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Bridging affine vertex algebras and affine Springer fibres

I would like to explain a program which aims to build relationships between affine Springer theory and representations of simple affine vertex algebras. We will discuss some explicit correspondence for admissible levels, and formulate some new conjectures about representations of simple affine vertex algebras at integer nonadmissible levels and their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan and Qixian Zhao.

Donaldson-Thomas invariants of \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\)

In this talk I will review the DT4 theory and talk about the computations of zero-dimensional DT invariants of the quotient stack \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\), confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for \( \mathbf{A}_{r-1}×\mathbb{C}^2\) and an explicit determination of orientations of Hilbert schemes of points on \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\).