FORTY YRS ALGEBRAIC GROUP, ALGEBRAIC GEOMETRY & REPRESENT .‪.‬

Professor Xihua Cao (1920–2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned 'ECNU School' in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad. Contents: Dedications and Reminiscences: Professor Xihua Cao and His School Dedications Reminiscences of Family Members Research Papers: A Particular Class of Properly Stratified Algebras (Xueqing Chen and Vlastimil Dlab) Quantized Cohomological Hall Algebra of the d-Loop Quiver Revisited (Neil J Y Fan, Changjian Fu and Liangang Peng) Generating Algebraic Groups by Small Subgroups and Equivariant Representations (Cuipo Jiang and Zongzhu Lin) On Quadri-Bialgebras (Xiang Ni and Chengming Bai) Partial Orderings on Affine Weyl Groups (Jianpan Wang) Surveys and Overviews: On Toroidal Vertex Algebras (Fulin Chen, Shaobin Tan, Qing Wang and Nina Yu) A Survey on Infinite-Dimensional Representations of Reductive Algebraic Groups with Frobenius Maps (Xiaoyu Chen, Junbin Dong and Nanhua Xi) Inductive Conditions of Alperin Weight Conjecture (Zhicheng Feng and Jiping Zhang) Regular Representations of Quantum Supergroups Uv(glm|n) and Uv (qn) (Haixia Gu and Zhongguo Zhou) BGG Category θ and ℤ-Graded Representation Theory (Jun Hu) Moduli Spaces of Vector Bundles on a Nodal Curve (Wenyao Hu and Xiaotao Sun) Automorphic Descents (Dihua Jiang) The Mazur Conjecture, the Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture (Hourong Qin) A Survey on the Cell Theory of Weighted Coxeter Groups (Jianyi Shi) On Bases of Quantum Affine Algebras (Jie Xiao, Han Xu and Minghui Zhao) Lecture Notes: Lectures on Dualities ABC in Representation Theory (Li Luo and Weiqiang Wang) Appendix: A Brief Biography of Xihua Cao Readership: Researchers and postgraduate students interested in algebras, especially in algebraic groups, quantum groups, Lie algebras, algebraic geometry and algebraic number theory; General readers of mathematics and mathematical history. Key Features: Vivid and enlightening depictions in the appendix and photo collection of Professor Xihua Cao's tortuous life and his touching lifelong pursuit of mathematics Virtuous descriptions in scholar dedications of the Chinese scholars' diligent pursuit and hard work in scientific development since the reform and opening-up of China, which not only have historical value, but also have a good educational effect on young generations 16 academic articles reflecting the academic achievements of Chinese scholars at home and abroad, from noted and esteemed authors, including two academicians of the Chinese Academy of Sciences (Nanhua Xi from Chinese Academy of Sciences and Jiping Zhang from Peking University) and a winner of the American Mathematical Society Chevalley Prize in Lie Theory (Weiqiang Wang, who won Chevalley Prize together with Huanchen Bao, both from the University of Virginia)