Organizers: Lin Chen, Will Donovan, Penghui Li, Peng Shan, Changjian Su, Wenbin Yan

Time:  3:30-4:30 pm, Monday

Place: B627, Shuangqing Complex Building（双清综合楼B627教室), Yau Mathematical Sciences Center,  Tsinghua University

双清综合楼地址：北京市海淀区逸清南路西延6号院1号，双清公寓马路对面、清华附小（双清校区）西侧。

official seminar webpage,  which contains the zoom information for online talks

03/02 Shiyixin Liang 梁石易新 (清华求真书院) 

A coherent version of geometric Satake equivalence for type A

The celebrated geometric Satake equivalence establishes an equivalence between the perverse sheaves category on the affine Grassmannian and the representation category of the dual group. In a series of works, S. Cautis and H. Williams established a framework on studying an abelian Coulomb branch category, and a particular case concerns about coherent sheaves on the (spherical) affine Steinberg variety. They conjectured a coherent version of geometric Satake for this setting. In this talk, I will discuss my work on this conjecture in the case of type A.

• 03/09 Rahul Singh (YMSC)

q-Opers and Bethe Ansatz for Open Spin Chains

Opers and their q-deformations provide a geometric way to encode solutions of the Bethe Ansatz equations arising in quantum integrable systems such as Gaudin models and closed spin chains. In this talk, we discuss the extension of this picture to open spin chains, which leads to a class of q-opers satisfying a reflection-invariance condition. We begin with a brief introduction to spin chains and then explain how this class of q-opers can be described using certain QQ-systems and how their nondegenerate solutions reproduce the Bethe Ansatz equations for open XXZ spin chains. This is a joint work with Peter Koroteev and Myungbo Shim.

• 03/16 Hiraku Nakajima (Kavli IPMU)

Involutions on quiver varieties and twisted Yangian 

Take smooth quiver varieties of finite ADE type, and consider involutions on them defined as composite of transpose of linear maps, reflection functors for the longest element, and a diagram automorphism. After Yiqiang Li, we consider stable envelopes and K-matrices which satisfy reflection equations, defined on the fixed point loci of involutions. They will give twisted Yangian by a variant of RTT construction. I will explain this construction, and give a few examples.

• 03/23 Cédric Bonnafé (IMAG, Université de Montpellier)

Exceptional varieties built from exceptional reflection groups

There are many ways for a variety to be "exceptional", but we focus in this talk on some old problems in classical algebraic geometry: finding curves and surfaces with many singular points, with big automorphism groups, finding surfaces with many lines or conics, finding K3 surfaces with big Picard numbers... We present a series of examples, all built from invariants of exceptional complex reflection groups, which are up to now the records in their own categories.

• 03/27 2pm-3pm, Jianrong Li (University of Vienna)

Tropical symmetries of cluster algebras

In this talk, I will present joint work with James Drummond and Ömer Gürdoğan on tropicalizations of quasi-automorphisms of cluster algebras. We study tropicalizations of quasi-automorphisms of cluster algebras and show that their induced action on g-vectors can be realized by tropicalizing their action on the homogeneous X-variables (\hat{y}-variables) of a chosen initial cluster. This perspective allows us to interpret the action on g-vectors as a change of coordinates in the tropical setting. Focusing on Grassmannian cluster algebras, we analyze tropicalizations of quasi-automorphisms in detail. In particular, we derive tropical analogues of the braid group action and the twist map, both on g-vectors and on tableaux. We also introduce the notions of unstable and stable fixed points for quasi-automorphisms. As an application, we show that the number of prime non-real tableaux with a fixed number of columns in SSYT(3, [9]) and SSYT(4, [8]), arising from the braid group action on stable fixed points, is governed by Euler’s totient function. Furthermore, we apply our results to scattering amplitudes in physics, providing a new interpretation of the square root appearing in the four-mass box integral in terms of stable fixed points of quasi-automorphisms of the Grassmannian cluster algebra C[Gr(4, 8)].

• 04/03 1:30pm-3:00pm, Andrei Okounkov (Columbia)

      Room C548, Shuangqing Complex Building

Quantum critical cohomology

This will be a report on joint work with Yalong Cao, Yehao Zhou, and Zijun Zhou. As a part of our recent work on critical stable envelopes, we are able to determine the quantum critical cohomology for symmetric quivers with potential. Moreover, a certain asymmetry of the framing of the quiver is allowed, making geometries like the Hilbert scheme of points in the affine 3-space an example of our general theory.

• 04/13  

• 04/20 Chi-Ming Chang (YMSC)

• 04/27 Marc Besson (YMSC)

• 05/11

• 05/18 Chin Hang Eddie Lam (The Chinese University of Hong Kong)

• 05/25 Yalong Cao 曹亚龙 (MCM）

Stable envelopes for critical loci 

Stable envelopes are geometric objects introduced by Maulik-Okounkov on cohomology of Nakajima quiver varieties. They can be used to construct R-matrices and Yangian type quantum groups. 

In a joint work with Andrei Okounkov, Yehao Zhou and Zijun Zhou. We extend the theory of stable envelopes to critical cohomology of symmetric quiver varieties with potentials. We prove that critical stable envelopes are compatible with dimensional reductions, specializations, Hall products, and other geometric constructions. 

Applications to enumerative geometry of critical loci have been elaborated in the talk of Andrei in this seminar series. Applications to geometric representation theory of shifted quantum groups will be explained more in Yehao’s talk next week.

• 06/01 Yehao Zhou 周业浩 (SIMIS)

• 06/08

• 06/15

• 06/22

• ...

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