Titles and Abstracts:

Abstract: This series of lectures is based on my papers arXiv: 2603.03386 and arXiv: 2511.08576. In the first part, I will introduce 2-dimensional cohomological Hall algebras of quivers and explain their relations to Yangians. In the second part, I will introduce cohomological Hall algebras associated with smooth surfaces, focusing in particular on the case where the surface is a minimal resolution of an ADE singularity. I will then discuss the relation between these algebras and the Bridgeland space of stability conditions on such resolutions.

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.