Resumo: This article is a broad-brush survey of two areas in differential geometry. While these two areas are not usually put side-by-side 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 differential-geometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a Kähler-Einstein 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 far-ranging results. By contrast, the theory of manifolds of exceptional holonomy is a wide-open 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.

Resumo: 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 "mean-field" 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.

Resumo: 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, Faber-Zagier, 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.

Resumo: 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 30-page 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.

Resumo: 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 (self-avoiding and loop-erased walk, Brownian loop measures, Schramm–Loewner evolution (SLE) and SLE-type loops, Gaussian free field, Liouville quantum gravity). I will also describe some analogous problems in other spatial dimensions.

Resumo: 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.

Resumo: The theory of curvature-dimension 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 non-Riemannian 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.

Resumo: I will discuss a number of results taken from a cross-section 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 shear-induced chaos. From there I will proceed to some applications of dynamical systems ideas, to epidemics control and computational neuroscience.

Resumo: We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de 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.

Resumo: 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.

Resumo: Over the past four decades, input from geometry and analysis has been central to progress in the field of low-dimensional topology. This talk will focus on one aspect of these developments, namely the use of Yang-Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds 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 3-dimensional 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 less-explored world of spatial graphs.

Resumo: Over the past four decades, input from geometry and analysis has been central to progress in the field of low-dimensional topology. This talk will focus on one aspect of these developments, namely the use of Yang-Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds 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 3-dimensional 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 less-explored world of spatial graphs.

Resumo: 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 term-scale, 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 long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions are proposed.

Resumo: 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

Resumo: 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.

Resumo: Machine learning is the sub-field 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.

Resumo: 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.

Resumo: 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.

Resumo: Geometric aspects play an important role in the construction and analysis of structure-preserving 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 low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.

Resumo: 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.

Resumo: 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 gradient-based optimization from a continuous-time, variational point of view. We go beyond classical gradient flow to focus on second-order dynamics, aiming to show the relevance of such dynamics to optimization algorithms that not only converge, but converge quickly.

Resumo: 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 G-shtukas with multiple modifications and the geometric Satake equivalence.