An important geometric inequality relating the two fundamental notions of surface area and volume,  is the isoperimetric inequality. While volume is an affine invariant, surface area is not. This has led  geometers, e.g., Blaschke, to introduce a notion of affine surface area. We describe properties of the affine surface area, in particular its corresponding affine isoperimetric inequality which relates volume and affine surface area.

We  show that affine surface area  is intimately connected with a geometric construction, namely the floating body. We present the construction of the floating body and discuss its properties. The floating body is related to a famous problem, Ulam’s problem  from the Scottish Book asks:  Is a solid of uniform density which floats  in water in equilibrium  in every position necessarily a sphere?  This is an old problem going back to the 1930's and while there have been partial answers, in full generality the problem is still open. 

In recent years, extension of affine surface area to the more general $L_p$ affine surface areas have been established. Again, we discuss their properties, $L_p$ affine isoperimetric inequalities among them.A very recent result is the analog of the Steiner formula in the setting of $L_p$ affine surface area.

Abstract:

Mahler's Conjectures date back to the 1930's while Bourgain's Conjectures to the 1980's. They both speculate that the cube and simplex are minimizers of certain functionals related to volumes of convex bodies. While these problems are phrased purely in terms of (real) Euclidean space, in recent years it has been realized that they are deeply related to fundamental notions of several complex variables as well as complex geometry. In these lectures I will attempt to describe key ideas from this beautiful story spanning over 80 years of mathematics.

References: 

https://arxiv.org/abs/2304.14363

https://arxiv.org/abs/2206.06188

F. Nazarov, The Hormander proof of the Bourgain-Milman theorem, in: Geometric Aspects of Functional Analysis (B. Klartag, S. Mendelson, V.D. Milman, Eds.), Springer, 2012, pp. 335–343.

B. Berndtsson, Complex integrals and Kuperberg’s proof of the Bourgain–Milman theorem, Adv. Math. 388 (2021), 107927.

Z. Blocki, A lower bound for the Bergman kernel and the Bourgain–Milman inequality, in: Geometric Aspects of Functional Analysis (B. Klartag, E. Milman, Eds.), Springer, 2014, pp. 53–63.

Z. Blocki, On Nazarov’s complex analytic approach to the Mahler conjecture and the Bourgain–Milman inequality, in: Complex Analysis and Geometry (F. Bracci et al., Eds.), Springer, 2015, pp. 89–98.

Abstract:  Random polytopes are generated as the convex hull of finitely many independent random points. The random points are chosen according to some ‚natural’ distribution. In most cases the distribution function is the uniform measure on a convex set, or the Gaussian distribution.

In the first part of the mini-course we will introduce some background material, necessary tools, and discuss some classical approaches and results in an introductory way.

In the second part we will present an overview of more recent results with an emphasize on (central) limit theorems for random polytopes, and the connection to convex hull peeling.

Reitzner, M.: Random polytopes.

In: New perspectives in stochastic geometry, 45–76.

Oxford University Press, Oxford, 2010

https://academic.oup.com/book/26196?

Hug, D.: Random polytopes.

In: Stochastic geometry, spatial statistics and random fields, 205–238.

Lecture Notes in Math., 2068

Springer, Heidelberg, 2013

https://link.springer.com/chapter/10.1007/978-3-642-33305-7_7

Sampling from convex bodies (and more generally from log-concave distributions) is a fundamental algorithmic problem that arises in a variety of applications.  Perhaps the most celebrated applications are the efficient randomized algorithms for approximating volumes of convex bodies: the first such algorithm was obtained by Dyer, Frieze and Kannan, over three decades ago. Over the last three decades, a variety of random walks have been proposed and analyzed as a method for solving the sampling problem: the

proposals are sometimes much older than the analyses.

The first part of the mini course will survey some of the most-well analyzed random walks, especially the ball walk, and the hit-and-run walks, and the tools used for analyzing them: the notion of conductance, connections to isoperimetry, and the "localization lemma" of Lovasz and Simonovits.

The second part will focus on some more recent works on variants of the "hit-and-run" walk, and discuss some open problems. 

