Abstract: This is a continuation of the previous talk about skein lasagna modules. We review some features of the Khovanov homology and its Lee deformation. We examine the resulting skein lasagna modules with these two theories as inputs, extract a lasagna version of Rasmussen’s s-invariant, and state some formal properties. We then show that Khovanov/Lee skein lasagna modules and lasagna … Read More

Abstract: Skein lasagna modules are invariants of smooth 4-dimensional manifolds capable of detecting exotic phenomena. Wall-type stabilization problems ask about the behavior of exotic phenomena under various topological operations. In this talk, we will describe the invariants we use and the necessary properties. We describe, with Wall-type external stabilization problems as motivation, a method for computing the Khovanov skein lasagna module of $S^2 … Read More

Abstract: In 2017, Braverman-Finkelberg-Nakajima gave a precise definition of the Coulomb branch of a 3d N = 4 supersymmetric gauge theory of cotangent type associated to a complex reductive group G and a finite dimensional complex representation N. In our first part of this talk, we will recall the definition and basic properties of such Coulomb branches, as well as … Read More

A basic question in classical gauge theory is to describe moduli spaces of solutions to PDEs arising in that context (such as Bogomolny equations on R^3 or anti-self-duality equations on R^4), at least as complex manifolds, in explicit finite-dimensional terms. In the case of Bogomolny equations, a powerful means to do this is provided by the scattering data assigned to … Read More

Instanton scattering matrices constructed in the previous talk give evidence for the existence of an R-matrix (re)construction of quantized Coulomb branch algebras of 3d N=4 quiver gauge theories using the (shifted) Yangian associated to the quiver. In this talk I will report on progress in putting this in a more standard context for geometric construction of Yangian actions, using critical … Read More

Abstract: We introduce a new class of smooth correspondences between Nakajima quiver varieties called split parabolic quiver varieties, and study their properties. We also construct some interesting operators in  equivariant K-theory of these varieties, and relate them to K-theoretic Hall algebra  and double Dyck path algebra. This is a joint work with Nicolle Gonzalez and Jose Simental.

Abstract: I will review the construction of the action of quantized Coulomb branches on critical cohomology of quasimap spaces from the previous talk, emphasizing essential features. Then I will sketch some conjectural features of this action, in particular the compatibility of geometric R-matrices with monopole and instanton scattering matrices. This is based on ongoing joint work with Tommaso Botta.

Abstract: The categorical dimension of the second fundamental representation of \mathfrak{sl}_4 is q^4 + q^2 + 2 + q^{-2} + q^{-4}, which is the graded dimension of a vector space spanned by all partitions fitting in a 2 \times 2 box. The transpose operation preserves that set of partitions, and if instead of counting partitions (the trace of the identity) … Read More

Abstract: Cyclotomic KLRW algebras are known to categorify tensor products of irreducible representations of quantum groups. I will explain how these algebras arise as endomorphism algebras of Lagrangians in additive Coulomb branches. There are generalizations of KLRW algebras that categorify arbitrary tensor products of Verma modules and irreducible representations of quantum groups. I will define these algebras (which were previously … Read More

Abstract: Beilinson’s resolution and full strong exceptional collection of line bundles for the derived category of projective space are incredibly useful computational tools. In recent work with many collaborators, we have shown that much of this structure persists for more general toric varieties. Namely, they admit short explicit resolutions of the diagonal by direct sums of line bundles and their derived … Read More