Minicourses
A convex body $K$ in ${\mathbb R}^n$ is called isotropic if it has volume one, center of mass at the origin, and there exists a constant $L_K>0$ such that $\int_K\langle y,\theta\rangle^2dy =L_K^2$ for every $\theta\in S^{n-1}$. The well--known slicing problem asks if there exists an absolute constant $C>0$ such that $L_K\leq C$ for every isotropic convex body in any dimension. We will describe recent results of Klartag, Paouris and others about the distribution of volume on isotropic convex bodies. The plan of the talks will be the following:
- Overview and the unconditional model
- $L_q$-centroid bodies
- Concentration of volume, subgaussian estimates and mean width
- Random points and isotropic constant of random polytopes
I'll survey the proof of the celebrated theorem of Dvoretzky stating that symmetric convex bodies have central sections, of dimension tending to infinity with the dimension of the body, which are arbitrarily close to Euclidean balls. I'll also talk on some relatively recent developments regarding the evaluation of the dimensions of these Euclidean balls. I may also speak on the existence of other high dimensional nice sections of convex bodies.
We start by presenting a selection of largely classical results about random matrices. This will involve various ensembles (classical groups, Gaussian, Wigner etc.) and various regimes (global and local). We will exemplify differences between limit results and fixed dimension results, as well as between large deviation, small ball, and concentration estimates.
Next, we will sketch proofs of some of those results. This will be mostly based on isoperimetric inequalities and the theory of Gaussian processes, but we may touch on free probability or transport of measures. We will also include some applications to functional analysis, mostly via Random Banach spaces.
Finally, we will discuss some topics concerning pseudo-random matrices, i.e., matrices that ``look like" random ones, but are (in the appropriate sense) explicit. This will entail links to properties $T$ and $\tau$ for groups, and to expander graphs.
With time, more detailed versions of this summary should be (hopefully) available at
the URL
[http://www.cwru.edu/artsci/math/szarek/abstracts/Bedlewo.pdf]
Random spaces were introduced in the Banach space theory in 1981 by E. Gluskin who used them to show that there exist two $n$-dimensional normed spaces whose Banach-Mazur distance is larger than or equal to $cn$ (where $c >0$ is an absolute constant). Thanks to subsequent work by several authors, including--besides Gluskin himself--P. Mankiewicz and S. Szarek, it very quickly became evident that these random spaces (whose unit balls were random linear images of the $\ell_1^N$ balls) satisfy many other extremal or nearly-extremal properties. We shall illustrate this phenomenon by theorem by Gluskin and Szarek on the basis constant.
Actually, every finite-dimensional (non-Euclidean) normed space carries in its structure seeds of some "pathological" behaviour. This was first noted by J. Bourgain (1987) and further developed by Mankiewicz and N. Tomczak-Jaegermann. This direction will be illustrated by the solution of finite-dimensional homogenous space problem.
Remaining time will be spent on discusssion of results from recent years on random polytopes of an arbitrary number of points and determined by different random distributions.
Invited talks
In connection with an unsolved problem of Bang (1951) we provide a lower bound for the sum of the base volumes of cylinders covering a given convex body in terms of the relevant basic measures of the body. As an application we establish a lower bound on the number of lines needed to cover all the integer points in the body.
We shall discuss some problems in Asymptotic Geometric Analysis concerning the study of samples of random variables with log-concave density and their interplay with random matrix theory.
Short Communications
I will present some recent results concerning random Toeplitz matrices, with emphasis on the behaviour of the operator norm. I will compare them with other, more classical models of random matrices and state some natural open problems arising in this context.
For every natural number $n\geq 3$ we construct an $n$-dimensional Banach space $E_n$ satisfying the following condition: there exists $\alpha>0$ such that for every linear subspace $L\subset E_n$ with codim$(L)=1$ there exists a projection from $E_n$ onto $L$ with the norm equal to $1+\alpha$, but there does not exist a projection from $E_n$ onto $L$ with the norm smaller than $1+\alpha$. This answers a question asked by S. Rolewicz in 1980's.
Let $K$ be a convex body; that is, a compact convex set in $\mathbb{R}^d$ with non–empty interior. We discuss problems of the following general form: given a series $K_1, K_2, \dots$ of smaller positive homothets of $K$ (that is, $K_i = \lambda_i K$, where $0< \lambda_i< 1$). What are sufficient or necessary conditions for the existence of a series of translations $t_1, t_2, \dots$ such that the family $K_1+t_1, K_2+t_2, \dots$ covers $K$. In this setting, one may measure the ``magnitude'' of the given series of homothets in terms of their total volume or the sum of their homothety ratios. Naturally, one may consider also the cardinality of the given series of homothets. The number of smaller positive homothetic copies of $K$ necessary to cover $K$ is called the illumination number of $K$. We introduce a fractional version of this quantity and present a weaker version of the Boltyanski--Hadwiger Conjecture (according to which, $2^d$ copies suffice). Finally, we discuss the relationship between the cardinality of homothetic copies covering $K$ and their homothety ratios.
I will discuss the problem of bounding the mean-width of a general isotropic convex body and report on some new observations in this direction.
Recent progress on study of cube-tilings and packings will be reported.
A new formula for the Demyanov's metric in the family of convex, compact subsets of Rn based on lexicographical order permits to extend this metric to arbitrary bounded, convex, not necessarily closed sets. Every bounded, convex set $A \subset R^n$ can be identified with a function whose domain is some subfamily of orthonormal systems of vectors $(e_1,\ldots,e_k)$ for $1 \le k \le n$ and the zero vector. The values correspond to lexicographical maxima in A with respect to these systems. This permitts to embed the space of bounded, convex sets with Demyanov's metric as a subspace of certain complete function space with the sup metric. The subspace, however, is not complete. A candidate for its closure within the whole space is known but this is a conjecture (proved only in $R^2$).
VC-dimension is a combinatorial notion introduced by Vapnik and Cervonenkis in 1971 for applications in probability theory. In geometric language, the VC-dimension of a family of shapes is a measure of how ably that family can express the distinctions among some chosen points. M\'arton Nasz\'odi has asked, in connection with the Boltyanski--Hadwiger illumination conjecture, whether the VC-dimension of the translates of a convex body is bounded (independently of the choice of body). We show that such a bound exists in the plane (it's 3), but not in higher dimensions.
I will show new concentration inequalities obtained for \ell_p balls. These inequalities generalize Talagrand's concentration for the exponential measure, and can be shown to be, in a sense, optimal. Also, I will show a concentration inequality stronger than Talagrand's for sets lying far away from the origin. The communication will be based on joint work with Rafał Latała.
The Christoffel problem asks for necessary and sufficient conditions for a given Borel measure on the sphere to be the first surface area measure of a convex body. The problem was solved in the late 1960's by Firey and Berg. We use Fourier transform techniques to present a new perspective on Berg's solution of Christoffel's problem.