Date/Time : 01 Dec 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Wenkui Du, Massachusetts Institute of Technology

In this talk, I will discuss the singularity theory of mean curvature flow. As the most natural evolution equation in extrinsic geometry, mean curvature flow has striking applications in geometry, topology and image processing. A central challenge for these applications is understanding the structure of singularities. This talk surveys recent progress on the classification of singularity models of mean curvature flow and their applications. In particular, ancient noncollapsed solutions naturally arise when we consider blow-up limits near singularities. Several results about classification of ancient noncollapsed solutions will be presented. I will also mention my future plan of singularity analysis in geometric variational problems. The talk is based on my joint works with collaborators B. Choi, Daskalopoulos, Haslhofer, Šešum, Zhao and Zhu.

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Date/Time : 28 Nov 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Koulik Khamaru, Rutgers University

Modern decision-making increasingly relies on adaptive experimentation, particularly in settings such as A/B testing, multi-armed bandits, and reinforcement learning. While these methods enable more efficient learning and allocation of resources, they fundamentally challenge traditional statistical inference. Classical i.i.d.-based tools often break down under adaptive data collection, resulting in biased estimators and misleading confidence intervals.

This talk offers an overview of statistical inference in these adaptive environments. We highlight the pitfalls of naive inference through concrete examples and introduce the concept of stability—originally formulated by Lai and Wei (1982)—as a unifying principle for valid inference under adaptivity. A key advantage of stability is that it allows us to recover classical inferential guarantees—such as asymptotic normality and valid confidence intervals—even when the data arise from highly adaptive algorithms. In other words, stability serves as the bridge that lets i.i.d.-based tools remain valid in non-i.i.d. bandit settings.

Next, we discuss the stability properties of two widely used algorithms: the Upper Confidence Bound (UCB) and Thompson Sampling. We argue that while UCB is stable, Thompson Sampling is not. Finally, we propose a modification of Thompson Sampling that regains stability while maintaining near-optimal regret. Key illustrations include quantitative central limit theorems of empirical mean in stochastic bandits and least square estimators in contextual bandits. We also present a new proof technique for analyzing regret and establishing stability, which we believe has broader applicability and may be of independent interest.

The talk is based on a series of joint works with Cunhui Zhang, Qiyang Han, Budhaditya Halder, Subhayan Pan, and Samya Praharaj.

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Date/Time : 26 Nov 2025 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Leonardo Nagami Coregliano, University of Chicago

In the Netflix Prize Competition (2006–2009), we are given a finite set $U$ of users, a finite set $M$ of movies and a partial function $f: U\times M\rightharpoonup R$, where $f(u,m)$ indicates how user $u$ rates movie $m$ and we are tasked with completing $f$ to a total function to predict how all users $U$ rate all movies in $M$. Although some algorithms did fairly well in the competition, giving a satisfactory theoretical explanation for their success has been difficult.

In this talk I will discuss how this learning problem can be seen as an instance of a framework that we call “sample completion learning” and how sample completion learning is completely characterized by the model-theoretic notion of $k$-dependence introduced by Shelah (which can be seen as a high-dimensional version of the Vapnik–Chervonenkis dimension). In turn, this gives a full theoretical characterization of when the Netflix problem is “solvable”.

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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 : 25 Nov 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Davide Ravotti, Université de Lille

Translation flows are a simple kind of motion on geometric surfaces built by gluing polygons edge-to-edge. In certain special cases, called self-similar, the flow repeats its behavior in a precise way when the surface is stretched and folded by a symmetry known as a pseudo-Anosov map. When this happens, we can understand the flow by studying this symmetry. A powerful method for doing this, pioneered in a work by Giulietti and Liverani, uses a tool called transfer operator. By analyzing how the transfer operator acts on a carefully chosen space of distributions, it is possible to deduce fine dynamical and statistical properties of self-similar flows on compact surfaces. This strategy was successfully carried out by Faure, Gouëzel, and Lanneau.

