Aug 02 (Thu) 
Speaker/Title 
8:309:30 Plenary Lecture 1 Chair: Eric Friedlander 
Simon Donaldson (UK/USA) Some recent developments in Kähler geometry and exceptional holonomy
Abstract: This article is a broadbrush survey of two areas in differential geometry. While these two areas are not usually put sidebyside in this way, there are several reasons for discussing them together. First, they both fit into a very general pattern, where one asks about the existence of various differentialgeometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a KählerEinstein metric. In the other we seek a metric of exceptional holonomy on a manifold of dimension $7$ or $8$. Second, as we shall see in more detail below, there are numerous points of contact between these areas at a technical level. Third, there is a pleasant contrast between the state of development in the fields. These questions in Kähler geometry have been studied for more than half a century: there is a huge literature with many deep and farranging results. By contrast, the theory of manifolds of exceptional holonomy is a wideopen field: very little is known in the way of general results and the developments so far have focused on examples. In many cases these examples depend on advances in Kähler geometry.

9:4010:40 Plenary Lecture 2 Chair: Cédric Villani 
Sylvia Serfaty (France/USA) Systems of points with Coulomb interactions
Abstract: Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the "meanfield" derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

10:5011:50 Plenary Lecture 3 Chair: Shigefumi Mori 
Rahul Pandharipande (Switzerland) Geometry of the moduli space of curves
Abstract: The moduli space of curves, first appearing in the work of Riemann in the 19th century, plays an important role in geometry. After an introduction to the moduli space, I will discuss recent directions in the study of tautological classes on the moduli space following ideas and conjectures of Mumford, FaberZagier, and Pixton. Cohomological Field Theories (CohFTs) play an important role. The talk is about the search for a cohomology calculus for the moduli space of curves parallel to what is known for better understood geometries. My goal is to give a presentation of the progress in the past decade and the current state of the field.

Aug 03 (Fri) 
Speaker/Title 
8:309:30 Plenary Lecture 4 Chair: Kenneth Ribet 
Andrei Okounkov (Russia/USA) On the crossroads of enumerative geometry and geometric representation theory
Abstract: The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example – that of $Hilb(\mathbb{C}^2,n)$, hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.

9:4010:40 Plenary Lecture 5 Chair: Stanislav Smirnov 
Gregory Lawler (USA) Conformally Invariant Measures on Paths and Loops
Abstract: There has been incredible progress in the last twenty years in the study of fractal paths and fields that arise in planar statistical physics. I will give an introduction to the area and discuss some recent results, focusing on some of the main characters (selfavoiding and looperased walk, Brownian loop measures, Schramm–Loewner evolution (SLE) and SLEtype loops, Gaussian free field, Liouville quantum gravity). I will also describe some analogous problems in other spatial dimensions.

10:5011:50 Plenary Lecture 6 Chair: Pierre Arnoux 
Carlos Gustavo Moreira (Brazil) Dynamical systems, fractal geometry and diophantine approximations
Abstract: We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra and generalizations in Dynamical Systems and Differential Geometry.

Aug 04 (Sat) 
Speaker/Title 
8:309:30 Plenary Lecture 7 Chair: Carlos Kenig 
Luigi Ambrosio (Italy) Calculus, heat flow and curvaturedimension bounds in metric measure spaces
Abstract: The theory of curvaturedimension bounds for metric measure structure has several motivations: the study of functional and geometric inequalities in structures which are very far from being Euclidean, therefore with new nonRiemannian tools, the description of the “closure” of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces. In the last few years, with crucial inputs coming from the theory of optimal mass transportation, we have seen a spectacular progress in all these directions, which also stimulated the development of new calculus tools in metric measure spaces. The lecture is meant both as a survey and as an introduction to this quickly developing research field.

9:4010:40 Plenary Lecture 8 Chair: Enrique Pujals 
LaiSang Young (USA) Dynamical systems evolving
Abstract: I will discuss a number of results taken from a crosssection of my work in Dynamical Systems theory and applications. The first topics are from the ergodic theory of chaotic dynamical systems. They include relation between entropy, Lyapunov exponents and fractal dimension, statistical properties and geometry, physically relevant invariant measures, and strange attractors arising from shearinduced chaos. From there I will proceed to some applications of dynamical systems ideas, to epidemics control and computational neuroscience.

10:5011:50 Plenary Lecture 9 Chair: Michael Rapoport 
Peter Scholze (Germany) Period maps in $p$adic geometry
Abstract: We discuss recent developments in $p$adic geometry, ranging from foundational results such as the degeneration of the Hodgetode Rham spectral sequence for “compact $p$adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of “universal” $p$adic cohomology theories. We finish with some speculations on how a theory that combines all primes $p$, including the archimedean prime, might look like.

Aug 06 (Mon) 
Speaker/Title 
8:309:30 Plenary Lecture 10 Chair: Ingrid Daubechies 
Ronald Coifman (USA) Harmonic analytic geometry on subsets in high dimensions – Empirical models
Abstract: We describe a recent evolution of Harmonic Analysis to generate analytic tools for the joint organization of the geometry of subsets of $\mathbb{R}^n$ and the analysis of functions and operators on the subsets. In this analysis we establish a duality between the geometry of functions and the geometry of the space. The methods are used to automate various analytic organizations, as well as to enable informative data analysis. These tools extend to higher order tensors, to combine dynamic analysis of changing structures. In particular we view these tools as necessary to enable automated empirical modeling, in which the goal is to model dynamics in nature, ab initio, through observations alone. We will illustrate recent developments in which physical models can be discovered and modelled directly from observations, in which the conventional Newtonian differential equations, are replaced by observed geometric data constraints. This work represents an extended global collaboration including, recently, A. Averbuch, A. Singer, Y. Kevrekidis, R. Talmon, M. Gavish, W. Leeb, J. Ankenman, G. Mishne and many more.

