Krishnamurthi Ramasubramanian: The History and Historiography of the Discovery of Calculus in India International Mathematical Union 2022-07-13 | Couched in rich poetic verses in the Sanskrit language, the history of mathematics in India provides a fertile field for researching the evolution of mathematical thinking. During the talk, starting with snippets from the work of Āryabhaṭa (c. 499 CE) we shall try to present how certain important breakthroughs lead to the pioneering contribution of Mādhava (c. 1340) of the Kerala School, which has a more direct bearing on calculus. Towards the end, we would also like to highlight some of interesting facets in the historiography pertaining to development of calculus in India.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/218-Krishnamurthi%20Ramasubramanian.pdf
Bruno Klingler: Hodge theory, between algebraicity and transcendence International Mathematical Union 2023-02-16 | The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, obtained mainly through the use of o-minimal geometry.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/151-Bruno%20Klingler.pdfA previous version of this video was viewed 1,038 times
Amnon Neeman: Finite approximations as a tool for studying triangulated categories International Mathematical Union 2023-02-16 | 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, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories. And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a major generalization of a theorem of Rouquier’s, a short, sweet proof of Serre’s GAGA theorem, and a proof of a conjecture by Antieau, Gepner and Heller.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/159-Amnon%20Neeman.pdfA previous version of this video was viewed 353 times
Thomas Nikolaus: Frobenius homomorphisms in higher algebra International Mathematical Union 2023-02-16 | A previous version of this video was viewed 701 times
Jennifer Hom: Homology cobordism and Heegaard Floer homology International Mathematical Union 2023-02-16 | A previous version of this video was viewed 1,017 times
Rolf Nevanlinna Prize 2014 Subhash Khot International Mathematical Union 2022-08-23 | 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, analysis and geometry.
Fields Medal Winner 2014 Artur Avila International Mathematical Union 2022-08-22 | 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.
Fields Medal Winner 2014 Manjul Bhargava International Mathematical Union 2022-08-22 | 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.
Fields Medal Winner 2014 Martin Hairer International Mathematical Union 2022-08-22 | 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.
Fields Medal Winner 2014 Maryam Mirzakhani International Mathematical Union 2022-08-22 | Maryam Mirzakhani is awarded the Fields Medal for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.
Fields Medals 2022 June Huh International Mathematical Union 2022-08-16 | 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 Fermats Last Theorem International Mathematical Union 2022-08-16 | ...
Alexander Kuznetsov: Homological algebraic geometry International Mathematical Union 2022-07-22 | 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, touching on some of its most fascinating aspects: (1) Semiorthogonal components with interesting properties and their geometric significance; (2) Categorical extensions of classical geometric constructions (homological projective duality, categorical joins and cones, categorical resolutions of singularities); and (3) Completely new constructions such as categorical absorptions of singularities.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/38-Alexander%20Kuznetsov.pdf
Michel van den Bergh: Noncommutative crepant resolutions International Mathematical Union 2022-07-22 | ...
Kai Cieliebak: Lagrange multiplier functionals International Mathematical Union 2022-07-22 | 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: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/17-Kai%20Cielibak.pdf
Nathalie Wahl: Homological Stability: a tool for computations International Mathematical Union 2022-07-22 | ...
Alexander Gamburd: Arithmetic and dynamics on varieties of Markoff type International Mathematical Union 2022-07-22 | ...
