Analysis Meets Geometry

This book is dedicated to the memory of Mikael Passare, an outstanding Swedish mathematician who devoted his life to developing the theory of analytic functions in several complex variables and exploring geometric ideas first-hand. It includes several papers describing Mikael’s life as well as his contributions to mathematics, written by friends of Mikael’s who share his attitude and passion for science. A major section of the book presents original research articles that further develop Mikael’s ideas and which were written by his former students and co-authors. All these mathematicians work at the interface of analysis and geometry, and Mikael’s impact on their research cannot be underestimated. Most of the contributors were invited speakers at the conference organized at Stockholm University in his honor. This book is an attempt to express our gratitude towards this great mathematician, who left us full of energy and new creative mathematical ideas.

This volume contains the refereed proceedings of the Special Session on Geometric Analysis held at the AMS meeting in Philadelphia in October 1991. The term ``geometric analysis'' is being used with increasing frequency in the mathematical community, but its meaning is not entirely fixed. The papers in this collection should help to better define the notion of geometric analysis by illustrating emerging trends in the subject. The topics covered range over a broad spectrum: integral geometry, Radon transforms, geometric inequalities, microlocal analysis, harmonic analysis, analysis on Lie groups and symmetric spaces, and more. Containing articles varying from the expository to the technical, this book presents the latest results in a broad range of analytic and geometric topics.

This book contains the proceedings of a special session held during the Summer Meeting of the Canadian Mathematical Society in 1990. The articles collected here reflect the diverse interests of the participants but are united by the common theme of the interplay among geometry, global analysis, and topology. The topics covered include applications to low dimensional manifolds, control theory, integrable systems, Lie algebras of operators, and algebraic geometry and provide an insight into some recent trends in these areas.

This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kähler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in R3, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace–Beltrami operators.

This book is an outgrowth of the Workshop on "Regulators in Analysis, Geom etry and Number Theory" held at the Edmund Landau Center for Research in Mathematical Analysis of The Hebrew University of Jerusalem in 1996. During the preparation and the holding of the workshop we were greatly helped by the director of the Landau Center: Lior Tsafriri during the time of the planning of the conference, and Hershel Farkas during the meeting itself. Organizing and running this workshop was a true pleasure, thanks to the expert technical help provided by the Landau Center in general, and by its secretary Simcha Kojman in particular. We would like to express our hearty thanks to all of them. However, the articles assembled in the present volume do not represent the proceedings of this workshop; neither could all contributors to the book make it to the meeting, nor do the contributions herein necessarily reflect talks given in Jerusalem. In the introduction, we outline our view of the theory to which this volume intends to contribute. The crucial objective of the present volume is to bring together concepts, methods, and results from analysis, differential as well as algebraic geometry, and number theory in order to work towards a deeper and more comprehensive understanding of regulators and secondary invariants. Our thanks go to all the participants of the workshop and authors of this volume. May the readers of this book enjoy and profit from the combination of mathematical ideas here documented.

The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.

This book presents recent advances in the field of shape analysis. Written by experts in the fields of continuous-scale shape analysis, discrete shape analysis and sparsity, and numerical computing who hail from different communities, it provides a unique view of the topic from a broad range of perspectives. Over the last decade, it has become increasingly affordable to digitize shape information at high resolution. Yet analyzing and processing this data remains challenging because of the large amount of data involved, and because modern applications such as human-computer interaction require real-time processing. Meeting these challenges requires interdisciplinary approaches that combine concepts from a variety of research areas, including numerical computing, differential geometry, deformable shape modeling, sparse data representation, and machine learning. On the algorithmic side, many shape analysis tasks are modeled using partial differential equations, which can be solved using tools from the field of numerical computing. The fields of differential geometry and deformable shape modeling have recently begun to influence shape analysis methods. Furthermore, tools from the field of sparse representations, which aim to describe input data using a compressible representation with respect to a set of carefully selected basic elements, have the potential to significantly reduce the amount of data that needs to be processed in shape analysis tasks. The related field of machine learning offers similar potential. The goal of the Dagstuhl Seminar on New Perspectives in Shape Analysis held in February 2014 was to address these challenges with the help of the latest tools related to geometric, algorithmic and numerical concepts and to bring together researchers at the forefront of shape analysis who can work together to identify open problems and novel solutions. The book resulting from this seminar will appeal to researchers in the field of shape analysis, image and vision, from those who want to become more familiar with the field, to experts interested in learning about the latest advances.​

The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.