Plenary Speakers

Isabelle Gallagher

École Normale Supérieure de Paris

On the dynamics of dilute gases

Abstract. The evolution of a gas can be described by different models depending on the scale of observation. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions.... In the case of gases of hard spheres, Lanford showed in 1974 that the Boltzmann equation appears as a law of large numbers in the low density limit, at least for very short times. In this talk we will present Lanford's result, and some recent extensions to understand fluctuations and large deviations around the Boltzmann equation. Read more

Trevor Hastie

Stanford University

Statistical Learning with Sparsity

Abstract. In a statistical world faced with an explosion of data, regularization has become an important ingredient. Often data are "wide" - we have many more variables than observations - and the lasso penalty and its hybrids have become... increasingly useful. This talk presents a general framework for fitting large scale regularization paths for a variety of problems. We describe the approach, and demonstrate it via examples using our R package GLMNET. We then outline a series of related problems using extensions of these ideas.
* Joint work with Jerome Friedman, Robert Tibshirani, and many students, past and present. Read more

Greg Moore

Rutgers University

Quantum Field Theory And Invariants Of Smooth Four-Dimensional Manifolds

Abstract. The talk will begin with some very general remarks on the topic of "physical mathematics."
Quantum Field Theory offers an interesting perspective on many topological and geometric invariants in mathematics. In general, the path integrals of QFT have not been defined with full mathematical rigor.... Nevertheless, there are well-defined rules for manipulating them, and insights from physics can lead to well-defined, and, sometimes, new and unexpected predictions about concrete mathematical quantities. These predictions are statements amenable to rigorous analysis. This talk focuses on the renowned example of Donaldson and Seiberg-Witten invariants in the theory of four-manifolds. In this case physics predicts an interesting equality between a path integral of an "ultraviolet" (UV) and an "infrared" (IR) quantum field theory. The physical prediction is that these two path integrals are exactly equal. The path integral of the UV theory localizes to an integral over the (finite-dimensional) moduli space of anti-self-dual instantons on a principal G bundle over an oriented, compact four-manifold X without boundary.
The path integral of the IR theory localizes to an integral over the (finite-dimensional, smooth, and compact) moduli space of solutions to the Seiberg-Witten equations on X.
The physical relation between the UV and IR theories allows one to derive the relation between Donaldson and Seiberg-Witten invariants for manifolds with b₂⁺>0. They key to the analysis is a finite-dimensional integral called the "u-plane integral." The u-plane integral was analyzed, precisely for this purpose, by Moore and Witten in 1997. After reviewing that basic example the talk will - time permitting - describe several recent developments. Some of the recent developments which might be described are:
1. Deeper understanding of the role of mock modular forms,
2. Generalizations that interpolate between Donaldson and Vafa-Witten invariants,
3. Extensions to 5-dimensional super-Yang-Mills and the so-called "K-theoretic Donaldson invariants," and
4. Extensions to families of four-manifolds.​ Read more

Zeev Rudnick

Tel Aviv University

A number theorist's adventures in the land of spectral theory

Abstract. I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections between random matrix theory, the zeros of the Riemann zeta function, and spectral statistics on the moduli space of hyperbolic surfaces.​​​

Claire Voisin

Collège de France

On the complex cobordism classes of hyper-Kähler manifolds

Abstract. Hyper-Kähler manifolds are symplectic holomorphic compact Kähler manifolds, a particular class of complex manifolds with trivial canonical bundle. They exist only in even complex dimension..., and there are two main series of known deformation classes of hyper-Kähler manifolds, with one model in each even dimension, that I will describe. I will discuss in this introductory talk a result obtained with Georg Oberdieck and Jieao Song on the complex cobordism classes of hyper-Kähler manifolds, and present a number of open questions concerning their Chern numbers.​ Read more

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Morning Speakers

Natalia Bochkina

Edinburgh University

Bernstein - von Mises theorem for regular and non-regular statistical models

Abstract. It is well known that the sum of a large number of independent identically distributed random variables with finite variance, appropriately recentred and rescaled, converges to a normal distribution (known as the Central Limit Theorem). This result underpins asymptotic classical statistical inference, implying, for instance that the estimator of an unknown parameter that maximises... the likelihood (distribution of data given parameter) asymptotically has normal distribution. A similar result also holds for Bayesian inference that combines likelihood with prior information about the unknown parameter into the posterior distribution of the parameter given observed data. For a large sample size, the posterior distribution is also approximately normal: this is known as the Bernstein-von Mises theorem. In particular, this result implies that for a large sample size, Bayesian inference is asymptotically equivalent to the classical statistical inference.
The original Bernstein-von Mises theorem applies to regular correctly specified statistical models with a finite dimensional parameter. There have been many extensions to other types of models such as high dimensional and misspecified models. I will also discuss the case a statistical model is nonregular, such as a linear regression model with one-sided errors, where the limiting posterior distribution and the rate of convergence differ from those under a regular model. I will also show our latest results for a linear regression model with one-sided errors where the distribution of the errors is unknown. These results will be illustrated on modelling procurement auctions. Read more

Ana Caraiani

Imperial College

Reciprocity laws and torsion classes

Abstract. At the heart of the Langlands programme lies reciprocity, the conjectural relationship between Galois representations and modular, or automorphic forms. A famous instance of this is the modularity of elliptic curves over the rational numbers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some recent progress in the Langlands programme, with a focus on new reciprocity laws over imaginary quadratic fields.​​​

