For a Lagrangian submanifold in a CY3, Ekholm and Shende defined a wavefunction living in the HOMFLY-PT skein module of the Lagrangian, which encodes open Gromov-Witten invariants in all genus. In this talk, we study a skein-valued cluster theory that generalizes quantum cluster theory and allows us to compute these wavefunctions in a range of examples. Our results agree with the physical prediction known as the topological vertex. Along the way we introduce a skein dilogarithm and prove a pentagon relation, generalizing previously known forms of the pentagon identity. This talk is based on joint works with Schrader, Zaslow, and Shende.

We will discuss  the ergodicity of anosov diffeomorphisms and the geodesic flow in negative curvature, talk about pathological foliations and Fubini’s nightmare. 

In this talk, I will present the stabilization phenomenon of cohomology groups and Kac polynomials associated with moduli spaces of quiver representations. Specifically, for Q and chosen dimension vectors d and e satisfying reasonable conditions, the cohomologies of various types of quiver varieties associated with dimension vectors d+ne stabilize as e tends to infinity. I will provide explicit generating functions for these stabilized dimensions and explain the implications for root multiplicities of Kac-Moody Lie algebras.

Schwartz’s pentagram map is a dynamical system defined on moduli spaces of polygons by intersecting diagonals. It is an integrable system, meaning that in appropriate coordinates, the map becomes a family of translations on complex tori. Some natural generalizations of the pentagram map produce integrable systems, but numerical experiments by Khesin-Soloviev suggest that others do not. In this talk, we use tools from dynamical systems to prove that the “skew” pentagram map is non-integrable.

We will examine the multiplicative structure of two skein algebras—the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang (generated by loops and arcs).  In joint work with Chloe Marple, we found an unexpected homomorphism between the usual skein algebra for a closed torus and the Roger-Yang skein algebra for a twice-punctured annulus.  In this talk, I’ll discuss how we  used the homomorphism to help compute representations and structural constants of the Roger-Yang skein algebra for a twice-punctured annulus, and  whether there might be similar relationships between skein algebras for other surfaces. 

I’ll discuss how to leverage symbolic dynamics to understand Anosov diffeomorphisms of surfaces, starting with the most basic case of linear total automorphisms. The key idea is the construction of so-called Markov partitions that code the dynamics of the diffeomorphism. 

More recently, the same ideas have been extended to prove new results about general diffeomorphisms of manifolds. 

I will discuss joint work with Terry Song on the calculation of the virtual Hodge numbers (i.e. Hodge—Deligne polynomial, Hodge-Euler characteristic, etc.) of the moduli space of degree-d maps to projective space from smooth n-pointed curves of genus g. Up to Brill—Noether loci in genus g>= 3, I will show how to reduce the calculation to the corresponding invariants of M_g,n. This reduction implies a strong stability statement for the virtual Hodge numbers as functions of the degree d, and suggests homological stability properties of the moduli space generalizing known statements in genus zero.  As an intermediate result, I will outline an analogous calculation for the universal Jacobian over M_g,n. As I will discuss, the theory of symmetric functions is fundamental to our approach.

I will talk about twisted traces on quantized Coulomb branches. Any Verma module over a quantized Coulomb branch gives rise to a twisted trace. For conical Coulomb branches there is also a trace introduced by Gaiotto and Okazaki (“sphere trace”). I will show how the sphere trace allows us to compute the graded character of Verma modules in certain cases. Based on joint work in progress with Vasily Krylov.

Finite-N effects in large-N gauge theories, such as trace relations, are expected to be holographically dual to non-perturbative phenomena in string theory, such as Giant Graviton branes. A convenient setting to study these effects are supersymmetric indices of U(N) gauge theories. The finite-N indices can be reproduced by a series of corrections to the infinite-N result, known as the Giant Graviton expansion.

In this talk I will present a generalization of the Molien-Weyl formula computing generating functions of invariants of supergroups U(N|M), which arise as gauge groups of brane/negative brane systems in string theory. The formula leads to a new expansion relating finite-N and infinite-N indices of U(N) gauge theories. I will comment on its relation to Murthy’s Giant Graviton expansion and suggest a physical interpretation in terms of branes and negative branes. This talk is based on arXiv:2509.20451 and work in progress with Davide Gaiotto.

We will discuss the dynamics and ergodicity of Anosov diffeomorphisms and related systems.