ensure that the absence of these properties does not lead to attacks. Modern automated protocol analysis tools are able to provably exclude large classes of attacks on complex real-world protocols such as TLS 1.3 and 5G. However, their abstraction of signatures (implicitly) assumes much more than existential unforgeability, thereby missing several classes of practical attacks. We give a hierarchy of new formal models for signature schemes that captures these subtleties, and thereby allows us to analyse (often unexpected) behaviours of real-world protocols that were previously out of reach of symbolic analysis. We implement our models in the Tamarin Prover, yielding the first way to perform these analyses automatically, and validate them on several case studies. In the process, we find new attacks on DRKey and SOAP’s WS-Security, both protocols which were previously proven secure in traditional symbolic models.

Title: Eigenpolytopes

Abstract: Each eigenspace of a graph gives rise to a real convex polytope. This connection works best for highly regular graphs - distance-regular graphs or, more generally, walk-regular graphs. I will discuss this relationship and give some applications, including a proof of the Erdos-Ko-Rado theorem.

Title: Parameter robust preconditioning

Abstract: The discretization of a partial differential equation (PDE) results in a linear system and iterative solvers are typically used to solve these linear systems, especially if these linear systems are large. Krylov subspace methods are an important class of iterative methods but for these methods to be effective they must be combined with a preconditioner. However, finding a good preconditioner for a given discretization of a PDE is a nontrivial task and so in the first part of this talk I will summarize some useful results from the literature that use a Functional Analysis framework to identify preconditioners for symmetric PDEs.

Title:The satisfiability threshold and solution space of random uniquely extendable CSPs

Abstract: A random constraint satisfaction problem (CSP) consists of a set of variables and a set of randomly chosen constraints. Many commonly studied CSPs are constructed by choosing constraints of a specific form. One such problem is $k$-UE-SAT, where each constraint is chosen from the set of uniquely extendable (UE) constraints. A conjecture due to Molloy and Connamacher gives an exact value for the high probability satisfiability threshold of $k$-UE-SAT problems. We make progress towards this conjecture by showing that a subclass of random CSPs with UE constraints has the conjectured satisfiability threshold. We also describe the solution space geometry for this class of CSPs, and make further conjectures about the general $k$-UE-SAT problem.This talk is based on joint work with Jane Gao.

Title: Tilings of Benzels (and other finite regions) in the hexagon grid

Abstract: In 1990, Conway and Lagarias introduced tilability criteria for tilability for finite regions of the hexagon and square grids. In the same year, Thurston expanded upon their work, introducing the height function criterion. We will discuss some new results in tilability: A new tilability criteria via the SL_2(C) double dimer model, and enumeration of tilings of special regions called Benzels, introduced in 2020 by Propp, using a technique called compression. If time allows, we will also discuss ongoing work that expands Thurston's height function to stone-and-bone tilings of the hexagon grid.

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

Title: Recurrence of unitary and stochastic quantum walks

Abstract: Recurrence means a return of the dynamical system to its initial state. Classical result of Polya [1] from 1920’s shows that a random walk on a line and a 2D grid returns to the origin with certainty, while it is transient on higher-dimensional lattices. For quantum walks, detection of recurrence requires partial measurement after each step, yielding a conditional quantum dynamic. We review the method to study quantum recurrence based on generating functions [2], focusing on the quantum walk on a line. Combination of measurement induced effects and faster spreading implies that a quantum walk on a line can escape to infinity without ever returning to the origin. Finally, we present a recent extension of the study of recurrence to quantum stochastic walks [3], which interpolates between quantum and classical walk dynamics [4]. Surprisingly, we find that introducing classical randomness can reduce the recurrence probability --- despite the fact that the classical random walk returns with certainty --- and we identify the conditions under which this intriguing phenomenon occurs.