## Geometric Representation Seminar

## 报告人简介

**2019-11-15**

**Speaker:** 李鹏辉 Penghui Li(YMSC)

**Title:** Whittaker
sheaf, commuting scheme and Geometric Langlands conjecture

**Abstract:** It is an long standing open problem that whether
the scheme of commuting matrices is reduced. We suggest a way to tackle this
problem via Langlands duality. In the talk, we briefly recall the definition of
commuting scheme, and how it is related to Ben-Zvi--Nadler's Betti Geometric
Langlands (BGL) conjecture. Then we summarize recent progresses on BGL
conjecture, and sketch a proof of reduceness of invariant function on commuting
schemes, based on the conjecture in genus 1. This work is based on a joint work
with David Nadler.

** **

**2019-11-1**

**Speaker:** 王彬 Bin Wang (YMSC)

**Title:** Parabolic Hitchin Maps and Their Generic Fibers: GL_{n}-Case

**Abstract: **In this
talk, we first recall Hitchin maps and Beauville-Narashimhan-Ramanan's
correspondence. Then in GL_{n} case, we talk about a parabolic analogue of
Beauville-Narashimhan-Ramanan's correspondence which in particular implies that
generic fibers of parabolic Hitchin maps are still Picard varieties. We will
also calculate the dimension of global nilpotent cone in this case. This is a
joint work with Xiaoyu Su and Xueqing Wen.

** **

**2019-10-25**

**Speaker:** Michael Ehrig (Beijing Institut of Technology)

**Title:** Lie Superalgebras via Schur-Weyl duality and Categorification

**Abstract: **In this
talk, I will outline an approach to understand and describe the category of
finite dimensional representations of a classical Lie superalgebra. Due to the
non semi simplicity, methods different from the ones for semi-simple Lie
algebras need to be applied to describe this category. Using variations of
Schur-Weyl duality, respectively the fundamental theorems of invariant theory,
we formulate the problem of understanding it in terms of centralizer algebras.
These centralizer algebras are then described via methods from categorification
of quantum groups and link invariants, yielding the description of the category
of finite dimensional representations for some of the classical Lie
superalgebras. This is joint work with Catharina Stroppel.

** **

**2019-10-11**

**Speaker:** Matthew Young (MPI Bonn)

**Title:** Twisted loop transgression and categorical character theory

**Abstract: **This
talk will be an introduction to the Real categorical representation theory of a
finite 2-group, in which a graded group acts by autoequivalences or
anti-autoequivalences of a category. In particular, I will discuss the
geometric character theory of such representations and its formulation in terms
of unoriented mapping stacks. Time permitting, I'll discuss applications to
topological field theory and monoidal categories.

** **

**2019-6-6**

**Speaker:** 苏长剑 Su Changjian(University of Toronto)

**Title:** Categorification of K-theory stable bases of the Springer resolution

**Abstract: **The
K-theoretic Maulik—Okounkov stable basis depends on the choice of an alcove. In
this talk, we compare the stable bases of the Springer resolution associated to
different alcoves. We prove that the change of alcoves operators are given by
the Demazure—Lusztig operators in the affine Hecke algebra. We then show that
these bases are categorified by the Verma modules of the Lie algebra, under the localization of
Lie algebras in positive characteristic of Bezrukavnikov, Mirkovic and Rumynin.
Joint work with Gufang Zhao and Changlong Zhong.

** **

**2019-5-24**

**Speaker:** 覃帆 Qin Fan (Shanghai Jiaotong University)

**Title:** Bases for upper cluster algebras and tropical points

**Abstract: **It is
known that many (upper) cluster algebras possess very different good bases
which are parametrized by the tropical points of Langlands dual cluster
varieties. For any given injective reachable upper cluster algebra, we describe
all of its bases parametrized by the tropical points. In addition, we obtain
the existence of the generic bases for such upper cluster algebras. Our results
apply to many cluster algebras arising from representation theory and higher
Teichmuller theory.

** **

**2019-5-17**

**Speaker:** Gus Lehrer
(University of Sydney)

**Title:** Tangle categories and extension of the Temperley-Lieb category
equivalence for quantum $\mathfrak{sl}_2$

**Abstract: **Let
$U_q$ be the quantum group of
$\mathfrak{sl}_2$. It is classically known that there is an equivalence between the category of representations of
the form $V^r:=V\otimes V\otimes...\otimes V$ of $U_q$, where $V$ is the
2-dimensional simple representation, and the Temperley-Lieb category $TL(q)$,
which is described in terms of diagrams. I shall describe an extension of this
equivalence to the Temperley-Lieb category $TLB(q,Q)$. The corresponding
representation category consists of certain infinite dimensional
representations of $U_q$. This is joint work with Ruibin Zhang and Kenji
Iohara.

** **

**2019-4-26**

**Speaker:** 卢明Lu
Ming (Sichuan University)

**Title: **Hall
Algebras and i-Quantum groups

**Abstract: **A
quantum symmetric pair consists of a quantum group and its coideal subalgebra
(called an i-quantum group). A quantum group can be viewed as an example of i-quantum
groups associated to symmetric pairs of diagonal type. In this talk, we present
a new Hall algebra construction of i-quantum groups. This relies on the
framework of modified Ringel-Hall algebras defined with Liangang Peng. Our
approach leads to monomial bases, PBW bases, and braid group actions for
i-quantum groups. In case of symmetric pairs of diagonal type, our work reduces
to a reformulation of Bridgeland’s Hall algebra realization of a quantum group,
which in turn was a generalization of earlier constructions of Ringel and
Lusztig for half a quantum group. This is joint work with Weiqiang Wang.

** **

**2019-4-19**

**Speaker:** 华诤 Hua Zheng (Hong Kong University)

**Title: **On
quivers with analytic potentials

**Abstract: **Given a
finite quiver, an element of the complete path algebra over the field of
complex number is called analytic if its coefficients are bounded by a
geometric series.

We may develop a parallel construction of Jacobi algebra and Ginzburg algebra for a quiver with an analytic potential. Analytic potential occurs naturally in the deformation theory of sheaves on projective Calabi-Yau manifold. It plays a central role in the construction of critical cohomological Hall algebra. It turns out that analytic potentials admit much richer structures in noncommutative differential calculus compared with the formal ones. I will give a brief introduction to some of my recent work on this topic.

** **

**2019-3-22**

**Speaker:** Gwyn Bellamy (University of Glasgow)

**Title: **Resolutions of symplectic quotient singularities

**Abstract: **In this talk I will explain how one can explicitly construct
all crepant resolutions of the symplectic quotient singularities associated to
wreath product groups. The resolutions are all given by Nakajima quiver
varieties. In order to prove that all resolutions are obtained this way, one
needs to describe what happens to the geometry as one crosses the walls inside
the GIT parameter space for these quiver varieties. This is based on joint work
with Alistair Craw.

** **

**2019-3-8**

**Speaker: **颜文斌 Yan Wenbin (YMSC)

**Title:** From S^1-fixed points to admissible representations

**Abstract: **I will present some observations on the relation between fixed
points on the moduli space of Hitchin system and admissible representations of
affine Kac-Moody algebra. I will mainly explain the one to one correspondence
between fixed points and admissible representations through examples and
extract a general statement. Other consequences will also be discussed.