Probabilistic and geometric aspects of classical and non-commutative lp-type spaces in high dimensions
type: CreativeWork
description: This cumulative dissertation contains selected contributions to the field of asymptotic geometric analysis and high-dimensional probability. It is divided into two chapters: Chapter 1 explains some of the necessary theoretical background. In Section 1.1 it first gives a very concise history of asymptotic geometric analysis in general and then of the objects under study in particular, setting out some cornerstones in the discovery of the functional-analytic, geometric, and probabilistic properties of the spaces under consideration. The next section (1.2) gives the precise definitions and very basic properties of the three lp-type spaces that play a role in the contributed articles: the classical lp-sequence spaces, the mixed-norm sequence spaces, and the Schatten-classes Sp, each in its infinite- and finite-dimensional version. Section 1.3 is dedicated to the interplay between geometry and probability, expounding the general idea, introducing a few of the common tools, and exemplifying these on two kinds of limit theorems: Schechtman-Schmuckenschläger-type results and Poincaré-Maxwell-Borel lemmas. The first chapter concludes with Section 1.4, addressing a small sample of open questions pertaining to the contributed articles which are not answered in said articles and may be the interest of future research. The entirety of Chapter 2 consists of the contributed articles.
abstract: This cumulative dissertation contains selected contributions to the field of asymptotic geometric analysis and high-dimensional probability. It is divided into two chapters: Chapter 1 explains some of the necessary theoretical background. In Section 1.1 it first gives a very concise history of asymptotic geometric analysis in general and then of the objects under study in particular, setting out some cornerstones in the discovery of the functional-analytic, geometric, and probabilistic properties of the spaces under consideration. The next section (1.2) gives the precise definitions and very basic properties of the three lp-type spaces that play a role in the contributed articles: the classical lp-sequence spaces, the mixed-norm sequence spaces, and the Schatten-classes Sp, each in its infinite- and finite-dimensional version. Section 1.3 is dedicated to the interplay between geometry and probability, expounding the general idea, introducing a few of the common tools, and exemplifying these on two kinds of limit theorems: Schechtman-Schmuckenschläger-type results and Poincaré-Maxwell-Borel lemmas. The first chapter concludes with Section 1.4, addressing a small sample of open questions pertaining to the contributed articles which are not answered in said articles and may be the interest of future research. The entirety of Chapter 2 consists of the contributed articles.
author: Juhos, Michael
keywords: functional analysis, asymptotic geometric analysis, limit theorems, lp-space, Schatten class
date published: 2024-08-28
date modified: 2026-07-24T03:45:10Z
provider: Universität Passau
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Universität Passau