Spring 2022

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. This is joint work with Vidit Nanda (Oxford).


Quantum theory  is a mathematical framework that has been incredibly successful at describing the physical world at very small scales. We will give a gentle introduction to this formalism from an algebraic point of view (no physics required!), and explain how representation theory can be used to address questions at the heart of quantum theory.


Quantum computation is defined to be any computational model based upon the theoretical ability to manufacture, manipulate, and measure quantum states. (2+1)-dimensional topological phases of matter (TPM) promise a route to quantum computation where quantum information is topologically protected against decoherence. In this talk, we will explore the underling mathematical theory that is driving the classification of these TPM. We will mainly focus on the algebraic/categorical structure behind such phases and explain where this structure fits in describing TPM. Finally, we will discuss some recent developments in the mathematical classification of TPM, namely the classification of modular categories.

The talk is intended for a general mathematical audience. No prior knowledge of physics is required.



In this talk, I will present an overview of plethysm covering some basic definitions, different approaches, and some of the known results. 


Many constructions with square matrices behave in surprising and beautiful ways as the dimension grows very large. In this talk we will introduce two examples that look very different: the large N behavior of Gaussian random matrices (discovered by Wigner and subsequently an active research area at the interface of probability, analysis, combinatorics, and the moduli of Riemann surfaces) and the Loday-Quillen-Tsygan theorem, which relates the cyclic (co)homology of an associative algebra A to the Lie algebra (co)homology of gl_N(A). We will not assume people are familiar with either topic and will introduce what we need. We will then explain how these two examples relate, via joint work with Ginot, Hamilton, and Zeinalian.


Classical invariant theory studies the ring of invariants k[x_1,..., x_n]^G under the action of a group G on a commutative polynomial ring k[x_1,..., x_n]. To extend this theory to a noncommutative context, we replace the polynomial ring with an Artin-Schelter regular algebra $A$ (that when commutative is isomorphic to a commutative polynomial ring), and study the invariants A^G under the action of a finite group, or, more generally, a semisimple Hopf algebra. In this talk we will present a survey of some results that generalize classical results: (1) when G is a reflection group, (2) when G is a finite subgroup of SL_n(C), and (3) that provide bounds on the degrees of the generators of A^G.



In his seminal 1996 paper, Kuperberg gives presentations for the categories of finite-dimensional representations of quantum groups associated to rank 2 simple complex Lie algebras (as braided pivotal categories). Such presentations underly various invariants in low-dimensional topology; in particular, they serve as a "foundation" for link homology theories. Kuperberg also poses the following problem: to find analogous descriptions of these categories for quantum groups associated with higher rank Lie algebras. In 2012, Cautis-Kamnitzer-Morrison solved this problem in type A using skew Howe duality, a technique that does not extend to give a solution in other types. In this talk, we will present a solution to Kuperberg's problem in type C. Our proof combines results on pivotal categories and quantum group representations with diagrammatic/topological analogues of theorems concerning reduced expressions in the symmetric group. Time permitting, we'll discuss some future directions. This work is joint with Elijah Bodish, Ben Elias, and Logan Tatham.



I will discuss the decategorification of the higher actions on bordered (sutured) Heegaard Floer strands algebras arising from joint work with Raphael Rouquier (and motivated by Douglas-Manolescu's constructions in cornered Heegaard Floer homology). I will also discuss a new perspective on the sutured surfaces to which these algebras are assigned, namely the interpretation of these sutured surfaces as open-closed cobordisms, and try to explain a more flexible gluing theorem (related to open-closed TQFT) for the decategorifications of the algebras, recovering the decategorification of the cornered-Floer or higher-representation-theoretic gluing theorem as a special case.




Coxeter groups were famously proven to be automatic by Brink and Howlett in 1993 and the automaticity of these groups has been an area of continued interest since. In this talk, we give a brief history and summary of recent developments in this area, and we introduce the theory of Regular Partitions of Coxeter groups. We show that Regular Partitions are essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group and explain how it gives a fundamentally free construction of automata. As a further application, we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi’s classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. (Joint work with James Parkinson)


Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula as a positive sum over excited diagrams of products of hook-lengths. In 2018, Morales, Pak, and Panova gave a $q$-analogue of Naruse's formula for semi-standard tableaux of skew shapes. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes gave their $q$-analogue. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called the \emph{minimal SSYT}, that are equinumerous with excited diagrams via a new description of the Hillan-Grassl bijection and have a version of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape.
This is joint work with Alejandro Morales and Greta Panova.



Web categories are a graphical to represent categories or subcategories of Rep for various groups. In particular, we will focus on the category Web(SL_{2k+1}^{-}), whose idempotent completion is equivalent to the category of tilting modules for SL_{2k+1}. We will show the existence of functors from this category to several categories with structure given by an affine building of type A_2k and consider actions of a group on this building to discover a class of Web(SL_{2k+1}^{-} module categories. 



One of the main objectives in quantum field theory is to compute and understand Feynman integrals. In this talk I will show how these integrals can be interpreted as solutions to A-hypergeometric systems of partial differential equations as defined by Gelfand, Kapranov and Zelevinsky. This in turn means that to every Feynman integral there is an associated toric ideal which comes with a large combinatorial toolbox. These combinatorial tools allow us to show that the toric ideals to some families of Feynman integrals are projectively normal. This especially means that the A-hypergeometric system associated to these integrals have parameter independent singularities and that the dimension of the solution space is always the expected one.