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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Date/Time : 27 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Wil Schilders, President of ICIAM, Eindhoven University of Technology & TU Munich-Institute for Advanced Study

Scientific machine learning (SciML) has been taking the academic world by storm as an interesting blend of traditional scientific modeling with machine learning (ML) methodologies like deep learning. While traditional machine learning methodologies have difficulties with scientific issues like interpretability, and enforcing physical constraints, the blend of ML with numerical analysis and differential equations has evolved into a novel field of research which overcome these problems while adding the data-driven automatic learning features of modern machine learning. Many successes have already been demonstrated, with tools like physics-informed neural networks, universal differential equations, deep backward stochastic differential equation solvers for high dimensional partial differential equations, and neural surrogates showcasing how deep learning can greatly improve scientific modeling practice. Consequently, SciML holds promise for versatile application across a wide spectrum of scientific disciplines, ranging from the investigation of subatomic particles to the comprehension of macroscopic systems like economies and climates.

However, despite notable strides in enhancing the speed and accuracy of these methodologies, their utility in practical and specifically industrial settings remain constrained. Many domains within the scientific community still lack comprehensive validation and robustness testing of SciML approaches. This limitation is particularly pronounced when confronted with complex, real-world datasets emanating from interactions between machinery and environmental sensors as usually addressed in industry. Still if appropriately addressed, SciML with its promise to accelerate innovations and scientific discoveries by orders of magnitudes, offers unique opportunities to address the insatiable desire for faster and more accurate predictions in many fields.

This presentation is dedicated to exploring recent advancements in the implementation of SciML techniques. We will discuss how methodologies can be refined to ensure their practical viability and scalability, particularly in industrial sectors where digital and physical components converge.

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Date/Time : 09 Jan 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jian Ding, Peking University

I will describe various percolation models in statistical physics, with emphasis on phase transition, critical phenomenon as well as disorder effects. In particular, I will start my talk with Bernoulli percolation, and then present a few important examples of correlated percolation models. While I try to be a “messenger” communicating progress accomplished in the community as a whole, the talk is inevitably biased by my personal interests (for instance, I will spend some time surveying recent progress on the random field Ising model and Gaussian free field related percolation models).

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Date/Time : 08 Jan 2025 15:30
Venue : IMS executive seminar room (S17 Level 3)
Speaker : Jeremie Szeftel, Sorbonne Universite

I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.

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Date/Time : 08 Jan 2025 14:00
Venue : IMS executive seminar room (S17 Level 3)
Speaker : Wendelin Werner (Recipient of the Fields Medal in 2006), University of Cambridge

Physical theories are often described in terms of fields and properties of their correlation functions. I will discuss several older and recent instances, where the understanding of related random underlying geometric objects and their properties turned out to be interesting and fruitful.

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Date/Time : 04 Dec 2024 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In the 80’s two type of related rigidity results emerged, one of geometric flavor and the other dynamical, but both have a dynamical background. Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. On the other hand, Shub ans Sullivan for expanding maps and later de la LLave, Marco and Moriyon for Anosov diffeomorphisms in dimension 2 showed that the marked Lyapunov spectrum determines the smooth isomorphism type of the system. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theory can be developed. This project is joint with A. Gogolev and M. Leguil.

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