Georgia Institute of Technology College of Sciences

The colored Jones polynomial is a quantum knot invariant which can be constructed as a Reshetikhin–Turaev invariant using representations of $U_q(sl_2)$. Khovanov homology categorifies the Jones polynomial and by extension categorifies the representation theory of $sl_2$. Of particular interest is sutured annular Khovanov homology, which admits a structure as an $sl_2$-module. We will discuss a result of Grigsby–Licata–Wehrli that this structure is a representation-theoretic invariant of an annular link. Time permitting, we will discuss some of the structure of this representation, and extend the result to $sl_n$.

Quantum computing is concerned with harnessing the peculiar properties of quantum mechanics, in order to perform information-processing tasks beyond the capabilities of classical computers. Graph states are a family of quantum states, the resources for quantum computers. Graph states exhibit complex forms of quantum entanglement, implying for example that a quantum computer based on graph states is as powerful as any other quantum computer. But, unlike general quantum states, graph states are very easy to describe thanks to their one-to-one correspondence with mathematical graphs. This correspondence implies that many tools from graph theory can be applied to problems in quantum computing.

This talk aims to provide a gentle introduction to graph states, directed toward graph theorists. I will discuss two main applications of graph states, quantum networks and measurement-based quantum computing, and relate these applications to well-known graph-theoretical concepts, in particular vertex-minors. Finally, I will discuss the problem of classifying graph states, and the recent progress achieved through the development of new graph-theoretical tools.

Quiver varieties provide a fundamental bridge between representation theory, enumerative geometry, and physics.  From 3d mirror symmetry, any quiver variety comes with a dual variety known as the Coulomb branch.  A conjecture proposed by Bullimore-Dimofte-Gaiotto-Hilburn-Kim and, independently, Okounkov, asserts that the cohomology of the moduli space of quasimaps to a quiver variety admits a canonical action by the quantized coordinate ring of the dual BFN Coulomb branch.  In this talk, I will report on progress on refining this conjecture and proving it.  The construction relies on a -1 shifted symplectic structure on the moduli space of quasimaps and the theory of cohomological Hall algebras.  Based on work in preparation with Spencer Tamagni.

Moduli spaces are central objects in modern topology and geometry, serving as powerful tools for extracting invariants from underlying manifolds. Gauge theory provides a prolific source of such spaces, utilizing techniques from geometry, analysis, and algebra to probe their structure. In this talk, we survey key gauge-theoretic moduli spaces with an emphasis on how $S^1$-actions can be used to study their topological properties. In particular, we apply these methods to hyperkähler ALE spaces, discussing their applications to the McKay correspondence and symplectic and contact homologies.