【Seminar 1: 9:00-10:00, 18 October, 2024】

Speaker: Yanpeng Li (Sichuan University)

Title: POLYNOMIAL INTEGRABLE SYSTEMS FROM CLUSTER STRUCTURES

Abstract: We present a general framework for constructing polynomial integrable systems with respect to linearizations of Poisson varieties that admit log-canonical coordinate systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures. As special cases, we consider an arbitrary standard complex semi-simple Poisson Lie group $G$ with the Berenstein-Fomin-Zelevinsky cluster structure, nilpotent Lie subgroups of $G$, identified with Schubert cells in the flag variety of $G$ with the standard cluster structure, and the dual Poisson Lie group of $\GL(n, \CC)$ with the Gekhtman-Shapiro-Vainshtein generalized cluster structure. In each of the three cases, we show that every extended cluster in the respective cluster structure gives rise to a polynomial integrable system on the respective Lie algebra with respect to the linearization of the Poisson structure at the identity element. TBased on joint work with Jang-Hua Lu and Yu Li.

【Seminar 2: 10:00-11:00, 18 October, 2024】

Speaker: Shizhuo Yu (Nankai University)

Title: Bott-Samelson atlas and Lusztig's total positivity on a flag variety

Abstract: The Bott-Samelson atlas is an atlas on a flag variety constructed via Kazhdan-Lusztig maps. When equipping with the standard Poissin structure, the Bott-Samelson atlas makes a flag variety covered by of symmetric CGL extensions. Moreover, all shifted big cells can be realized as 'cut' of the symmetric CGL extensions，which induce the  Lusztig's total positivity simultaneously but different cluster structures separately.  In particular, each coordinate function inside the Bott-Samelson atlas is positive. This is a joint work with Jiang-Hua Lu.

【Seminar 3: 11:00-12:00, 18 October, 2024】

Speaker: Chenchang Zhu (University of Goettingen)

Title: iCFO of Higher Derived Groupoids and Shifted Symplectic Structures

Abstract: In this talk, based on many previous works, we will introduce a helpful new tool for differential geometers using:

• higher: to deal with quotient singularities

• derived: to heal transversality problems

• shifted: for more flexible symplectic situations.

At the same time as being as complete as possible, we also make the framework as explicit as possible using groupoid models. We build an incomplete category of fibrant objects (iCFO) for higher derived Lie groupoids. In the end, we will explain in some concrete examples how this theory is used. This is based on a joint work in progress with Miquel Cueca Ten, Florian Dorsch and Reyer Sjamaar.