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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Date/Time : 13 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Melvin Leok, University of California, San Diego

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

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Date/Time : 13 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Guillaume Bal, The University of Chicago

Surprising asymmetric transport phenomena along interfaces separating insulating bulks have been observed in many areas of applied sciences, e.g., electronics, photonics, and geophysics. Such transport displays quite strong robustness to perturbations as an obstruction to Anderson localization. In fact, the phenomenon affords a topological origin: systems in the same topological class display the same robust, quantized, edge/interface transport. Perturbations may then be interpreted as continuous deformations that preserve the topological class.

This talk considers such systems modeled by general elliptic partial differential operators on the Euclidean plane. We review a recent simple topological classification, which provides an explicit computation of a topological invariant, technically the index of a Fredholm operator obtained by means of confining domain walls. We next introduce a physical observable that allows us to quantify the asymmetry of the edge transport. The evaluation of such an observable is challenging in practice. We then present a bulk-edge correspondence, a pillar of topological phases of matter in the physical literature, which shows that the interface current observable is in fact equal to the aforementioned simple topological invariant.

The theoretical findings are illustrated with examples ranging from electronics applications to geophysical fluid flows.

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Binyong Sun, IASM, Zhejiang University

After introducing some basic notions in the representation theory of classical Lie groups, I will explain three results in this theory: the multiplicity one theorem for classical groups, the conservation relation for local theta correspondence, and the non-vanishing hypothesis at infinity for Rankin-Selberg convolutions. Some relevant or consequent results will also be explained.

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Date/Time : 21 Aug 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Shahar Mendelson, Australian National University

Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of the natural function class extension of a random projection of a subset of 𝑅^𝑑 : for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the corresponding coordinate projection of 𝐹 is the set 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 } ⊂ 𝑅^n.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem.

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