Representation theory of finite groups started and developed by Ferdinand Georg Frobenius from1896, and contributions were also made by Dedekind, Burnside, Schur, Noether, and others. Foundations of the modular representation theory were laid out almost singlehandedly by Richard Brauer, started in 1935 and continued over the next few decades. Today the Representation Theory of Finite Groups is a thriving subject, with many fascinating and deep open problems, and some recent great successes. In 1963 Richard Brauer formulated a list of deep conjectures and problems about ordinary and modular representations of finite groups. These have lead to many new concepts and methods, but basically all of his main conjectures are still unsolved to the present day. We discuss some basic problems and current approaches and recent progress on some of these problems.

In the late 1990s, E. E. Enochs asked whether each covering class of modules is closed under direct limits. This problem is still open in general. In this talk, a brief introduction to Enochs conjecture will be given and some results on this subject will be reviewed and discussed.

Key Words: Covering class; Direct limit; Coherent ring; Absolutely pure module.

In the seminal work, Beilinson-Lusztig-MacPherson gave a beautiful realization for quantum $\mathfrak{gl}_n$ via a geometric setting of quantum Schur algebras. We will talk about BLM realization of quantum affine $\mathfrak{gl}_n$ and its relation with affine quantum Schur-Weyl duality. Furthermore, we will talk about some applications of BLM realization of quantum affine $\mathfrak{gl}_n$ . This talk is mainly based on joint works with Bangming Deng and Jie Du.

The notion of weighted projective lines is introduced by Geigle-Lenzing in 1987 which gives a geometric treatment to the representation theory of the canonical algebras in the sense of Ringel. Due to Geigle-Lenzing'87, Chen-Chen-Zhou'15, etc., the category of coherent sheaves over a weighted projective line of tubular type is equivalent to the category of equivariant coherent sheaves over some elliptic curve with respect to a certain cyclic group action. Recently, J. Chen, X. Chen and S. Ruan showed that the categories of coherent sheaves over weighted projective lines of tubular type can be related to each other via the equivariantization with respect to certain finite group actions. In particular, the notion of admissible homomorphisms between the string groups of weighted projective lines is introduced which plays an important role to find the group and determined the explicit group action. In this talk, I will introduce the joint work with J. Chen, S. Ruan and H. Zhang which shows that how to relate the weighted projective lines via equivariantization by using admissible homomorphisms. As applications, we classify all the coherent sheaves categories over the weighted projective lines of tubular type and of domestic type in the sense of admissible homomorphisms.

Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $\mathbb{Z}Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. We introduce the $n$-translation quiver $\mathbb{Z}|_{n-1} Q$ as a generalization of the $\mathbb{Z} Q$ construction in studying $n$-translation algebras. We find a class of algebras $\Gamma$ of global dimension $n$, called $n$-slice algebras, for which the $\tau_n$-closure of its dual and $\tau_n^{-1}$-closure of it can be truncated from $\mathbb{Z}|_{n-1} Q$, as the preinjective and preprojective components do from $\mathbb{Z} Q$ for the hereditary algebra.

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $\Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $\textrm{Diff}_q(\Omega_q)$ defined over $\Omega_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $\mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $\Omega_q(m|n)$. An interested point of this approach here is that even though $\mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This leads to one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $\Omega_q$ is made into the $\mathcal U_q(\mathfrak g)$-module (super)algebra structure, $\Omega_q=\Omega_q(m|n)$ for $q$ generic, or $\Omega_q(m|n, \mathbf{1})$ for $q$ root of unity, and $\mathfrak g=\mathfrak{gl}(m|n)$ or $\mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $\mathcal U_q(\mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $\Omega_q^!$ as $\mathcal U_q(\mathfrak g)$-module algebra. In this paper, some examples of pointed Hopf algebras arise from the QDOs, whose idea is an expansion of the spirit noted by Manin. Actually, more attention on a suitable QDO approach is deserved to be paid. This is a joint work with Ge Feng, Meirong Zhang and Xiaoting Zhang.

In this talk, we study the derived representations of Calabi-Yau algebras, and show that there is a shifted symplectic structure on their derived representation schemes. Applications to Calabi-Yau categories and noncommutative crepant resolutions of low dimensional singularities will also be discussed. This talk is based on a joint work with Eshmatov.

A characterization will be given of subgroups of alternating groups of odd index, which particularly shows that except for $n=7,8$ or 9, all non-solvable composition factors of each subgroup of $\mathrm{Alt}(n)$ of odd index are alternating. Then an application is given of a classification of a family of 2-arc-transitive graphs of odd order.

An $(n,k,t,\lambda,\mu)$-directed strongly regular graph is a directed graph with $n$ vertices satisfying (i) each vertex has $k$ out-neighbors and $k$ in-neighbors, including $t$ neighbors counted as both in- and out-neighbors of the vertex; and (ii) the number of paths of length two from a vertex $x$ to another vertex $y$ is $\lambda$ if there is a directed edge from $x$ to $y$, and is $\mu$ otherwise. Such graphs were introduced by Duval in 1988 as one of the possible generalization of classical strongly regular graphs to the directed case. Cayley graphs on dihedral groups are called dihedrants. In this talk, a class of directed strongly regular dihedrants will be characterized.

Let $G$ be a finite abelian group and $\mathbb{F}$ a finite field of characteristic $p$. We give an algorithm to write down a set of generators of $K_2(\mathbb{F}[G])$ via a simple presentation of it and obtain some formulae for counting these generators. Using Witt decomposition, we also determine the group structure of $K_3(\mathbb{F}[C_{p^n}])$. we also determine the $K_2$-group of the group algebra $\mathbb{F}[G\times T^q]$ where $ T^q$ is any finitely generated free abelian group of rank $q$. We give a basis of the Bass Nil-groups $N^mK_1(\mathbb{F}[G])$ and $N^mK_2(\mathbb{F}[G])$, respectively. We also prove that $N^mK_1(\mathbb{F}[G])$ is a finitely generated $\mathbb{Z}[\mathbb{N}_+^m]$-module, but $N^mK_2(\mathbb{F}[G])$ is not.This talk is based on joint works with Hao Zhang and Hang Liu.