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 : 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 : 08 Jul 2025 13:30
Venue : S17 Level 3 (IMS Executive Seminar Room)
Speaker : Lou van den Dries, The University of Illinois Urbana-Champaign

This will be a survey of our work on these topics. No familiarity with transseries or Hardy fields is required, but I mention here that both are natural settings for solving ordinary differential equations and studying their asymptotics (and for much more). Transseries are formal objects, and Hardy fields consist of actual functions (or rather germs of such). Around 2015 we showed that the differential field of transseries has a good model theory. We went on to prove that sufficiently rich Hardy fields have the same (complete) model theory, with interesting consequences even on the level of linear differential equations. More recently we extended this even to analytic Hardy fields. Some of the ideas involved go back to Sturm and Liouville (1830’s).

This is joint work with Aschenbrenner and Van der Hoeven.

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Date/Time : 19 Mar 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Robert Lazarsfeld, Stony Brook University

Recall that two algebraic varieties of the same dimension n are birationally equivalent if they contain isomorphic Zariski-open subsets. In particular, a variety is rational if it is birational to projective space. In the past few decades there has been a great deal of interest and progress in the study of rationality, but most varieties aren’t rational. I will survey a body of recent work going in a complementary direction, namely: how can one measure and control “how irrational” a given variety might be? More generally, how can one quantify the “birational distance” between two varieties?

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Date/Time : 18 Feb 2025 10:00
Venue : S17 Level 3 (IMS executive seminar room)
Speaker : Qiang Du, Columbia University

Recent applications and theoretical developments of models of integral equations using nonlocal operators have shown promise as effective alternatives to local models, especially in the presence of singularities and anomalies. These models also serve as continuum limits for large-scale discrete models used in data learning and network analysis. We present these models on bounded spatial domains and discuss related modeling, analysis, and computational issues.

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Date/Time : 05 Feb 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : JongHae Keum, Korea Institute for Advanced Study

The surfaces in the title are complex algebraic surfaces with the Betti numbers of the complex projective plane, and are called (rational) homology projective planes. If such a surface has only quotient singularities (a kind of mild singularities), then its minimal resolution is a 2-dimensional complex manifold with zero geometric genus and irregularity. They are 2-dimensional generalizations of the complex projective line (the Riemann sphere), and are closely related to the theory of 4-manifolds. Fake projective planes and the complex projective plane are smooth examples of a rational homology projective plane.

My presentation will begin with basic definitions and examples, and then describe recent progress in the study of such surfaces, singular ones and fake projective planes.

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Date/Time : 05 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Grigorios Pavliotis, Imperial College London

In this talk we present recent results on the quantitative study of stochastic interacting particle systems and of their mean field limit. We will start by presenting links between uniform propagation of chaos, the absence of phase transitions and Gaussian fluctuations around the mean field limit. We will present different methodologies for inferring parameters in the mean field (McKean) stochastic differential equation from observations of particle paths. Finally, we show how we can identify stationary states of the mean field PDE and how to steer the dynamics towards a chosen steady state using optimal control methodologies.

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