I will explain how the self-duality of equivariant K-theoretic DT/PT counts of local threefold curves implied by 3d Mirror symmetry is related to a conjectural correspondence between the sheaf counts and GW invariants of fivefolds. We will see that for fibrewise anti-diagonal torus actions on the local curve both sides of the correspondence are explicitly computable and indeed match. Regarding more general group actions I will explain the current obstructions towards establishing the correspondence. The talk is based on joint work in progress with Monavari and with Giacchetto and Pandharipande.

Bow varieties, introduced by Nakajima and Takayama, provide a natural framework for studying mirror symmetry in affine type A. In this talk, I will review several aspects of mirror symmetry using the language of bow varieties, including duality of vertex functions, difference equations, and elliptic interfaces. I will end with some speculations toward a general proof. Based on joint work, including work in progress, with Rimányi and Dinkins.

I will survey recent results describing certain moduli of quasimaps from P^1 to Nakajima varieties as critical loci of explicit functions, and application of these results to geometric construction of modules for quantized Coulomb branches and quantum groups. If time permits I will explain some applications to 3d mirror symmetry of quasimap counts. This is based in part on joint work with Tommaso Botta.

In this talk, I review the quantum differential equation for the Hilbert scheme of points in the complex plane and its K-theoretic generalization. Next, I discuss a new description of these equations arising from the 3D self-mirror symmetry of the Hilbert scheme.

Elliptic stable envelope is introduced by Aganagic–Okounkov as a generalization of stable envelopes in equivariant elliptic cohomology. One of their important properties is that they satisfy the 3d mirror symmetry — elliptic stable envelopes of a 3d mirror pair are transpose to each other. Further observation indicates that those classes may arise from a common class, called duality interface, on the product of the mirror pair. In this talk, I will discuss the existence of duality interface in terms of an inductive construction. This is based on joint work in progress with H. Nakajima and A. Okounkov.

3d mirror symmetry asserts that quasimap counts to dual varieties are the same after the appropriate identifications. In this talk, I will address the problem of extending this picture 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. Joint work in progress with Dinkins and Tamagni.

I will present a description of certain moduli spaces of quasimaps from higher genus curves to Nakajima varieties, as critical loci of another explicit function, which is strongly suggestive of a certain local-to-global principle governing quasimap counts. Then I’ll outline some local-to-global conjectures and speculate on how the above results could be applied to proving them.