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 : 26 Aug 2025 10:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Marcel Nutz, Columbia University

Entropic optimal transport — the optimal transport problem regularized by KL divergence — is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of dimensionality suffered by unregularized optimal transport. The flip side of smoothness is overspreading: the entropic coupling always has full support, whereas the unregularized coupling that it approximates is usually sparse, even given by a map. Regularizing optimal transport by less-smooth divergences such as L^p regularization is known to allow for sparse approximations, but is often thought to suffer from the curse of dimensionality as the couplings have limited differentiability and the dual is not strongly concave. We refute this conventional wisdom and show, for a broad family of divergences, that the key empirical quantities converge at the parametric rate, independently of the dimension. More precisely, we provide central limit theorems for the optimal cost, the optimal coupling, and the dual potentials induced by i.i.d. samples from the marginals. (Joint work with Alberto Gonzalez-Sanz and Stephan Eckstein.)

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Date/Time : 21 Aug 2025 TBA
Venue : TBA
Speaker : Liu Sheng, Stanford University

While modern machine learning models and systems, including most advanced large language models, have achieved impressive results, their success is still predicated on massive collections of human-curated, carefully filtered data. This reliance exposes a fundamental vulnerability: when deployed in the real world, these models encounter data that is noisy, incomplete, or ambiguous, leading to unpredictable and unreliable behavior. This challenge is more urgent in high-stakes domains such as medicine, where robustness and trustworthiness are paramount.
In this talk, I will present how we can move beyond the limitations of curated data by developing machine learning algorithms and theory that enable models to learn and adapt directly from the complexity of real-world environments. I will share new insights into the underlying mechanisms of failure for current models under imperfect data, and introduce principled methods that address these vulnerabilities. Finally, I will illustrate how these advances empower reliable AI systems for critical medical applications, including early Alzheimer’s disease detection and automated radiotherapy treatment planning.

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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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