Spring 2023
03/22 – William Clark & Keegan Stump (Davidson College): New Progress on Shuffle-Compatibility
Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. In 2018, Gessel and Zhuang formalized this notion of a shuffle-compatible permutation statistic and developed an algebraic framework for shuffle-compatibility, based on quasisymmetric functions, for a large family of statistics called descent statistics. Since then, shuffle-compatibility has become an active area of research, and this talk will feature two undergraduate research projects on this topic. William Clark will present new results on kernels of descent statistics, which were defined and first studied by Grinberg in the context of shuffle-compatibility. Keegan Stump will discuss an extension of a bijective approach for shuffle-compatibility (created by Baker-Jarvis and Sagan for ordinary permutations) to the setting of colored permutations.03/29 – Joe Johnson (NCSU): Promotion and Rowmotion
Rowmotion maps an order ideal of a finite poset to the order ideal generated by the minimal elements of the complement. On certain graded posets, the orbits of rowmotion are particularly small and are related to promotion on certain tableaux via an equivariant map. We discuss one instance of this phenomenon.
04/05 – Andrew Harder (Lehigh University): The geometry of 2-loop Feynman integrals
In particle physics, many quantities of interest are expressed in terms of Feynman integrals. These integrals are attached to combinatorial objects called Feynman graphs, and can be expressed as integrals over (infinite) domains inside the real plane. In examples, one often finds that Feynman integrals are equal to special values of functions that are of interest to algebraic and arithmetic geometers. For instance, multiple zeta functions, elliptic dilogarithms and the like. One explanation for this comes from the work of Bloch-Esnault-Kreimer, and subsequent work of Brown, which shows that Feynman integrals can be interpreted as periods of smooth algebraic varieties in the sense of Kontsevich and Zagier. In the literature, almost all examples that have been worked out from this perspective belong to a rather restricted class of graphs called "primitively divergent" graphs.
I will talk about recent work with Doran and Vanhove which studies an infinite class of (non-primitively divergent) graphs with first betti number equal to 2. We show that in this case, the Feynman integrals which appear are constructed from algebraic functions and periods of hyperelliptic curves.
04/08/2022 - Triangle Lectures in Combinatorics at NCSU
04/12 – Sean Sanford (The Ohio State University): Real Fusion Categories
Fusion categories are algebraic gadgets that have seen many applications in topology and mathematical physics. In particular, they can be used to encode topological quantum field theories in the sense of Atiyah. Classical examples of fusion categories include C-Rep(G), the category of finite dimensional complex representations of a finite group G. Because of their connections to physics, much of the literature on fusion categories has focused on the case where the base field is the complex numbers. In this talk we will discuss some of the new phenomena that can be found when working over the real numbers, and we will give some examples of new fusion categories recently discovered in https://arxiv.org/abs/2303.17843 .
04/19 – Mrunmay Jagadale (CalTech): Towards the decorated TQFT for Z-hat
The Z-hat invariant, proposed by Gukov, Pei, Putrov, and Vafa, is a fascinating q-series invariant of three-manifolds related to various areas of mathematics and physics. We want to find a topological quantum field theory (TQFT) that computes this invariant to understand it better. This invariant cannot come from TQFT in the sense of Atiyah, as it depends on an additional structure on the three-manifold, viz. spin-c structures. We need decorated TQFTs, TQFTs that compute invariants that depend on additional structures. In this talk, we'll discuss decorated TQFTs, cutting and gluing rules in decorated TQFTs, and propose vector spaces assigned to two-dimensional surfaces in the decorated TQFT for Z-hat.
04/26 – Roeland Wiersema (University of Waterloo): Here comes the SU(N): multivariate quantum gates and gradients.
Variational quantum algorithms use non-convex optimization methods to find the optimal parameters for a parametrized quantum circuit in order to solve a computational problem. The choice of the circuit ansatz, which consists of parameterized gates, is crucial to the success of these algorithms. Here, we propose a gate which fully parameterizes the special unitary group SU(N). This gate is generated by a sum of non-commuting operators, and we provide a method for calculating its gradient on quantum hardware. In addition, we provide a theorem for the computational complexity of calculating these gradients by using results from Lie algebra theory. In doing so, we further generalize previous parameter-shift methods. We show that the proposed gate and its optimization satisfy the quantum speed limit, resulting in geodesics on the unitary group. Finally, we give numerical evidence to support the feasibility of our approach and show the advantage of our gate over a standard gate decomposition scheme. In doing so, we show that not only the expressibility of an ansatz matters, but also how it's explicitly parameterized.