Support is a fundamental concept to provide a geometric approach for studying all kinds of algebraic structure. Based on Quillen's work, the description of the algebraic varieties corresponding to the cohomology ring of a finite group, Carlson introduced a support variety of any finitely generated module over a finite group $G$ whose dimension is given by the complexity of the module. The support variety of a module is a powerful invariant in the modular representation theory of a finite group, and contains a lot of deep structure information about modular representations of finite groups.

The very active direction in this field is to develop and generalize the theory of support varieties to finite dimensional algebras (due to Snashall and Solberg), and read off homological information on the module itself. This needs a non-trivial assumption which is known as the (Fg) condition.

• (Fg1) There exists a noetherian graded subring $H$ of ${\rm HH}^*(A)$.
• (Fg2) ${\rm Ext^*}_{A}(X, Y)$ is a finitely generated $H$-module for all $X, Y\in A-{\rm mod}$.
In this talk, we will give a new approach to support varieties. As applications, we will show that all skew-gentle algebras satisfy the ({\bf Fg}) condition, and the support variety of any indecomposable module over a skew-gentle algebra is point or a curve. It follows that the complexity of any module over a skew-gentle algebra is at most one. This is a joint work with Steffen Koenig.

In this talk, we give a sufficient condition for a module to satisfy the Mittag-Leffler condition. It is shown that every module in the left orthogonal class of $\mathcal{B}$ %$M\in{^\bot \mathcal{B}}$ is strict $\mathcal{B}$-stationary for any class $\mathcal{B}$ closed under direct sums. As applications, we will talk about some open problems on Gorenstein modules. This talk is a report on joint work with Guocheng Dai.

In this talk, I'll introduce Schur algebra and its Lusztig subalgebra associated to affine flag variety of type $D$. We show that the i-quantum group corresponding to subalgebra of the Lusztig algebra forms quantum symmetric pairs together with $\mathrm{U_q(\widehat{\frak{sl}}_n)}$.We further construct monomial and canonical bases of its idempotented version and prove the positivity properties of the canonical basis with respect to multiplication and the bilinear pairing. This is a join work with Quanyong Chen.

Over its one-hundred year history, the theory of Schur--Weyl duality and its quantum analogue have had and continue to have profound influences in several areas of mathematics such as Lie theory, representation theory, invariant theory, combinatorial theory, and so on. Beilinson--Lusztig--MacPherson (BLM) gave a geometric construction of quantum $\frak{gl}_n$ by exploring further properties coming from the quantum Schur--Weyl duality. In this lecture, we will talk about quantum Schur--Weyl duality, BLM's theory and their connections.

We characterised linear grid categories that have an abelian structure.

The notion of Hecke-Clifford superalgebra is a super analog of Hecke algebra of type A and its representation theory is parallel to that the Hecke algebras in many aspects. In 1992, Olshanski introduced the notion of quantum queer superalgebras as a $q$-deformation of the universal enveloping of the queer Lie superalgebras and established duality between Hecke-Clifford superalgebras and quantum queer superalgebras, as a super analog of Jimbo-Schur-Duality. In this talk, I will discuss some recent progress on the representation theory of these two algebras including a Frobenius character formula for irreducible representations of Hecke-Clifford algebras and a realization of quantum queer superalgebras via Hecke-Clifford superalgebras. This is partially based on a joint with Jie Du, Haixia Gu and Zhenhua Li, respectively.

In this talk we mainly introduce the notion of weak algebraic quantum hypergroups and develop Pontryagin duality unifying the corresponding theory for regular weak Hopf algebra with enough integrals given by Van Daele and the speaker.

Ganter-Kapranov定义了有限群的2-表示，其设定与Balmer关于群表示叠形（stack）的研究一致，均基于纤维范畴（fibred category）和叠形理论。同一框架中的相关工作还有Kashiwara-Schapira的模叠形的Morita等价定理，以及（更加广义的）Asashiba的导出等价黏合定理。我们尝试抓住其共通之处，探讨有限范畴的2-表示理论。

We study the dominant dimension, tilting and cotilting subcatecories for resentations of a quiver on a ring. This talk is based on a joint work with Mohammad Keshavarz and Yefei Ren and another joint work with Mohammad Keshavarz.