Aug 21, 2017
Monday

09:00 AM  09:10 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:10 AM  10:10 AM


Geometry of isotropic convex bodies
Apostolos Giannopoulos (National and Kapodistrian University of Athens (University of Athens))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 wellknown open questions regarding the distribution of volume in highdimensional convex bodies, such as the slicing problem, the thin shell conjecture and the KannanLovászSimonovits conjecture.
 Supplements


10:10 AM  11:00 AM


Break  View the Eclipse

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Deviations of random matrices and applications
Roman Vershynin (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Mixing in product spaces
Elchanan Mossel (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Geometric classification
Monika Ludwig (Technische Universität Wien)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A fundamental theorem of Hadwiger classifies all rigidmotion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusionexclusion 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.
 Supplements




Aug 22, 2017
Tuesday

09:30 AM  10:30 AM


Gaussian convexity
Ramon Van Handel (Princeton University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
It has long been known that Gaussian measures possess unique convexity properties within the class of logconcave 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.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


BrunnMinkowski theory, functionalization of geometric notions and inequalities 1. Volume inequalities
Shiri Artstein (Tel Aviv University)

 Location
 MSRI: Simons Auditorium
 Video

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



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Mixing in product spaces
Elchanan Mossel (Massachusetts Institute of Technology)

 Location
 MSRI: Atrium
 Video

 Abstract
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.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Geometry of isotropic convex bodies
Apostolos Giannopoulos (National and Kapodistrian University of Athens (University of Athens))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 wellknown open questions regarding the distribution of volume in highdimensional convex bodies, such as the slicing problem, the thin shell conjecture and the KannanLovászSimonovits conjecture.
 Supplements

Notes
869 KB application/pdf


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Aug 23, 2017
Wednesday

09:30 AM  10:30 AM


Geometric classification
Monika Ludwig (Technische Universität Wien)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A fundamental theorem of Hadwiger classifies all rigidmotion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusionexclusion 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.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Deviations of random matrices and applications
Roman Vershynin (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements




Aug 24, 2017
Thursday

09:30 AM  10:30 AM


Mixing in product spaces
Elchanan Mossel (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Geometry of isotropic convex bodies
Apostolos Giannopoulos (National and Kapodistrian University of Athens (University of Athens))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 wellknown open questions regarding the distribution of volume in highdimensional convex bodies, such as the slicing problem, the thin shell conjecture and the KannanLovászSimonovits conjecture.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Gaussian convexity
Ramon Van Handel (Princeton University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
It has long been known that Gaussian measures possess unique convexity properties within the class of logconcave 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.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


BrunnMinkowski theory, functionalization of geometric notions and inequalities 2. Duality and Central symmetry
Shiri Artstein (Tel Aviv University)

 Location
 MSRI: Simons Auditorium
 Video

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




Aug 25, 2017
Friday

09:30 AM  10:30 AM


BrunnMinkowski theory, functionalization of geometric notions and inequalities 3. Covering numbers
Shiri Artstein (Tel Aviv University)

 Location
 MSRI: Simons Auditorium
 Video

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



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Deviations of random matrices and applications
Roman Vershynin (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Geometric classification
Monika Ludwig (Technische Universität Wien)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A fundamental theorem of Hadwiger classifies all rigidmotion invariant and continuous functionals on convex bodies (that is, compact convex sets) in ${\mathbb R}^n$ that satisfy the inclusionexclusion 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
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Gaussian convexity
Ramon Van Handel (Princeton University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
It has long been known that Gaussian measures possess unique convexity properties within the class of logconcave 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.
 Supplements



