The classical problem about  the possible eigenvalues of a product of unitary matrices AB given the eigenvalues of A and B has relations to many areas of geometry, representation theory and combinatorics. There is a polytope controlling this problem whose faces can be described in terms of the quantum cohomology of Grassmannians. The problem of determining the vertices of the polytope remained open for a long time, but recently I gave an inductive framework for understanding  vertices on a given face: some vertices come explicitly from  cycle classes of geometrically defined loci,  and the remaining vertices come from an induction procedure from smaller groups. I want to describe this method, and generalization to the case of arbitrary groups (joint work with Joshua Kiers).  If time permits, I will state connections to the theory of unitary rigid local systems.

In 1999 Lawrence and Zagier pioneered a connection between modular forms and invariants of 3-manifolds arising from quantum topology and physics. Specifically, they provided a holomorphic function on the unit disk that unifies the Witten-Reshetikhin-Turaev invariants of the Poincaré homology sphere and is a quantum modular form. Using a recently developed invariant by Akhmechet, Johnson, and Krushkal (AJK), we realize the series of Lawrence and Zagier as one instance in an infinite family of quantum modular invariants whose radial limits toward roots of unity may be thought of as deformations of WRT invariants. As part of our process, we provide the first calculation of the AJK invariant for an infinite family of 3-manifolds. This talk is based on joint work with Eleanor McSpirit.

This talk is an introduction to supersymmetric quantum mechanics. We discuss the origin of this subject in quantum field theory and some interesting applications to other fields of mathematics and physics. We focus on the finite-dimensional case where everything can be formulated in terms of linear algebra. Our main applications are to a tantalizing framework relating graph theory, Lie theory and quantum computational complexity theory. 

Modern mathematical research has been enriched by exploring the bijection between Higgs bundles and local systems over noncompact Riemann surfaces, an incredible feat accomplished profoundly by C. Simpson by introducing weighted filtrations. In this discussion, our focus will be on further unraveling this relationship by delving into the correlation between the Dolbeault and de Rham moduli spaces for generally complex reductive structure groups, utilizing the influential language of parahoric group schemes as initially established by Bruhat-Tits. This presentation is based on collaborative work with Georgios Kydonakis, Pengfei Huang, and Hao Sun.

In this talk, I’ll review the role of Lie algebras in variational quantum computing and in quantum dynamics and control. One of the main obstacles to training parameterized quantum circuits is the presence of flat regions in the parameter landscape, called barren plateaus. To diagnose such barren plateaus, we obtain an exact expression for the variance of the cost function in terms of the dynamical Lie algebra generated by the circuit. Second, we classify all dynamical Lie algebras of 1-dimensional translation-invariant spin chains. The talk will be based on the papers arXiv:2309.09342 and arXiv:2309.05690.

Unlike Relativity, which can be derived from a few physical postulates, Quantum Mechanics is defined from a mathematical "recipe" that is known to "work," in the sense that it leads to a theory whose predictions agree with experiment exactly, but no one really understand what the physical meaning of the recipe is.  One approach that has been tried to clarify the meaning of the "recipe" is to modify parts of it to see how the resulting theory changes.  For instance, one can change the field over which the state space is defined from the complex numbers to other fields or division algebras, e.g. real numbers or quaternions.  In this talk, I will discuss what would happen if we change that field to a finite field Fq.  We find that the correlations in the resulting toy theory do not violate the CHSH inequality though its predictions cannot be reproduced by a hidden variable theory.  Furthermore, we argue that the q->1 limit, in which the field becomes the "field with one element," the theory becomes "classical."

The generalized tangent bundle, the direct sum of the tangent and cotangent bundle of a manifold, is an object of great interest in both differential geometry and physics. Skew-symmetric endomorphisms that square to negative the identity are known as generalized almost complex structures and their integrability is equivalent to the vanishing of the Courant-Nijenhuis tensor. Integrability conditions for more general skew-symmetric endomorphisms are not well-understood. In joint work with Sergio Da Silva and Daniele Grandini, we describe all possible tensorial integrability conditions in terms of a shifted version of the Courant-Nijenhuis tensor.

A classical invariant in Algebraic Geometry is the Picard group of a smooth projective curve C. This group is the quotient of the group of divisors on C (aka the free abelian group on all points of the curve) modulo divisors of rational functions on C. Over an algebraically closed field this group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the Jacobian variety of the curve. In higher dimensions there is an analog of this group, called the Chow group of 0-cycles, for which many things generalize from the case of curves. The main issue that makes the situation much more chaotic though is that the Abel-Jacobi map is in general no longer an isomorphism. In fact, over large fields like the complex numbers and for many important classes of varieties (abelian varieties, K3 surfaces, products of curves of positive genus,...) the kernel of the Abel-Jacobi can be really enormous. On the other extreme a famous conjecture of Beilinson from the mid 80's predicts that over the algebraic closure of the rational numbers the Abel-Jacobi map is an isomorphism unconditionally on the variety.  This conjecture is very hard to establish and in fact, there are no interesting examples in the literature known to satisfy this conjecture. In this talk I will present recent joint work with Jonathan Love, where we make substantial progress on this conjecture for abelian surfaces.  First, we describe a very rich collection of relations in the kernel of the Abel-Jacobi arising from hyperelliptic curves, that is, curves that are given by an equation of the form y^2=f(x). Second, we will give a geometric construction that produces large countable collections of such curves mapping to a given abelian surface.

Between 1990 and 2006, Birman and Menasco wrote a sequence of papers with the running major title “Stabilisation in the braid groups”, that investigated the question of what stabilisation accomplishes in Markov’s theorem. One key result coming out of that sequence of papers was the discovery of a new isotopy -- the exchange move – which also preserves braid index and writhe, but could jump between conjugacy classes. Thus, it was established that using just a sequence of braid isotopies, exchange moves and destabilisations, one could go from any closed braid representative of the unlink to the representative of minimal index – the trivial closed braid. It was also shown that using just braid isotopies and exchange moves, one could take a closed braid representing a split or composite link to one where it was “obviously” split or composite, respectively.

In the setting of plat presentations of links, we now ask the same question: what does stabilisation accomplish? We introduce two new isotopy moves that preserve the bridge index: the pocket move and the flip move, which can be thought of as being analogous to an exchange move of the closed braid setting. Equipped with these isotopies, we prove results analogous to Birman-Menasco’s results, in the plat setting.

A special case of certain diagrammatics, when G is the group of affine transformations of a line, can be related to the infinitesimal dilogarithm and to the Shannon entropy of finite probability distributions. I will explain the diagrammatics and their connections to infinitesimal dilogarithms and entropy.

This is joint work in progress with Mikhail Khovanov.