IMU Award Ceremony 2022 International Mathematical Union 2022-07-22 | International Mathematical Union IMU Award Ceremony 5 July 2022 at 10.00 (EEST)Venue: Aalto University, Runeberginkatu 14–16, Aalto Töölö, 00100 Helsinki PROGRAM (times are given according to EEST) 10:00–10:10 Opening by a state representative of Finland 10:10–10:20 Address by IMU President Carlos E. Kenig Awards: 10:20–10:35 Announcement of the Fields Medalists by IMU President | Presentation of the Awards 10:35–12:00 Videos and Laudatios for all Fields Medalists 12:00–12:15 Musical entertainment 12:15–12:30 Announcement of the IMU Abacus Medalist by IMU President Presentation of the Award | Video12:30–12:45 Laudatio for IMU Abacus Medalist 12:45–14:15 Lunch 14:15–14:30 Announcement of the Chern Medal Award by IMU President Presentation of the Award | Video14:30–14:45 Laudatio for Chern Medal Award 14:45–15:00 Musical entertainment 15:00–15:15 Announcement of the Carl Friedrich Gauss Prize by IMU President Presentation of the Award | Video 15:15–15:30 Laudatio for Gauss Prize 15:30–15:45 Announcement of the Leelavati Prize by IMU President Presentation of the Award | Video15:45–16:00 Laudatio for Leelavati 16:00–16:15 Some farewell words by IMU President
Olivier Wittenberg: Some aspects of rational points and rational curves International Mathematical Union 2022-07-21 | ...
Laure Saint-Raymond: Dynamics of dilute gases: a statistical approach International Mathematical Union 2022-07-19 | ...
Avi Wigderson: Symmetries, Computation and Math (or, can P ≠ NP be proved via gradient descent?) International Mathematical Union 2022-07-18 | ...
Svetlana Jitomirskaya: Small denominators and multiplicative Jensens formula International Mathematical Union 2022-07-14 | ...
Larry Guth: Decoupling estimates in Fourier analysis International Mathematical Union 2022-07-14 | Decoupling is a recent development in Fourier analysis, which has applications in harmonic analysis, PDE, and number theory. We survey some applications of decoupling and some of the ideas in the proof.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/49-Larry%20Guth.pdf
Umesh Vazirani: On the complexity of quantum many body systems International Mathematical Union 2022-07-14 | The ground state of a quantum system of n particles is the eigenvector of minimum eigenvalue of a matrix (the Hamiltonian) of dimension that scales exponentially in n. In this talk I will describe a recent body of work, inspired by concepts from quantum computation and information theory that shows that for a large class of 1D quantum systems the solution can be succinctly represented and computed in polynomial time on a classical computer.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/69-Umesh%20Vazirani.pdf
Francis Bach: Gradient descent on infinitely wide neural networks: Global convergence and... International Mathematical Union 2022-07-14 | Many supervised machine learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters, this often leads to convex problems for which many mathematical guarantees exist. Models which are non-linear in their parameters such as neural networks lead to non-convex optimization problems for which guarantees are harder to obtain. I will consider two-layer neural networks with homogeneous activation functions where the number of hidden neurons tends to infinity, and show how qualitative convergence guarantees may be derived.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/75-Francis%20Bach.pdf
Jan Philip Solovej: The Ground State of Quantum Gases International Mathematical Union 2022-07-14 | ...
