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Parabolic AGT correspondence from critical stable envelopes

Negut formulated and proved an parabolic AGT correspondence, namely, certain q-deformed W-algebra acts on the equivariant K-theory of the moduli space of parabolic sheaves on \(P^1\times P^1\) (a.k.a. affine Laumon space). The cohomological version remains open. In this talk I will outline a proof of the cohomological version of parabolic AGT in the rectangular case. A key ingredient in the proof is the critical stable envelope developed in my recent joint work with Yalong Cao, Andrei Okounkov, and Zijun Zhou.

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K-theoretic DT/PT counts for local curves in 3-folds are expected to exhibit a very nontrivial and powerful duality as a consequence of the conjectural 3-dimensional mirror symmetry for the Hilbert schemes of points in the plane, as well as from their conjectural relations to counts of membranes in 5-folds. Talks by Rahul Pandharipande, Andrey Smirnov, Yannik Schuler, Zijun Zhou, Tommaso Botta, Andrei Okounkov, and others will present various approaches to these conjectures and related topics. \(\Rightarrow\)website \(\Rightarrow\)poster

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 \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\).