Peking University

The cohomology of Shimura varieties plays an important role in Langlands program, serving as a link between automorphic forms and Galois representations. In this talk, we prove a vanishing result for the cohomology of Shimura varieties of abelian type with torsion coefficients, generalizing the previous results of Caraiani-Scholze, Koshikawa, Hamann-Lee, and others. Our proofs utilize the unipotent categorical local Langlands correspondence developed by Zhu and the Igusa stacks constructed by Daniels-van Hoften-Kim-Zhang. This is a joint work with Xinwen Zhu.

In recent years, some progress has been made in the development of finite element complexes, particularly in the discretization of BGG complexes in two and three dimensions, including Hessian complexes, elasticity complexes, and divdiv complexes. In this talk, I will discuss distributional complexes in two and three dimensions. These complexes are simply constructed using geometric concepts such as vertices, edges, and faces, and they share the same cohomology as the complexes at the continuous level, which reflects that the discretization is structure preserving. The results can be regarded as a tensor generalization of the Whitney forms of the finite element exterior calculus. This talk is based on joint work with Snorre Christiansen (Oslo), Kaibo Hu (Edinburgh), and Qian Zhang (Michigan).

In this talk, I will review some recent progress on random graph matching problems, that is, to recover the vertex correspondence between a pair of correlated random graphs from the observation of two unlabelled graphs. In this talk, I will touch issues of information threshold, efficient algorithms as well as complexity theory. This is based on joint works with Hang Du, Shuyang Gong and Zhangsong Li.

Deep learning continues to dominate machine learning and has been successful in computer vision, natural language processing, etc. Its impact has now expanded to many research areas in science and engineering. In this talk, I will mainly focus on some recent impacts of deep learning on computational mathematics. I will present our recent work on bridging deep neural networks with numerical differential equations, and how it may guide us in designing new models and algorithms for some scientific computing tasks. On the one hand, I will present some of our works on the design of interpretable data-driven models for system identification and model reduction. On the other hand, I will present our recent attempts at combining wisdom from numerical PDEs and machine learning to design data-driven solvers for PDEs and their applications in electromagnetic simulation.