Events

Title: Counting Substructures in Hypergraphs with Spectrum

Abstract: Jacobi’s classical result expresses the generating function for closed walks at a vertex of a graph as the ratio of two characteristic polynomials. We find a hypergraph analogue of this relationship, showing that the same rational function for a hypergraph also counts combinatorial substructures within it, called infragraphs. We use Viennot’s Heaps of Pieces framework to establish this result for the adjacency tensor of a hypergraph. As an immediate consequence, we obtain an alternative proof for the monotonicity of the principal eigenvalue of a hypergraph. This is based on joint work with Joshua Cooper and Utku Okur (arXiv: 2411.03567).

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

Title: Turnpike phenomena in optimal control

Abstract: This talk will begin with a concise introduction to optimal control, viewed as an optimization problem where the decision variables are subject to dynamical constraints. We will then discuss the turnpike phenomenon in optimal control problems and show how it can be characterized through the energy-based concept of dissipativity of the underlying dynamical system. The theoretical results will be illustrated in the context of linear–quadratic optimal control problems, both for discrete and continuous time dynamics, where a complete characterization can be derived in terms of the spectral properties of the system.

Title: Cover-free Families on Graphs

Abstract: A family of subsets of a t-set is a d-cover-free family or d-CFF if no subset in the family is contained in the union of any d other subsets. Let t(d, n) denote the minimum t for which there exists a d-CFF of a t-set with n subsets. t(1, n) is determined using Sperner’s theorem. For d ≥ 2, we rely on bounds for t(d, n). Erdős, Frankl, and Füredi proved 3.106 log(n) < t(2, n) < 5.512 log(n). 

Title:ROAST: Robust Asynchronous Schnorr Threshold Signatures

Abstract: In this talk, I will review ROAST, which is a practical wrapper that turns any threshold signature scheme into a robust and asynchronous threshold signature scheme, as long as the base scheme is semi-interactive with identifiable aborts and unforgeable under concurrent signing sessions. Since the well-known FROST scheme meets these conditions, applying ROAST to FROST yields a simple, efficient Schnorr-based threshold signature scheme.

Title: Minimising the Martin invariant

Abstract: The Martin invariant of an even degree regular graph is an integer, defined by a recurrence at a vertex. In this talk, we define
this invariant and review some of its properties, and then focus on the following question: which graphs minimise the Martin invariant?

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

Title: Combinatorics of causal set theory

Abstract: Causal set theory is an approach to quantum gravity where spacetime is a locally finite poset. This approach asks interesting questions about posets that are as of yet little explored in combinatorics. I'll explain how I got interested in this subject recently and some of the aspects that might particularly appeal to a discrete mathematician.

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.

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:Hybrid Signature Schemes

Abstract: The transition to post quantum cryptography comes with many challenges. On the one hand, classically secure algorithms are well-tested, and we have a high degree of confidence in them against classical adversaries, but they are vulnerable to quantum computers. On the other hand, we currently have no reason to believe that post quantum algorithms are vulnerable to either classical or quantum adversaries, however, they are still relatively new, and because of that, they have not been scrutinized to the degree of their classical counterparts. Additionally, the transition will not happen instantaneously, and there will be a period of time where some interfaces are able to support post quantum algorithms while others can only support classical ones. One possible solution to these challenges (on the digital signature side of things) is to use hybrid signature schemes, which, loosely speaking, are digital signature schemes based off of both a classical and a hybrid digital signature scheme, which are secure if at least one of the underlying signature schemes is secure. In this talk, I will cover a few different signature combiners, and compare and contrast both how they work and the properties they guarantee. This talk is based off of the papers "A Note on Hybrid Signature Schemes" by Nina Bindel and Britta Hale, and "Bird of Prey: Practical Signature Combiners Preserving Strong Existential Unforgeability" by Jonas Janneck.

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.