Date/Time : 25 Jan 2024 15:00
Venue : S16-06-118, Seminar Room
Speaker : Zhou Fan, Yale University

In statistical and probabilistic models, “mean-field” behavior arises when the aggregate effect of many variables is well-approximated by an average or effective field. Ideas from the analysis of mean-field models in statistical physics have been influential in our understanding of various problems of high-dimensional statistics in recent years. In this talk, I will argue that statistical applications can conversely drive the basic study of mean-field phenomena.

As a motivating example, I will discuss an application of high-dimensional mean-field theory to a nonparametric empirical Bayes procedure for PCA. I will then describe several developments of the theory beyond i.i.d. models, including extensions of Approximate Message Passing algorithms and analyses of spin glass models where the couplings matrices are orthogonally invariant in law.

Joint work w/ Yufan Li, Subhabrata Sen, Chang Su, Tianhao Wang, Yihong Wu, Xinyi Zhong.

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Date/Time : 24 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Bong Lian, Brandeis University

We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of $\P^1$ branched over 4 points. Two-fold covers of $\P^2$ branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. The new insight here is the idea of `abelian gauge fixing’ and `fractional complete intersections’ that leads to a new interpretation of those classical results. The idea further points to a construction of large class of Calabi-Yau mirror pairs.

The lecture is based on joint work with S. Hosono, T.-J. Lee, M. Romo, L. Santilli, H. Takagi, S.-T. Yau.

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Date/Time : 24 Jan 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Ziqing Xiang, Southern University of Science and Technology

We introduce a new notion called the quantum wreath product, which produces an algebra $B \wr_Q H(d)$ from a given associative algebra $B$, a positive integer $d$, and a choice $Q = (R, S, \rho, \sigma)$ of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product $\Sigma_m \wr \Sigma_2$ between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations. This is joint work with Daniel Nakano and Chun-Ju Lai.

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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 : 24 Jan 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Arturo Rodríguez Fanlo, Hebrew University of Jerusalem

In this talk I will present a new (first-order) logic that mixes positive logic (where negation is not allowed) and local logic (where quantifiers are bounded). We will discuss several basic notions of model theory for this logic such as compactness, positive closedness (existential closedness), completeness (irreducibility), type spaces and retractors (saturated models).

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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 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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