Peking University

Neurons interact via spikes, which is a pulse-like, discontinuous mechanism. Their mean-field PDE description gives Fokker-Planck equations with novel nonlinearities. From a probability point of view, these give rise to Mckean-Vlasov equations involving hitting times. Similar mechanisms also arise in models for systemic risk in mathematical finance, and the supercooled Stefan problem. In this talk, we will first present models for spiking neurons: both microscopic particle models and macroscopic PDE models, with an emphasis on the general mathematical structure. A central question for these equations is the finite-time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuronal network. We will discuss how to continue the solution physically after the blow-up, by introducing a new timescale. The new timescale also helps us to understand the long term behavior of the equation, as it reveals a hidden contraction structure in the hyperbolic case. Finally, we will present a recently developed numerical solver based on this framework. Numerical tests show that during the synchronization the standard microscopic solver suffers from a rather demanding time step requirement, while our macro-mesoscopic solver does not.

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.