School of Mathematics Events http://hdl.handle.net/1853/55955 2021-06-23T08:46:57Z 2021-06-23T08:46:57Z Concentration and Convexity - Part 3 Paouris, Grigoris http://hdl.handle.net/1853/62171 2020-01-12T08:04:13Z 2019-12-13T00:00:00Z Concentration and Convexity - Part 3 Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Presented on December 12, 2019 at 10:30 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech.; Workshop in Convexity and Geometric Aspects of Harmonic Analysis; Grigoris Paouris, Texas A&M University; Runtime: 49:20 minutes 2019-12-13T00:00:00Z Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Concentration and Convexity - Part 2 Paouris, Grigoris http://hdl.handle.net/1853/62170 2020-01-12T08:04:16Z 2019-12-01T00:00:00Z Concentration and Convexity - Part 2 Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Presented on December 12, 2019 at 9:10 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech.; Workshop in Convexity and Geometric Aspects of Harmonic Analysis; Grigoris Paouris, Texas A&M University; Runtime: 54:20 minutes 2019-12-01T00:00:00Z Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Concentration and Convexity - Part 1 Paouris, Grigoris http://hdl.handle.net/1853/62169 2020-01-12T08:04:17Z 2019-12-10T00:00:00Z Concentration and Convexity - Part 1 Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Presented on December 10, 2019 at 9:10 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech.; Workshop in Convexity and Geometric Aspects of Harmonic Analysis; Grigoris Paouris, Texas A&M University; Runtime: 52:47 minutes 2019-12-10T00:00:00Z Paouris, Grigoris The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles. Fourier Analysis in Geometric Tomography - Part 3 Koldobsky, Alexander http://hdl.handle.net/1853/62166 2020-01-09T22:04:36Z 2019-12-13T00:00:00Z Fourier Analysis in Geometric Tomography - Part 3 Koldobsky, Alexander Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions. Presented on December 13, 2019 at 9:10 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech.; Workshop in Convexity and Geometric Aspects of Harmonic Analysis; Alexander Koldobsky, University of Missouri-Columbia; Runtime: 61:06 minutes 2019-12-13T00:00:00Z Koldobsky, Alexander Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions.