Georgia Institute of Technology College of Sciences

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.

This talk provides an elementary introduction to skein algebras of surfaces, which serve as quantizations of $SL_n$-character varieties. For surfaces with boundary, we extend this framework to stated skein algebras, demonstrating how they provide simple and transparent geometric interpretations of various quantum group structures.

Specifically, we present a geometric realization of the dual canonical basis of $\mathscr{O}_q(\mathfrak{sl}_n)$ using skeins for $n=2$ and $n=3$. If time permits, we will also show how the skein algebra framework can be used to recover the Shapiro–Schrader embedding of the quantized enveloping algebra into a quantum torus algebra.

The moduli space of Higgs bundles lies at the crossroads of different areas of mathematics. Its cohomology plays a central role in Ngo's proof of the fundamental lemma of the Langlands program, and it is the subject of recent results such as topological mirror symmetry and the P=W conjecture. Even though these developments seem unrelated, they all ultimately rely on a (partial) understanding of the Decomposition Theorem for the associated Hitchin fibration. In this talk, I will report on a complete and uniform description of the Decomposition Theorem in the logarithmic case, fully generalizing Ngo's results beyond the elliptic locus. This is joint work in progress with Mark de Cataldo, Roberto Fringuelli, and Mirko Mauri.