In this talk, I will describe how we extend this approach to self-similar translation flows on certain non-compact surfaces, namely Abelian covers. By introducing a family of weighted transfer operators, we obtain precise formulas describing the long-term behavior of time averages of observables, for points that are typical with respect to the natural measure lifted to the cover (joint work with Henk Bruin, Charles Fougerons, and Dalia Terhesiu). If time allows, I will also discuss ongoing work (with Mauro Artigiani, Roberto Castorrini, and Yuriy Tumarkin) that goes beyond the natural area measure and treats a broader class of ergodic invariant measures, known as Maharam measures.

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Date/Time : 25 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Bjoern Bringmann, Princeton University

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

This is joint work with S. Cao.

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Date/Time : 25 Nov 2025 10:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Lo Chi-Heng , National University of Singapore

In this talk, I will discuss Jiang’s Conjecture, which predicts an upper bound for the wavefront set of any representation of Arthur type in terms of the local Arthur packet containing it. I will then describe its generalization to ABV-packets and to arbitrary irreducible representations (not necessarily of Arthur type), and show that all these formulations of Jiang’s Conjecture are in fact equivalent. Moreover, I will explain that these conjectures can be further reduced to the case of anti-tempered representations.

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Date/Time : 24 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Leonardo Nagami Coregliano, The University of Chicago

The classic de Finetti’s Theorem from 1931 says that an infinite sequence of random variables is invariant under permutations (exchangeable) if and only if it is a mixture of i.i.d. random variables. For a long time, this was considered a curiosity of classic probability theory. In the 80s, exchangeability was revived with the Aldous–Hoover Theorem which completely characterized exchangeable distributions of random matrices (i.e., distributions invariant under simultaneous permutations of rows and columns), and more generally tensors. Since the resulting Aldous–Hoover representation theorem yielded complicated representations, Hoover asked which distributions admitted simpler representations, called $\ell$-independent representations and noticed that a natural assumption, called $\ell$-locality, was necessary but not sufficient.

In this talk, I will cover surprising connections of exchangeability theory with combinatorics and learning theory. In the former connection, exchangeability can be used to construct semantic/geometric limits of dense combinatorial objects, which capture the asymptotic behavior of densities of subobjects; in turn, these limit objects are used to study quasirandomness and connect back to Hoover’s question on independence vs. locality. In the latter connection, exchangeability forms the backbone of high-arity probability approximately correct (PAC) learning theory, which is a natural learning theory for learning hypergraphs. In turn, high-arity PAC learning theory can be used to obtain better bounds in tame hypergraph regularity lemmas, which connect back to Aldous–Hoover representations of low rank (a notion dual to $\ell$-independence).

No background in exchangeability theory, limit theory or learning theory is required to follow the talk.  This talk is based on several joint works with Maryanthe Malliaris, Alexander Razborov, Caroline Terry and Henry Towsner.

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Date/Time : 24 Nov 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Wang, Chao, Southern University of Science and Technology

Variational Bayesian inference offers a powerful framework for tackling key challenges in tensor analysis and image restoration, which are critical problems in machine learning and computer vision. This talk presents innovative approaches to address these challenges by integrating Bayesian principles, variational inference, and physical modeling. First, we introduce a Bayesian framework for Tensor Robust Principal Component Analysis (TRPCA) to recover low-rank structures and characterize sparse noise in mixed-noise scenarios. By embedding a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior within a Bayesian framework, our method learns the optimal tensor nuclear norm and balances low-rank and sparse components automatically. Second, we propose a novel physics-informed optimization framework for jointly solving depth estimation and image restoration from a single defocused image. By modeling the defocused image as a function of a depth map and an all-in-focus (AiF) image based on optical physics, the framework leverages their intrinsic connections. The depth map guides AiF image recovery, while the AiF image regularizes the depth map reconstruction via reconstruction error. A variational inference algorithm, parameterized with deep neural networks, ensures flexibility and high performance.

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