Date/Time : 26 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Bjoern Bringmann, Princeton University

Gibbs measures were first introduced in statistical mechanics, where they have been used to describe statistical equilibria of physical systems. Over the last decades, Gibbs measures have also become central objects in probability theory, partial differential equations, and mathematical physics. In this talk, we are interested in the heat, wave, and Schrödinger dynamics corresponding to Gibbs measures. In the first part of the talk, we discuss the invariance of the $\Phi^4_3$-measure under the three-dimensional cubic nonlinear wave equation. In the second part, we then discuss the global well-posedness of the stochastic Abelian-Higgs model, which is the Langevin equation for the Abelian-Higgs measure.

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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 : 28 Aug 2025 10:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Huy Tuan Pham, California Institute of Technology

Sumset is a fundamental object in additive combinatorics. Since Freiman’s celebrated theorem which characterizes the structure of integer sets with small sumsets, the structure theory of sets with small sumsets has gathered significant development, breakthroughs and applications.

In this talk, I will discuss new combinatorial and probabilistic perspectives on sumsets. Motivated from the study of random Cayley graphs, these perspectives provide new ways to quantify the structure and complexity of sets with small sumsets that are robust, flexible and quantitative. Emerging from this picture is the rich interaction between additive combinatorics and probabilistic combinatorics, which leads to new understanding of Cayley graphs and random Cayley graphs in general groups, as well as the resolution of longstanding questions in additive combinatorics

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Date/Time : 20 Aug 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Andreas Wieser, Institute for Advanced Study

A classical question in number theory asks: which integers are represented by a given quadratic form? More generally, one can ask when an integral quadratic form q in m variables can be represented by another integral quadratic form Q in n > m variables—that is, whether there exists an integer matrix T such that Q(Tx) = q(x).

Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo N for every positive integer N. In the absence of local obstructions, does a (global) representation of q by Q necessarily exist?

A breakthrough result of Ellenberg and Venkatesh in 2008 used deep methods from homogeneous dynamics to prove such a local-global principle when the codimension n – m is at least 5. In this talk, we present joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi in which we establish a local-global principle in codimension n – m at least 3. The key new ingredient is an effective equidistribution result for periodic orbits of semisimple groups on compact adelic homogeneous spaces.

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Date/Time : 30 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Liu Jinpeng, Tsinghua University

Nonlinear dynamics play a prominent role in scientific computation and artificial intelligence. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. Furthermore, we design the first quantum algorithm for training classical sparse neural networks with end-to-end settings. We benchmark instances of ResNet with sparse pruning applied to Cifar-100 dataset and DPM-Solver for U-ViT applied to ImageNet-100 dataset, and we find that a quantum enhancement is possible at the early stage of learning. Our work shows that fault-tolerant quantum computing can contribute to the training and inference processes of most state-of-the-art large language models and diffusion models.

References:
[1]Efficient quantum algorithm for dissipative nonlinear differential equations. Proceedings of the National Academy of Science 118, 35 (2021).
[2]Towards provably efficient quantum algorithms for large-scale machine learning models. Nature Communications 15, 434 (2024)
[3]Towards efficient quantum algorithms for diffusion probability models. arXiv:2502.14252

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Date/Time : 23 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Jialun Li, Ecole Polytechnique de Paris

I will begin with the classical setting of the modular surface: the unit tangent bundle of the modular curve T^1(SL(2,Z)\H)=SL(2,Z)\SL(2,R), where periodic geodesics correspond to periodic orbits of the diagonal subgroup. Margulis and Bowen established dynamical frameworks for studying these orbits, while Selberg and Sarnak developed spectral and number-theoretic approaches.

We then turn to the higher-rank generalization, periodic diagonal orbits on SL(3,Z)\SL(3,R). In this case, each periodic diagonal orbit is a two-dimensional flat torus embedded in the space. Different counting problems emerge, depending on how one orders these tori–whether by dynamical ordering, geometric ordering or arithmetic ordering. The dynamical and geometric ordering are closely connected through the shapes of the flat tori. Finally, I will discuss my recent joint work with Thi Dang and Nihar Gargava on the density of shapes of periodic tori.

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Date/Time : 19 Mar 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jiajie Chen, New York University

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is a long-standing open problem in mathematical fluid mechanics. In this talk, I will begin by providing an overview of this problem and then introduce a framework for stable, nearly self-similar blowup, developed in joint work with Tom Hou (Caltech). Using this framework, we establish singularity formation in the incompressible Euler equations with smooth initial data and boundary. Additionally, I will briefly present recent results on constructing singularities in fluids and related equations using the self-similar blowup method.

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Date/Time : 05 Mar 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : James Farre, Max Planck Institute for Mathematics in the Sciences

The geodesic flow on the unit tangent bundle of a hyperbolic surface is a prototypical example of an Anosov flow, exhibiting rich, structurally stable dynamics. While invariant measures and closed sets for the geodesic flow can be quite complicated, the closely related horocycle flow exhibits striking rigidity. For example, on a closed surface, every horocycle is dense and equidistributed in its unit tangent bundle; this is very far from being true for geodesics.

Despite this rigidity, the behavior of horocycles remains mysterious in the setting of hyperbolic surfaces with infinite topology. In this talk, I will explore the dynamics of horocyclic trajectories for a particular class of infinite-area regular covers of closed hyperbolic surfaces. Surprisingly, these trajectories are intimately connected to an exceptional set for the geodesic flow: the distance-minimizing geodesic lamination. Both objects arise naturally in a geometric optimization problem: what does an optimal Lipschitz map to the circle in a given homotopy class look like?

This is joint work with Or Landesberg and Yair Minsky.

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : He Weikun, AMSS, Chinese Academy of Sciences

Diophantine approximation studies how well a real number can be approximated by rational numbers. Following a question asked by Mahler in 1984, we are interested in Diophantine properties of points in the middle third Cantor set or more generally fractal sets. I will survey briefly this topic and present a recent joint work with Timothée Bénard and Han Zhang, where we obtained a Khintchine dichotomy for the middle third Cantor set.

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