9:4010:40 Plenary Lecture 11 Chair: Paolo Piccione 
Peter Kronheimer (USA) and Tomasz Mrowka (USA) Knots, threemanifolds and instantons
Abstract: Over the past four decades, input from geometry and analysis has been central to progress in the field of lowdimensional topology. This talk will focus on one aspect of these developments, namely the use of YangMills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4manifolds beginning in 1982, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3dimensional topology and knot theory. In our exploration of this subject, a recurring question will be, "How can we detect knottedness?" Many mathematical techniques have found application to this question, but gauge theory in particular has provided its own collection of answers, both directly and through its connection with other tools. Beyond classical knots, we will also take a look at the nearby but lessexplored world of spatial graphs.

10:5011:50 Plenary Lecture 12 Chair: June BarrowGreen 
Catherine Goldstein (France) Longterm history and ephemeral configurations
Abstract: Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathematics has tended to focus on local topics, over a short termscale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased attention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a longterm history are illustrated and tested using a number of episodes in the nineteenthcentury history of Hermitian forms, and finally, some open questions are proposed.

Aug 07 (Tue) 
Speaker/Title 
8:309:30 Plenary Lecture 13 Chair: Robert Bryant 
Alexander Lubotzky (Israel) High dimensional expanders
Abstract: Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways. In the last decade, a theory of “high dimensional expanders” has begun to emerge. We will describe some paths of this new area of study.:w

9:4010:40 Plenary Lecture 14 Chair: Artur Avila 
Nalini Anantharaman (France) Delocalization of Schrödinger eigenfunctions
Abstract: A hundred years ago, Einstein wondered about quantization conditions for classically ergodic systems. Although a mathematical description of the spectrum of Schrödinger operators associated to ergodic classical dynamics is still completely missing, a lot of progress has been made on the delocalization of the associated eigenfunctions.

10:5011:50 Plenary Lecture 15 Chair: François Loeser 
Sanjeev Arora (USA) The mathematics of machine learning and deep learning
Abstract: Machine learning is the subfield of computer science concerned with creating programs and machines that can improve from experience and interaction. It relies upon mathematical optimization, statistics, and algorithm design. The talk will be an introduction to machine learning, and to its popular subarea, deep learning. Empirical success here currently outstrips mathematical understanding. We will survey some of the myriad mathematical mysteries of this field, and initial progress made towards understanding them.

Aug 08 (Wed) 
Speaker/Title 
8:309:30 Plenary Lecture 16 Chair: Roberto Imbuzeiro de Oliveira 
Assaf Naor (USA) Metric dimension reduction: A snapshot of the Ribe program
Abstract: The purpose of this article is to survey some of the context, achievements, challenges and mysteries of the field of metric dimension reduction, including new perspectives on major older results as well as recent advances.

9:4010:40 Plenary Lecture 17 Chair: Yuri Tschinkel 
Geordie Williamson (Australia/Germany) Representation theory and geometry
Abstract: One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theory of Lie groups and algebras. A recurring theme is the appearance of geometric techniques in seemingly algebraic problems.

10:5011:50 Plenary Lecture 18 Chair: Rolf Jeltsch 
Christian Lubich (Germany) Dynamics, numerical analysis and some geometry
Abstract: Geometric aspects play an important role in the construction and analysis of structurepreserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical lowrank approximation of timedependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.

Aug 09 (Thu) 
Speaker/Title 
8:309:30 Plenary Lecture 19 Chair: Yoshiharu Kohayakawa 
Gil Kalai (Israel) Noise Stability, Noise Sensitivity and the Quantum Computer Puzzle
Abstract: I will talk about two related puzzles involving mathematics and computation. The first puzzle is about errors made when votes are counted during elections, and I will present a theory of noise stability and noise sensitivity of voting rules and other processes. The second puzzle is: are quantum computers possible? I will discuss the sensitivity of noisy intermediate scale quantum (NISQ) systems and provide an argument for why quantum computers are not possible.

9:4010:40 Plenary Lecture 20 Chair: Helge Holden 
Michael Jordan (USA) Dynamical, symplectic and stochastic perspectives on gradientbased optimization
Abstract: Our topic is the relationship between dynamical systems and optimization. This is a venerable, vast area in mathematics, counting among its many historical threads the study of gradient flow and the variational perspective on mechanics. We aim to build some new connections in this general area, studying aspects of gradientbased optimization from a continuoustime, variational point of view. We go beyond classical gradient flow to focus on secondorder dynamics, aiming to show the relevance of such dynamics to optimization algorithms that not only converge, but converge quickly.

10:5011:50 Plenary Lecture 21 Chair: Alicia Dickenstein 
Vincent Lafforgue (France) Global Langlands parameterization and shtukas for reductive groups.
Abstract: We discuss recent developments in the Langlands program for function fields. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G over a function field, indexed by global Langlands parameters. The proof uses the cohomology of Gshtukas with multiple modifications and the geometric Satake equivalence.
