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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Date/Time : 12 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Maxim Raginsky, University of Illinois at Urbana-Champaign

The majorizing measure theorem of Fernique and Talagrand is a fundamental result in the theory of random processes. It relates the boundedness of random processes indexed by elements of a metric space to complexity measures arising from certain multiscale combinatorial structures, such as packing and covering trees. In this talk, based on joint work with Yifeng Chu, I will build on the ideas first outlined in a little-noticed preprint of Andreas Maurer to present an information-theoretic perspective on the majorizing measure theorem, according to which the boundedness of random processes is phrased in terms of the existence of efficient variable-length codes for the elements of the indexing metric space. I will close by discussing the relevance of these results to the analysis of statistical performance of machine learning algorithms through the lens of data compression and the Minimum Description Length principle.

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Date/Time : 31 Aug 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ofer Zeitouni, Weizmann Institute and NYU Courant

The extreme value theory for Gaussian logarithmically correlated fields has emerged in the last decade as a powerful tool in the analysis of interface models, quantum gravity, random matrices and in a myriad of other applications.

The two dimensional Gaussian free field (and its discrete analogue) is an important motivating example of such a field. In this lecture, I will describe the relation and differences between the extreme value theory for i.i.d. variables and that for Gaussian logarithmically correlated fields, and discuss a sample of non-Gaussian examples.

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Date/Time : 28 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Gao Ziyang, Leibniz Universitat Hannover

It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.

In my talk, I will explain the historical and recent developments of this problem according to the different grades. Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.

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Date/Time : 24 Aug 2023 14:00
Venue : S17 #04-06 (SR1) (Colloquium (joint with IMS))
Speaker : Yujiro Kawamata, University of Tokyo

We will describe infinitesimal deformations of complex manifolds to the direction of something having possibly non-commutative structure sheaves by using Hochschild cohomology. We will also describe global non-commutative deformations of some surfaces.

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Date/Time : 08 Aug 2023 15:00
Venue : S17 #05-12 (SR4)
Speaker : Barbara Wohlmuth, Technical University of Munich

In this talk, we discuss several aspects of time‐fractional differential operators. The non‐local nature of the operator is challenging from a theoretical and algorithmic point of view but allows us to account for memory effects in the mathematical model. We consider a wide class of time‐fractional PDE models, including gradient flow, viscoelasticity, random field generators, and polymeric fluids. Of special interest is the equivalence of the time fractional gradient flow model with an integer order system in a higher dimension. Exploiting this equivalence allows us to construct efficient algorithms for time fractional PDEs. Replacing the kernel by a finite sum of exponentials transforms the non‐local PDE into an integer‐order system for the modes. To balance the discretization error in time and space with the kernel approximation error, the number of modes has to grow logarithmically. However, we observe numerically that the number of modes is very moderate if the finite sum of exponentials is based on a rational approximation of the kernel function. In this talk, we present theoretical existence results, algorithmic aspects and show several applications in which fractional operators play an important role. Numerical examples illustrate the large flexibility of the proposed techniques.

Joint work with Marvin Fritz (RICAM, Austria), Brendan Keith (Brown University, USA), Ustim Khristenko (Safran, France) and Endre Süli (University of Oxford, UK).

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Date/Time : 27 Jul 2023 15:00
Venue : S17-04-06 (SR 1)
Speaker : Weinan E, AISI and Peking University

For many years, the lack of good algorithms has severely limited our ability to conduct scientific research. At its heart, the difficulty comes from the notorious “curse of dimensionality” problem. Deep learning is exactly the kind of tool needed to address this problem. In the last few years, we have seen a tremendous amount of scientific progress made as a result of the AI revolution, both in our ability to make use of the fundamental principles of physics and in our ability to make use of experimental data.

In this talk, I will analyze the origin of the AI for Science revolution, review some of the major progresses made so far, and discuss how it will impact the way we do scientific research.

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