Date/Time : 22 Apr 2024 13:30
Venue : S17 #04-06 (SR1)
Speaker : Kevin Buzzard, Imperial College London

Computer programs which understand the axioms of mathematics have existed for decades. However, it is only in the last few years that they have captured the imagination of mathematicians. In this talk, I will give an overview of where we are in 2024, what these systems can currently do, and what they cannot yet do. I will discuss how these systems have helped mathematicians, and ways in which they may help mathematicians in the future. I will be careful to distinguish facts from science fiction. The talk will be suitable for a general mathematical audience and will not assume any expertise in AI, computer science, or interactive theorem provers.

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Date/Time : 03 Apr 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Tomoyuki Arakawa, Kyoto University

The 4D/2D duality is a physical theory discovered rather recently by Beem et al, which gives a vertex algebra to any 4D N=2 superconformal field theory. The resulting vertex algebra is an important invariant of a 4-dimensional theory, which may be regarded as a refinement of the Higgs branch.

In this talk I will discuss the representation theory of vertex algebras arising from 4-dimensional theories.

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Date/Time : 14 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhiming Chen, Chinese Academy of Sciences

We consider high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems, which release the work of body-fitted mesh generation and allow us to design adaptive finite element methods for solving curved geometric singularities. We introduce new concepts of the large element and interface deviation to solve the small cut cell problem of unfitted finite element methods. We construct reliable algorithms to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. Numerical examples are presented to illustrate the competitive performance of the method. This talk is based on joint works with Ke Li, Yong Liu and Xueshuang Xiang.

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Date/Time : 08 Feb 2024 15:00
Venue : S17-05-11 (SR 5)
Speaker : Tan Bui-Thanh, The University of Texas at Austin

Digital twins (DTs) are digital replicas of systems and processes. At the core of a DT is a  physical/mathematical model which captures the behavior of the real system across temporal and spatial scales. One of the key roles of DTs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, and to calibrate parameters to match the actual process. In this talk we will present various (faster than) real time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, Euler, and Navier-Stokes equations will be presented.

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Date/Time : 06 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Dong Bin, Peking University

This presentation delves into the recent advancements in applying artificial intelligence (AI) to mathematical exploration. Initially, we will outline the current state of integrating AI methodologies, including machine learning, deep learning, and neural networks, into mathematical research, thereby fostering enhanced approaches and innovative concepts. Subsequently, we will highlight some preliminary results from ongoing research in this intersecting domain. Concluding the discussion, we will anticipate the future trajectory of research at the intersection of AI and mathematics, focusing on the potential of AI techniques to tackle new challenges and uncover novel problems in mathematical studies.

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Date/Time : 05 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Xiaoping Wang, The Chinese University of Hong Kong, Shenzhen

Topology optimization, which aims to find the optimal physical structure that maximizes mechanical performance,is vital in engineering design applications in aerospace, mechanical, and civil engineering. We introduce a deep generative model, based on diffusion models, to address the structure optimization problem. Combine with the threshold dynamics method, we present a successful frame for the topology optimization.

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Date/Time : 23 Jan 2024 17:00
Venue : S17 #04-06 (SR1)
Speaker : Philippe Rigollet, MIT

Since their introduction in 2017, Transformers have revolutionized large language models and the broader field of deep learning. Central to this success is the groundbreaking self-attention mechanism. In this presentation, I’ll introduce a mathematical framework that casts this mechanism as a mean-field interacting particle system, revealing a desirable long-time clustering behavior. This perspective leads to a trove of fascinating questions with unexpected connections to Kuramoto oscillators, sphere packing, and Wasserstein gradient flows. Primarily based on https://arxiv.org/abs/2312.10794 as well as more recent results from our group.

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Date/Time : 17 Jan 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yifeng Liu, IASM, Zhejiang University

BSD conjecture is one of the most influential problems in number theory, which concerns arithmetics of elliptic curves. It has been generalized to motives over number fields by Beilinson, Bloch and Kato in the 1980s. In this talk, we will survey recent progress of these conjectures for higher dimensional varieties.

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Date/Time : 22 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Gang Bao, Zhejiang University

Inverse problems are concerned with determining model parameters from the observed data, which arise in diverse application areas, such as nondestructive testing, seismic imaging, optical imaging, materials characterization, and medical imaging. A well-known representative inverse problem is electrical impedance tomography (EIT), also known as Calderón’s problem, which determines the electrical conductivity from the voltage to current map on the boundary. In general, inverse problems are known to be severely ill-posed, in particular, small perturbations in the measured data may lead to large errors in the reconstructions. Stable numerical solution of inverse problems remains a big challenge in computational inverse problems. In this talk, the speaker will discuss some new directions for solving inverse problems. Of particular interest are recursive linearization methods based on multiple frequency data, deep learning methods, and optimal transportation approaches. These methods are shown to be suitable for solving severely ill-posed problems. Numerical results will be presented to validate the effectiveness and feasibility of the methods.

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