Date/Time : 21 Nov 2025 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Matthew Shunshi Zhang, University of Toronto

We introduce a new method for analyzing midpoint discretizations of stochastic differential equations (SDEs), which are frequently used in Markov chain Monte Carlo (MCMC) methods for sampling from a target measure π \propto exp(−V). Borrowing techniques from Malliavin calculus, we compute estimates for the Radon-Nikodym derivative for processes which may anticipate the Brownian motion, in the sense that they may not be adapted to the filtration at the same time. Applying these to various popular midpoint discretizations, we are able to improve the regularity and cross-regularity results in the literature on sampling methods. We also obtain a query complexity bound of O(d^{1/4}) for obtaining a ε^2-accurate sample in 𝖪𝖫 divergence, under log-concavity and strong smoothness assumptions for ∇^2 V.

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Date/Time : 21 Nov 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Seonwoo Kim, Yonsei University

The main purpose of this lecture course is to introduce the recently-developed resolvent approach [Landim-Marcondes-Seo, JEMS ’25] to metastability and beyond. First, I will review the historical developments in the treatment of the metastability phenomenon and summarize the pros/cons of the previous methods. Next, I will state and prove the resolvent approach in a simple setting, focusing especially on its strength, along with some real-model applications. Finally, I will summarize my recent work with Jungkyoung Lee (KIAS) to extend and generalize this approach to non-compact metastable transitions.

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Date/Time : 20 Nov 2025 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Lénaïc Chizat, École Polytechnique Fédérale de Lausanne (EPFL)

Recent progress in AI has been fueled by training neural networks of ever-increasing size, a difficult craft guided largely by heuristics. A promising approach to provide principled foundations for this discipline is to analyze neural networks via their infinite-size limits. While the large-width limit is now relatively well understood and has already yielded practical tools adopted in industry (such as μP), the extension of the theory to large-depth models has remained elusive.

In this talk, I will present the first analysis that rigorously captures the joint infinite-width and infinite-depth limit for architectures used in practice. Our approach combines techniques from stochastic approximation, propagation of chaos and dynamical mean-field theory. The resulting framework provides phase diagrams, quantitative error bounds, and conceptual insights. For instance, our theory shows that models of infinite depth, regardless of their actual width, behave throughout training as though they were also infinitely wide. Overall, our analysis sheds new light on the structure of training dynamics and suggests a path toward principled methods for scaling deep neural networks.

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Date/Time : 20 Nov 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yang Daodao, Eastern Institute of Technology, Ningbo

The metric theory of Diophantine approximation investigates whether typical real numbers can be well approximated by rational numbers. In 1924, Khintchine proposed the first 0-1 law in metric Diophantine approximation: assuming that the approximation function has a certain monotonicity, this law provides a necessary and sufficient condition for typical real numbers to be well approximated by infinitely many rational numbers. In 1941, Duffin and Schaeffer attempted to generalize Khintchine’s theorem without assuming that the approximation function has monotonicity. Their proposed new 0-1 law, known as the Duffin–Schaeffer conjecture, became a central problem in Diophantine approximation. After considerable efforts by many number theorists, the Duffin–Schaeffer conjecture was finally resolved by Koukoulopoulos and Maynard in 2019. A remaining challenge is to establish a quantitative law, similar to the strengthening of Khintchine’s theorem by Erdős and Schmidt, to estimate the number of reduced fractions that can approximate the given real number sufficiently well. In 2022, Aistleitner, Borda and Hauke established a quantitative version of the Duffin–Schaeffer conjecture, but the error term they obtained only achieved a logarithmic power-saving. An open problem is to establish an asymptotic formula with a power-saving error term. In a recent joint work with Koukoulopoulos and Maynard, we prove a quantitative version of the Duffin–Schaeffer conjecture with an almost sharp error term.

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Date/Time : 19 Nov 2025 15:00
Venue : S16 #05-17
Speaker : Ely Sandine, University of California, Berkeley

The Euler-Poisson system describes the evolution of a self-gravitating compressible fluid. I will present recent work proving the existence of certain self-similar implosion profiles for this system. These solutions belong to a family numerically conjectured by Hunter. I will discuss related PDEs, previous results for this system and aspects of the proof.

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Date/Time : 19 Nov 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Seonwoo Kim, Yonsei University

The main purpose of this lecture course is to introduce the recently-developed resolvent approach [Landim-Marcondes-Seo, JEMS ’25] to metastability and beyond. First, I will review the historical developments in the treatment of the metastability phenomenon and summarize the pros/cons of the previous methods. Next, I will state and prove the resolvent approach in a simple setting, focusing especially on its strength, along with some real-model applications. Finally, I will summarize my recent work with Jungkyoung Lee (KIAS) to extend and generalize this approach to non-compact metastable transitions.

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Date/Time : 19 Nov 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Srecko Durasinovic, Nanyang Technological University

Mixture models, such as Gaussian mixture models (GMMs), are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein ($\was$) or the total variation ($\tv$) distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

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Date/Time : 19 Nov 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Arpan Kabiraj, Indian Institute of Technology Palakkad

In 80’s, Goldman introduced a Lie bracket on the linear span of the free homotopy classes of directed closed curves on a surface called the Goldman Lie bracket. In this talk we will discuss some properties of Goldman Lie bracket using hyperbolic geometry. We will use these properties to give characterizations of simple closed curves. This is a joint work with Moira Chas.

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Date/Time : 19 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Siqing Zhang, Yale University

The Hodge Theorem reveals a fundamental phenomenon in algebraic geometry: the de Rham cohomology can often be identified with the Dolbeault cohomology. The Non Abelian version of this phenomenon “identifies” the moduli of flat connections with the moduli of Higgs bundles. The arithmetic version of this phenomenon echoes these identifications in prime characteristics. In this talk, I will introduce recent advances on the interplay among the Non Abelian Hodge Theories in different characteristics. By bringing moduli theory into the prime characteristic version, we reveal new structures over the complex numbers.

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