Workshop

The aim of the talks will be to present the main results on the geometry of isotropic convex bodies in connection with a number of well-known open questions regarding the distribution of volume in high-dimensional convex bodies, such as the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture.

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This series of lectures will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in geometric functional analysis and data science, in particular to covariance estimation, signal recovery, and sparse regression. 

Mixing properties of dynamical systems are a central topic of study in mathematics. 

In the lectures we will take a modern, probabilistic, short horizon quantitative view of this problem and see that it involves a number of areas in modern mathematics including isoperimetric theory, noise stability and additive combinatorics. We will explain where are we in a quest for a unified theory as well as some of the applications of this theory. 

A fundamental theorem of Hadwiger classifies all rigid-motion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusion-exclusion principle, $$\operatorname{Z}(K)+ \operatorname{Z}(L) =\operatorname{Z}(K\cup L) +\operatorname{Z}(K\cap L)$$ for convex bodies $K$ and $L$ such that $K\cup L$ is convex. Under weak additional assumptions, such a functional $\operatorname{Z}$ is a finitely additive measure and hence Hadwiger's theorem is a counterpart to the classification of Haar measures. Hadwiger's theorem characterizes the most important functionals within Euclidean geometry, the $n+1$ intrinsic volumes, which include volume, surface area, and the Euler characteristic. In recent years, numerous further functions and operators defined on the space of convex bodies and more generally on function spaces were characterized by their properties. An overview of these results will be given:\\ \hspace*{16pt}\parbox{16pt}{(i)} Real and tensor valuations\\ \hspace*{16pt}\parbox{16pt}{(ii)} Minkowski valuation\\ \hspace*{16pt}\parbox{16pt}{(iii)} Valuations on function spaces.\\ The focus is on valuations that intertwine the SL$(n)$ and on connections to geometric functional analysis and analytic inequalities.

A fundamental theorem of Hadwiger classifies all rigid-motion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusion-exclusion principle, $$\operatorname{Z}(K)+ \operatorname{Z}(L) =\operatorname{Z}(K\cup L) +\operatorname{Z}(K\cap L)$$ for convex bodies $K$ and $L$ such that $K\cup L$ is convex. Under weak additional assumptions, such a functional $\operatorname{Z}$ is a finitely additive measure and hence Hadwiger's theorem is a counterpart to the classification of Haar measures. Hadwiger's theorem characterizes the most important functionals within Euclidean geometry, the $n+1$ intrinsic volumes, which include volume, surface area, and the Euler characteristic. In recent years, numerous further functions and operators defined on the space of convex bodies and more generally on function spaces were characterized by their properties. An overview of these results will be given:\\ \hspace*{16pt}\parbox{16pt}{(i)} Real and tensor valuations\\ \hspace*{16pt}\parbox{16pt}{(ii)} Minkowski valuation\\ \hspace*{16pt}\parbox{16pt}{(iii)} Valuations on function spaces.\\ The focus is on valuations that intertwine the SL$(n)$ and on connections to geometric functional analysis and analytic inequalities.

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This series of lectures will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in geometric functional analysis and data science, in particular to covariance estimation, signal recovery, and sparse regression. 

We will discuss fundamentals of Brunn Minkowski theory via the functional versions of basic geometric notions and inequalities.

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This series of lectures will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in geometric functional analysis and data science, in particular to covariance estimation, signal recovery, and sparse regression. 

A fundamental theorem of Hadwiger classifies all rigid-motion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusion-exclusion principle, $$\operatorname{Z}(K)+ \operatorname{Z}(L) =\operatorname{Z}(K\cup L) +\operatorname{Z}(K\cap L)$$ for convex bodies $K$ and $L$ such that $K\cup L$ is convex. Under weak additional assumptions, such a functional $\operatorname{Z}$ is a finitely additive measure and hence Hadwiger's theorem is a counterpart to the classification of Haar measures. Hadwiger's theorem characterizes the most important functionals within Euclidean geometry, the $n+1$ intrinsic volumes, which include volume, surface area, and the Euler characteristic. In recent years, numerous further functions and operators defined on the space of convex bodies and more generally on function spaces were characterized by their properties. An overview of these results will be given:\\ \hspace*{16pt}\parbox{16pt}{(i)} Real and tensor valuations\\ \hspace*{16pt}\parbox{16pt}{(ii)} Minkowski valuation\\ \hspace*{16pt}\parbox{16pt}{(iii)} Valuations on function spaces.\\ The focus is on valuations that intertwine the SL$(n)$ and on connections to geometric functional analysis and analytic inequalities

It has long been known that Gaussian measures possess unique convexity properties within the class of log-concave measures. In particular, a remarkable sharp form of Gaussian convexity was discovered by A. Ehrhard in the early 1980s, but has mostly remained somewhat of a beautiful curiosity. In recent work, however, this inequality, the theory surrounding it, and its utility in applications have become significantly better understood. My aim in these talks is to review and discuss in some detail several recent developments surrounding this theory and its applications to Gaussian concentration phenomena (by us as well as by other authors). I will also highlight some key mysteries that remain.