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A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We’ll begin with a quick review of some basic constructions, such as forming the Cauchy completion of a category with respect to a metric. And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric...

A previous version of this video was viewed 701 times

A previous version of this video was viewed 1,017 times

Subhash Khot is awarded the Nevanlinna Prize for his prescient definition of the “Unique Games” problem, and leading the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems; his work has led to breakthroughs in algorithmic design and approximation hardness, and to new exciting interactions between computational complexity, an...

Artur Avila is awarded a Fields Medal for his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

Manjul Bhargava is awarded a Fields Medal for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.

Martin Hairer is awarded a Fields Medal for his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations.

Maryam Mirzakhani is awarded the Fields Medal for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

June Huh is awarded the Fields Medal 2022 for bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling-Wilson conjecture for geometric lattices, the proof of the Heron-Rota-Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture. A previous version of this video was viewed 99,857 times

Frank Calegari: 30 years of modularity: number theory since the proof of Fermat's Last Theorem

The idea of studying the geometry of an algebraic variety through the structure of its derived category of coherent sheaves goes back to the pioneering works of Bondal and Orlov on the verge of the millennium. One of the central concepts of this approach is that of a semiorthogonal decomposition. In my talk I will overview the (rapidly developing) story of semiorthogonal decompositions, touchin...

Michel van den Bergh: Noncommutative crepant resolutions

Kathryn Mann: Groups acting at infinity

I will discuss the role of Lagrange multiplier functionals in mathematics and physics. The main focus is on Rabinowitz’ action functional and its usage in symplectic geometry, as well as recent applications in string topology and the study of closed geodesics. Slides: www.mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/17-Kai Cielibak.pdf

Danny Calegari: Sausages and butcher paper

Nathalie Wahl: Homological Stability: a tool for computations

Alexander Gamburd: Arithmetic and dynamics on varieties of Markoff type

IMU Award Ceremony 2022

Olivier Wittenberg: Some aspects of rational points and rational curves

Laure Saint-Raymond: Dynamics of dilute gases: a statistical approach

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

Xin Zhou: Mean Curvature and Variational Theory

Tianyi Zheng: Asymptotic behaviors of random walks on countable groups

Richard Nickl: Statistical inverse problems and PDEs: progress and challenges

Annette Imhausen: Some uses and associations of mathematics, as seen from a distant historical...

Bhargav Bhatt: Algebraic geometry in mixed characteristic

Scott Sheffield: What is a random surface?

Jian Ding, Julien Dubedat, Ewain Gwynne: The Liouville quantum gravity metric

David Kazhdan: On the Langlands correspondence for curves over local fields