Mathias Schacht: Restricted extremal problems in hypergraphs International Mathematical Union 2022-07-14 | Extremal problems for 3-uniform hypergraphs concern the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any l elements of V at least one triple is missing in E. This innocent looking problem is still open, despite a great deal of effort over the last 80 years. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E, which are motivated by the theory of quasirandom hypergraphs. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erdos and Sós in the 1980ies. We discuss a unifying framework for these problems and report recent progress.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/77-Mathias%20Schacht.pdf
Peter Zograf: Counting lattice points in moduli spaces of quadratic differentials International Mathematical Union 2022-07-14 | We show how to count lattice points represented by square-tiled surfaces in the moduli spaces of meromorphic quadratic differentials with simple poles on complex algebraic curves. We demonstrate the versatility of the lattice point count on three different examples, including evaluation of Masur-Veech volumes of the moduli spaces of quadratic differentials, computation of asymptotic frequencies of geodesic multicurves on hyperbolic surfaces, and asymptotic enumeration of meanders with a fixed number of minimal arcs.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/76-Peter%20Zograf.pdf
Zhifei Zhang: Hydrodynamic stability at high Reynolds number and Transition threshold problem International Mathematical Union 2022-07-14 | The hydrodynamic stability theory is mainly concerned with how the laminar flows become unstable and transit to turbulence at high Reynolds number. To shed some light on the transition mechanism, Trefethen et al. [Science 261(1993)] proposed the transition threshold problem. Many physical effects such as 3-D lift-up, inviscid damping, enhanced dissipation and boundary layer, play a crucial role in determining the transition threshold. In this talk, I will survey some important progress on linear inviscid damping and enhanced dissipation for shear flows. I will outline key ideas and main ingredients in our proof of transition threshold for 3-D Couette flow in a finite channel.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/79-Zhifei%20Zhang.pdf
Ye Tian: The arithmetic of quadratic twists of elliptic curves International Mathematical Union 2022-07-14 | We discuss the behavior of Selmer groups and L-values of elliptic curves under quadratic twists. The congruent number problem is a basic example in this topic. A brief overview of some progress towards the BSD conjecture will also be discussed.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/80-Ye%20Tian.pdf
Dmitriy Zhuk: Constraint Satisfaction Problem: what makes the problem easy International Mathematical Union 2022-07-14 | Many combinatorial problems, such as graph coloring or solving linear equations, can be expressed as the constraint satisfaction problem for some constraint language. In the talk we first discuss a proof of a famous conjecture stating that for any constraint language the problem is either solvable in polynomial time, or NP-complete. Then we consider other variants of this problem whose complexity is still not known. For instance, we could allow both universal and existential quantifiers, or require the input or the solution to satisfy an additional condition.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/78-Dmitriy%20Zhuk.pdf
Ronen Eldan: Revealing the simplicity of high-dimensional objects via pathwise analysis International Mathematical Union 2022-07-14 | ...
Lu Wang: Entropy in mean curvature flow International Mathematical Union 2022-07-14 | The entropy of a hypersurface is defined by the supremum over all Gaussian integrals with varying centers and scales, thus invariant under rigid motions and dilations. It measures geometric complexity and is motivated by the study of mean curvature flow. We will survey recent progress on conjectures of Colding—Ilmanen—Minicozzi—White concerning the sharp lower bound on entropy for hypersurfaces, as well as their extensions.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/70-Wang%20Lu.pdf
Joel Kamnitzer: Perfect bases in representation theory: three mountains and their springs International Mathematical Union 2022-07-14 | In order to give a combinatorial descriptions of tensor product multiplicites for semisimple groups, it is useful to find bases for representations which are compatible with the actions of Chevalley generators of the Lie algebra. There are three known examples of such bases, each of which flows from geometric or algebraic mountain. Remarkably, each mountain gives the same combinatorial shadow: the crystal B(∞) and the Mirković–Vilonen polytopes. In order to distinguish between the three bases, we introduce measures supported on these polytopes. We also report on the interaction of these bases with the cluster structure on the coordinate ring of the maximal unipotent subgroup.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/73-Joel%20Kamnitzer.pdf
Isabella Novik: Face numbers: the upper bound side of the story International Mathematical Union 2022-07-14 | What is the largest number of i-dimensional faces that a simplicial polytope of dimension d with n vertices can have? What about the largest possible number of i-dimensional faces that a triangulation of a (d - 1)-sphere can have? What are the maximizers? How common or rare are they? How do the answers change if the object in question must be centrally symmetric (i.e., endowed with a free action of Z/2Z)? This talk will be a survey of what is known and what is still unknown in this fascinating field, with an emphasis on some recent developments.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/64-Isabella%20Novik.pdf
Jennifer Tour Chayes: Graphons and Graphexes as Limits and Models of Large Sparse Graphs International Mathematical Union 2022-07-14 | ...