Vittoria Colizza

INSERM & Sorbonne Université

Big Data, models, and COVID-19 policies

Abstract. Years of preparedness and scientific progress have been harshly put at test in the most difficult health crisis of the last 100 years. Following the path of our experience in the first two years of the COVID-19 pandemic in France, I will discuss the pitfalls, challenges, and opportunities to improve the science informing COVID-19 policies.​​​

Jessica Fintzen

Duke & Cambridge University

Representations of p-adic groups

Abstract. The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with a focus on recent developments. I will also briefly discuss applications to other areas.​​​

Andrea Montanari

Stanford University

Mathematical problems in modern machine learning

Abstract. The last fifteen years have witnessed dramatic advances in machine learning. This progress was mainly driven by engineering advances: greater computing power, new programming abstractions, and larger availability of training data. Not only the collection of methods that emerged from this revolution are not well understood mathematically, but they actually appear to defy traditional mathematical theories of machine learning. I will focus on generalization, namely the desirable property that a model learnt on a certain set of data can be used to capture the behavior of yet unseen data.... Traditionally, this is viewed as a consequence of uniform convergence: when the training sample is large enough as compared to the model complexity, the training error concentrates uniformly around the test error over all the models in the class.
Deep learning models are ofter trained in a regime that is forbidden by this theory: the model complexity is often larger than the sample size and the train error does not concentrate around the test error. I will discuss some simple models in which this phenomenon can be rigorously understood and studied in detail. This analysis will require mathematical tools (random matrix theory, Gaussian processes inequalities) that are substantially different from the ones of more traditional learning theory. Read more

George Pappas

Michigan State University

Number theory and low dimensional topology

Abstract. There is a beautiful classical analogy between the theory of numbers and 3-dimensional topology in which prime numbers correspond to knots in the 3-sphere. For example, under this analogy, the Iwasawa characteristic power series for the tower of cyclotomic fields of prime power roots of unity corresponds to the Alexander polynomial of a knot. Recently, after an input of ideas from mathematical physics, the analogy has been expanded to include new objects and constructions. I will describe some of these developments and discuss certain natural questions about these constructions.

Maria Reguera

Birmingham University

Sparse domination on and beyond classical harmonic analysis.

Abstract. In this talk we discuss the domination of classical operators by much simpler averaging operators known as sparse operators. Sparse operators were first discovered by Andrei Lerner. We will see how these sparse operators appear associated to classical Calderon-Zygmund operators, square functions, even certain Fourier multiplier operators. They also appear in less expected contexts like associated to the Bergman projection on spaces of complex analytic functions. We will have a look at these and other recent results on sparse domination.

Ingo Runkel

Universität Hamburg

Topological field theory and higher categories

Abstract. One way to think about topological field theories is as a bridge between topology and geometry on the one side and algebra on the other side. One is led to consider higher categories in algebra, and one can use ideas from topology to get a new point of view on algebraic structures. In this talk I would like to give a first glimpse at this bridge and illustrate it with some examples from algebra and an outlook into mathematical physics.

Richard Samworth

University of Cambridge

Nonparametric inference under shape constraints

Abstract. Traditionally, we think of statistical methods as being divided into parametric approaches, which can be restrictive, but where estimation is typically straightforward (e.g. using maximum likelihood), and nonparametric methods, which are more flexible but often require careful choices of tuning parameters. The area of nonparametric inference under shape constraints sits somewhere in the middle, seeking in some ways the best of both worlds. I will give an introduction to this currently very active area, providing some history, recent developments and a future outlook.

Sakura Schafer Nameki

University of Oxford

Generalized Symmetries in QFT and Strings

Abstract. I will give an overview of recent progress on generalized symmetries (higher-form, higher-group, and more general categorical symmetries) in the context of Quantum Field Theories (QFTs). In instances when QFTs are strongly coupled and thus not accessible using perturbative descriptions it is paramount to realize them within String Theory. I will provide a string theoretic imprint of these symmetries, in particular in the context of compactifications on canonical three-fold singularities in M-theory.

Kristian Seip

Norwegian University of Science and Technology

Contractive norm inequalities

Abstract. Contractive inequalities involving norms of H^p spaces can be particularly useful when the objects at hand (like the norms and/or an underlying operator) lift in a multiplicative way from one (or few) to several (or infinitely many) variables. Another motivation for identifying such norm inequalities comes from a conjecture of Lieb and Solovej about Wehrl-type entropy inequalities. This conjecture was recently settled by Kulikov who thus established a remarkable extremal property of reproducing kernels in Hardy and Bergman spaces. I will give a survey on this and other contractive norm inequalities established recently; we will see interesting phenomena both in the transition from low to high dimension and from low to infinite dimension.

Karen Vogtmann

University of Warwick

Outer spaces

Abstract. The simplest infinite groups tend to have the most complicated - and interesting! - automorphism groups. Inner automorphisms are usually well understood, so it is natural to focus instead on the outer automorphism groups Out(G)=Aut(G)/Inn(G). I will explain how to construct “Outer spaces” whose symmetries capture Out(G), so that we may use geometric methods to study the algebraic structure of Out(G). I will illustrate with examples, some classical and some very new.