ICM 2022 Preview Talks

ICM 2022 will be held virtually this year.  Please join Camillo De Lellis, Ronen Eldan, and Avi Wigderson as they give live presentations of their ICM lectures.  The schedule and abstracts are as follows:

• 1:00 PM - Camillo De Lellis, IAS
• 2:00 PM - Ronen Eldan, IAS & Microsoft Research
• 3:00 PM - Break
• 4:00 PM - Avi Wigderson, IAS

Camillo De Lellis, IAS: Regular and singular minimal surfaces

Minimal surfaces are surfaces whose area is stationary under smooth perturbations: a well known example is given by minimizers of the area among those which span a given contour and the study of their shape and properties dates back at least to the work of Lagrange in the middle of the 18th century. It is known since long that such objects are, in general, not necessarily smooth and this very fact presents immediately an intriguing challenge: what should we understand with the words ``surface'' and ``area''? The mathematical literature has seen quite a few different approaches, all leading to concepts of ``generalized minimal surfaces'' which have distinctive features. A pivotal question is when and where these objects are smooth, how large the sets of their singularities can be, and which behavior they can possibly display at the singular points. In my talk, I will review a selection of classical works, recent results, and future challenges.

Ronen Eldan, IAS & Microsoft Research: Revealing the simplicity of high-dimensional objects via pathwise analysis.

A common motif in high dimensional probability and geometry is that the behavior of objects of interest is often dictated by their marginals onto a fixed number of directions. This is manifested in the fact that several classical functional inequalities are dimension free (i.e., have no explicit dependence on the dimension), the extremizers of those inequalities being functions or sets that only depend on a fixed number of variables. Another related example comes from statistical mechanics, where Gibbs measures can often be decomposed into a small number of "pure states" which exhibit a simple structure that only depend on a small number of directions in space.

In this talk, I will present an emerging technique that helps reveal phenomenona of this nature. This technique is based on pathwise analysis: We construct a stochastic process, driven by Brownian motion, associated with a given high-dimensional object. This process allows us to associate quantities related to the object with corresponding properties of the stochastic process, thus making the former tractable via the analysis of the latter (for example, through differentiation with respect to time).

I will try to explain how this technique works and will briefly discuss several results that stem from it, including functional inequalities in Gaussian space, concentration inequalities in high-dimensional convexity, concentration of measures on the discrete hypercube, as well structure theorems which ensure decomposition of Gibbs measures into product-like components.

Avi Wigderson, IAS:  Symmetries, Computation, and Math (or, can P ≠ NP be proved via gradient descent?)

This talk aims to summarize a project I was involved in during the past 6-7 years, with the hope of explaining our most complete understanding so far, as well as challenges and open problems. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

(1) The most basic tools of convex optimization in Euclidean space extend to a far more general setting of Riemannian manifolds that arise from the symmetries of non-commutative groups. We develop first-order and second-order algorithms, and analyze their performance in general. While proving convergence bounds requires heavy algebraic and analytic tools, convergence itself depends in an elegant way on natural ``smoothness’’ parameters, in analogy with the Euclidean (commutative) case.

(2) These algorithms can give exponential improvements in run-time for solving algorithmic many problems across CS, Math, and Physics. In particular, these include problems in algebra (e.g. testing rational identities in non-commutative variables), in analysis (testing the feasibility and tightness of Brascamp-Lieb inequalities), in quantum information theory (to the quantum marginals problem), in computational complexity (to derandomizing new special cases of the PoIynomial Identity Testing problem) and in optimization (to testing membership in large, implicitly described polytopes).

(3) The focus on symmetries exposes old and reveals new relations between the problems above, and between analysis, algebra, and algorithms. Essentially, they are all membership problems in null cones and moment polytopes of natural group actions on natural spaces. Invariant theory, which studies such group actions, plays an essential role in this development. In particular, a beautiful non-commutative duality theory (expending linear programming duality in the commutative case), and notions of geodesic convexity (extending the Euclidean one), and moment maps (extending the Euclidean gradient) are central to the algorithms and their analysis. Interestingly, most algorithms in invariant theory are symbolic/algebraic, and these new numeric/analytic algorithms proposed here often significantly improve on them.

Based on joint works with Zeyuan Allen-Zhu, Peter Burgisser, Cole Franks, Ankit Garg, Leonid Gurvits, Pavel Hrubes, Yuanzhi Li, Visu Makam, Rafael Oliveira and Michael Walter.