Titles and Abstracts:

Abstract: This series of five lectures is based on joint papers with Diaconescu, Porta, Schiffmann, and Vasserot, arXiv:2502.19445 and arXiv:2603.03386. The main goal of the course is to explain the relationship between the cohomological Hall algebra (COHA) of 1-dimensional sheaves on the minimal resolution of a Kleinian singularity, the COHA of the preprojective algebra of the corresponding affine ADE quiver, and the corresponding affine Yangian via the derived McKay correspondence. This relationship is established using several key ingredients, including braid group actions on COHAs and Yangians, Bridgeland stability conditions, and the construction of “limiting” COHAs.

A detailed plan of the lecture series is as follows. In the first lecture, I will introduce the construction of 2-dimensional cohomological Hall algebras (COHAs) of quivers and their nilpotent variants in a broad setting. In the second lecture, I will focus on the relationship between the 2-dimensional COHAs of quivers without edge loops and the corresponding multiparameter Yangians, and I will introduce braid group actions on both COHAs and Yangians. The third lecture will be divided into two parts. The first will cover the Lie theory associated with affine ADE quivers, including the various braid groups that arise in this context. I will also introduce affine Yangians, describe their classical limits, and explain their relationship with COHAs. In the second part, I will introduce COHAs of surfaces and their nilpotent variants. The fourth lecture will introduce the derived McKay correspondence and the associated braid group actions. I will then state the main theorem of the course and begin discussing Bridgeland stability conditions and limiting COHAs. The fifth and final lecture will be devoted to the proof of the main theorem.

Abstract: 3d N=4 gauge theories give rise to a pair of dual varieties known as the Higgs and Coulomb branches, and 3D mirror symmetry concerns a strong relationship between curve counts for these dual targets. A preliminary step toward a mathematical understanding of this statement is the appropriate construction of the corresponding curve-counting problems. The theory of bow varieties, introduced by Nakajima and Takayama, provides a GIT presentation of a large class of dual targets corresponding to affine type A quiver gauge theories, and hence a natural setting in which quasimap theory can be used to define curve counts and rigorously test the mirror symmetry conjecture. In this mini-course, I will review several aspects of mirror symmetry using the language of bow varieties, including the duality of vertex functions, difference equations, and elliptic interfaces. I will discuss the general proof in finite type A and conclude with some speculation toward a general argument. Additionally, I will address the problem of extending this conjecture to counts with nontrivial descendant insertions. Specifically, I will present a conjecture identifying vertex functions with descendants on one side with vertex functions for Hecke-modified quasimaps on the dual side. Based on joint work, including work in progress, with subsets of Dinkins, Rimányi, and Tamagni.