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 : 03 Sep 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Laura DeMarco, Harvard University

In holomorphic dynamical systems, one studies maps on the Riemann sphere (or other complex manifolds) with focus on their Julia sets and invariant measures. From this point of view, the Lattès maps — those that are quotients of maps on elliptic curves — are rather uninteresting; their dynamical features are well understood. But viewed algebraically, there are still many unanswered questions. I’ll begin the talk with some history of these maps. Then I’ll describe a recent question about the geometry of torsion points on elliptic curves and how it has led to interesting complex-dynamical questions about other families of maps and, in turn, new perspectives on the arithmetic side.

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Date/Time : 02 Sep 2025 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Yilin Wang, ETH Zürich

The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. After recalling basic facts about surfaces with constant curvature, their length spectrum, and Brownian motion on them, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. This is mainly based on a joint work with Yuhao Xue (IHES).

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Date/Time : 01 Sep 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wenqiang Xu, New York University

Are there infinitely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative? This is a classical question that goes back to Fekete and Polya in 1911. Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests that such characters are nevertheless plentiful. For instance, computations show that there are many more examples than may be expected by modeling such sums by a simple random walk. I will discuss joint work with Angelo and Soundararajan which obtains new upper bounds for the number of such characters, by studying a related problem on maxima of L-values (closely connected to the Fyodorov–Hiary-Keating conjectures). I will give a heuristic explanation for why such characters are plentiful, which suggests that our upper bounds may not be too far from the truth.

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Date/Time : 28 Aug 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Zhang Zhen, Southern University of Science and Technology

We propose a novel phase-field model for N-phase problems by using a dichotomic representation. Derived from Onsager’s variational principle, the proposed model enjoys mechanic, energetic, algebraic and dynamic consistencies and is shown to asymptotically converge to its sharp interface limit. An operator-splitting based numerical method is developed to simulate the model with both second-order accuracy and energy stability. Various numerical results are presented to validate theoretical derivation and show interesting phenomena.

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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 : 27 Aug 2025 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Sun Hongpeng, Renmin University of China

This paper introduces a preconditioned convex splitting algorithm enhanced with line search techniques for nonconvex optimization problems. The algorithm utilizes second-order backward differentiation formulas (BDF) for the implicit and linear components and the Adams-Bashforth scheme for the nonlinear and explicit parts of the gradient flow in variational functions. The proposed algorithm, resembling a generalized difference-of-convex function approach, involves a changing set of convex functions in each iteration. It integrates the Armijo line search strategy to improve performance. The study also discusses classical preconditioners such as symmetric Gauss-Seidel, Jacobi, and Richardson within this context. The global convergence of the algorithm is established through the Kurdyka-Łojasiewicz properties, ensuring convergence within a finite number of preconditioned iterations. Numerical experiments demonstrate its high efficiency. This is a joint work with Xinhua Shen and Zaijiu Shang.

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Date/Time : 27 Aug 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Guolei Zhong, Institute for Basic Science, Korea

Let X be a smooth projective variety and G an abelian subgroup of automorphisms of X over the field of algebraic numbers. Suppose that G is of positive entropy. It was proved by Dinh and Sibony that the group is then free abelian of rank no more than dim(X)-1. In the past decade, from algebraic dynamics, the geometry of the extremal case when the rank achieves the maximal has been well-understood by Zhang. In this talk, we study the maximal rank case in the aspect of arithmetic dynamics. More precisely, we construct a canonical height function associated with the group, corresponding to a nef and big divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi–Silverman conjecture for each group element. Some further discussion towards its application and the existence of Zariski dense orbits will also be presented. This is based on a joint work with Fei Hu.

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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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