Abstract:

Determinantal point processes are probabilistic models of point configurations that present repulsion. They appear naturally in areas as varied as random matrix theory, combinatorics, mathematical physics and machine learning. A particularly important class of functionals for point processes are linear statistics, defined as $\sum_i f(x_i)$ where the $x_i$'s denote the positions of the random points and $f$ is a function. 

I will survey some known results about the distribution of linear statistics and also present a new way of computing the asymptotic behaviour of variances of linear statistics, based on the celebrated work by Bourgain, Brezis and Mironescu on fractional Sobolev seminorms.

Statistics of real zeros of a random polynomial with i.i.d. coefficients have been studied for a long time, starting with Bloch and Polya (1930s), and the now well-known works of Kac, Littlewood, Offord, Erdos, Ibragimov and Maslova (1960s). Despite this, there have been many interesting results on this topic in the recent past (last 15-20 years). In these two lectures we give a survey of these recent results and some of the proof techniques. The list of references (TBA) below will give an idea of some of these results.

Abstract: 

This talk will begin with a brief survey of some applications of optimal transport towards geometric and functional inequalities, with an emphasis on the role played by relative entropy. We will thereafter specialise to dimensional Brunn--Minkowski inequalities for even log-concave measures when the admissible sets have additional structure. The use of optimal transport in this problem is new, yielding results beyond the convexity assumption on admissible sets that is essential for the prominent differential-geometric technique in the area. These new results will be presented, and if time permits, we will also see how this method allows us to improve the Gaussian logarithmic Sobolev inequality for even log-concave functions. Based on joint work with Liran Rotem.  

Abstract : 

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures μ and ν in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p+q as n tends to infinity.

We present a unification, generalization, and sharpening of two inequalities– a bound on the maxima of convolutions due to Bobkov and Chistyakov, and a bound on marginals of projections of product measures due to Livshyts, Paouris and Pivovarov (which in turn gave a quantitative form of a result due to Rudelson and Vershynin). Our unification is obtained as a consequence of general anticoncentration mechanisms for linear images in Euclidean spaces, combining Choquet-theoretic ideas of Rogozin and the use of rearrangement inequalities. The underlying mechanism is general enough to not only yield additional applications in Euclidean spaces, but are of potential promise in the general setting of locally compact groups. In the reverse direction, we discuss the relation (first recognized by Keith Ball) of inequalities for maxima of convolutions to Busemann’s theorem in convex geometry. The talk is based on joint work with James Melbourne (CIMAT, Mexico) and Peng Xu (Eastern Illinois University).

We consider determinantal point processes on d-complex-dimensional space. These are characterized by a correlation kernel constructed with complex multivariate orthogonal polynomials. For d=1 they describe the eigenvalues of random normal matrices, it is well-known (and recently proved in generality) that the boundary of the limiting spectrum of eigenvalues exhibits a universal error function behavior as the matrix dimension becomes large. We show that such error function behavior extends to higher dimension d, for a particular "elliptic" subclass, and expect its universality to extend to more general higher dimensional models.

Abstract: I shall discuss recent developments concerning random polytopes of convex bodies. In particular I shall discuss the joint paper with

M. Reitzner and E. Werner The convex hull of random points on the boundary of a simple polytope.

Abstract: Studying excursion sets of random fields has a long history with motivations coming from statistical hypothesis testing, cosmology, stereology, and integral geometry, to name a few. Questions pertaining to characterisation of excursion sets are often considered very relevant, and very challenging. For instance, one is often interested in obtaining precise statistical description of the volume, (cumulative) surface area, perimeter of excursion sets of random fields. In this talk, we shall present an asymptotic description of (some) such geometric functionals of excursion sets of random fields in different scenarios. Based on joint works with D. Marinucci, M. Kratz, T. R. Reddy and D. Yogeshwaran.

Abstract: Loewner theory is a key ingredient in the construction of Schramm-Loewner-Evolutions. We consider Loewner chains driven by semimartingale drivers and establish existence and simpleness of the trace under fairly general conditions. As an application, we show that stochastic Komatu Loewner evolutions are generated by curves. This talk is based on a joint work with Yizheng Yuan and Vlad Margarint.