Guozhen Wang, Zhouli Xu: Stable homotopy groups of spheres and motivic homotopy theory International Mathematical Union 2022-07-14 | The computation of stable homotopy groups of spheres is one the most fundamental problems in topology. It has connections to many topics in topology, such as cobordism theory and the classification of smooth structures on spheres. In this talk, we will survey some classical methods, explain their difficulty via Mahowald’s Uncertainty Principles, and describe a new technique using motivic homotopy theory. This new technique yields streamlined computations through previously known range, and gives new computations through dimension 90. We will also discuss questions and conjectures for future study.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/63-Zhouli%20Xu.pdf
Amir Mohammadi: Finitary analysis in homogeneous spaces International Mathematical Union 2022-07-14 | ...
Tamas Hausel: Enhanced mirror symmetry for Langlands dual Hitchin systems International Mathematical Union 2022-07-14 | ...
Benoit Collins: Weingarten calculus and its applications International Mathematical Union 2022-07-14 | A fundamental property of compact groups and compact quantum groups is the existence and uniqueness of a left and right invariant probability - the Haar measure. This is a very natural playground for classical and quantum probability, provided that it is possible to compute its moments. Weingarten calculus addresses this question in a systematic way. The purpose of this talk is to survey recent developments, describe some salient theoretical properties of Weingarten functions, as well as applications of this calculus to random matrix theory, quantum probability and algebra, mathematical physics, and operator algebras.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/59-Benoit%20Collins.pdf
Alessandro Giuliani: Scaling limits and universality of Ising and dimer models International Mathematical Union 2022-07-14 | After having introduced the notion of universality in statistical mechanics and its importance for our comprehension of the macroscopic behavior of interacting systems, I will review recent progress in the understanding of the scaling limit of lattice critical models, including a quantitative characterization of the limiting distribution and the robustness of the limit under perturbations of the microscopic Hamiltonian. I will focus on results obtained for two classes of non-exactly-solvable two- dimensional systems: non-planar Ising models and interacting dimers. Based on joint works with Giovanni Antinucci, Rafael Greenblatt, Vieri Mastropietro, Fabio Toninelli.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/58-Alessandro%20Giuliani.pdf
Rupert Frank: Lieb-Thirring Inequalities: What we know and what we want to know International Mathematical Union 2022-07-14 | Lieb-Thirring inequalities are functional inequalities that generalize Sobolev inequalities and that have proved to be powerful tools in several questions from mathematical physics, PDEs and functional analysis. Recently, in the spirit of Lieb-Thirring inequalities, certain inequalities in harmonic analysis were extended to the setting of orthonormal functions. In this talk we give a gentle introduction to classical aspects of the subject, some recent progress and some open problems.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/66-Rupert%20Frank.pdf
Keita Yokoyama: Reverse mathematics from multiple points of view International Mathematical Union 2022-07-14 | ...
Nicholas J. Higham: Numerical Stability of Algorithms at Extreme Scale and Low Precisions International Mathematical Union 2022-07-14 | As computer architectures evolve and the exascale era approaches, we are solving larger and larger problems. At the same time, much modern hardware provides floating-point arithmetic in half, single, and double precision formats, and to make the most of the hardware we need to exploit the different precisions. How large can we take the dimension n in matrix computations and still obtain solutions of acceptable accuracy? Standard rounding error bounds are proportional to p (n) u , with p growing at least linearly with n. We are at the stage where these rounding error bounds are not able to guarantee any accuracy or stability in the computed results for extreme-scale or low-accuracy computations. We explain how rounding error bounds with much smaller constants can be obtained. The key ideas are to exploit the use of blocked algorithms, which break the data into blocks of size b and lead to a reduction in the error constants by a factor b or more; to take account of architectural features such as extended precision registers and fused multiply–add operations; and to carry out probabilistic rounding error analysis, which provides error constants that are the square roots of those of the worst-case bounds. Combining these different considerations provides new understanding of the limits of what we can compute at extreme scale and low precision in numerical linear algebra.Slides: mathunion.org/fileadmin/IMU/ICM2022/Presentation-slides/61-Nick%20Higham.pdf
