Date/Time : 24 Jan 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Kelei, Wuhan University

Since the seminal work of Alt-Caffarelli in 1981, the one phase free boundary problem has been studied by many people. To study the regularity and singularity of free boundaries, blow up analysis is a standard method. It turns out for this free boundary problem, the nondegeneracy condition is crucial for the application of this method. Although this condition holds for energy minimizers, it’s not true for general solutions. In this talk, I will discuss a proof of the nondegeneracy for stable solutions. This is based on a joint work with N. Kamburov.

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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 : 22 Jan 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yang Lei, Institute for Advanced Study

I will talk about recent progress in the study of quantitative equidistribution of unipotent orbits in homogeneous spaces, namely, effective versions of Ratner’s equidistribution theorem. I will sketch the main idea of my proof for unipotent orbits in SL(3,R)/SL(3,Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the quantitative behavior of unipotent orbits is closely related to a Kakeya model.

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Date/Time : 19 Jan 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Linfeng Zhang, AI for Science Institute, Beijing, and DP Technology

The rapid development of artificial intelligence is driving significant changes in the field of molecular and materials modeling, simulation, and design. In this report, we will explore the breakthroughs and progress in AI-assisted molecular modeling, with a focus on the shift from multiscale methods to large pre-trained models and the resulting software, data, and model ecosystem. In particular, we will investigate the possibility of a large atomic model that covers the entire periodic table and, based on this, envision the future of the materials genome.

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Date/Time : 18 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Biao Ma, Technion - Israel Institute of Technology

Let Mod(S) be the mapping class group of a connected, closed, oriented surface S of genus g ≥ 2. Based on ideas of Margulis, Athreya-Bufetov-Eskin-Mirzakhani gave an asymptotic formula for orbit growth of Mod(S) on the Teichmuller space with respect to the Teichmuller metric. In this talk, I will explain how we can get asymptotic formulas for orbit growth of certain subgroups of Mod(S) based on Roblin’s ideas. More precisely, let G be a non-elementary subgroup of Mod(S) so that the associated Bowen-Margulis measure is finite. We will discuss an asymptotic growth formula for G with respect to the Teichmuller metric. This talk is based on a joint work with Ilya Gekhtman.

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Date/Time : 18 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Ruixiang Zhang, University of California, Berkeley

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\”{o}rmander asked if a more general class of operators (known as H\”{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\”{o}rmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\”{o}rmander’s question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\”{o}rmander type operator satisfying Bourgain’s condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain’s condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

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Date/Time : 18 Jan 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Yongqi Liang, University of Science and Technology of China

Scharaschkin and Skorobogatov conjectured that when smooth projective curves violate the Hasse principle then the violations are explained by the Brauer-Manin obstruction. Colliot-Thelene and Sansuc have similar conjecture for degree 4 del Pezzo surfaces. We use this obstruction to construct explicite algebraic families of degree 4 del Pezzo surfaces and smooth projective curves of genus congruent to 1 modulo 4, such that they violate the Hasse principle. In particular, over every number field,  we find explicite families of elliptic curves whose Tate-Shafarevich groups are non-trivial. This gives an explicite answer to a question of Colliot-Thelene and Poonen asked about 20 years ago, when they proved the existence of such a family of elliptic curves over the field \mathbb{Q} of rational numbers.

This is a joint work with K. Huang. 

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Date/Time : 17 Jan 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Sam Stechmann, University of Wisconsin–Madison

Sea ice floats on the surface of the Arctic Ocean, and plays an important role in Earth’s climate. Traditionally, sea ice has been modeled as a material, as a continuum, with various rheological models proposed. From a more detailed perspective, sea ice is a collection of many individual blocks of ice, called “floes.” In this talk, I will discuss approaches for floe-based (i.e., particle-based) models of sea ice, including multi-scale models to bridge the gap between floe-based and continuum-based models.

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Date/Time : 17 Jan 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruixiang Zhang, University of California, Berkeley

Fourier restriction type problems form a class of important problems in harmonic analysis. They are also related to PDE, analytic number theory, geometric measure theory and mathematical physics. In the past 15 years, much progress has been made on these problems. In this talk, we will introduce Fourier restriction type problems and survey some recent results, and then conclude by mentioning some personal favorite future directions.

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