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Cyclic Cohomology at 40: Achievements and Future Prospects
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher ca...
Microlocal Analysis and Spectral Theory
The NATO Advanced Study Institute "Microlocal Analysis and Spectral The ory" was held in Tuscany (Italy) at Castelvecchio Pascoli, in the district of Lucca, hosted by the international vacation center "11 Ciocco" , from September 23 to October 3, 1996. The Institute recorded the considerable progress realized recently in the field of Microlocal Analysis. In a broad sense, Microlocal Analysis is the modern version of the classical Fourier technique in solving partial differential equa tions, where now the localization proceeding takes place with respect to the dual variables too. Precisely, through the tools of pseudo-differential operators, wave-front sets and Fourier integral operators, the general theory of the lin ear partial differential equations is now reaching a mature form, in the frame of Schwartz distributions or other generalized functions. At the same time, Microlocal Analysis has grown up into a definite and independent part of Math ematical Analysis, with other applications all around Mathematics and Physics, one major theme being Spectral Theory for Schrodinger equation in Quantum Mechanics.
Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations
This volume focuses on recent developments in non-linear and hyperbolic equations. It will be a most valuable resource for researchers in applied mathematics, the theory of wavelets, and in mathematical and theoretical physics. Nine up-to-date contributions have been written on invitation by experts in the respective fields. The book is the third volume of the subseries "Advances in Partial Differential Equations".
Parabolicity, Volterra Calculus, and Conical Singularities
Partial differential equations constitute an integral part of mathematics. They lie at the interface of areas as diverse as differential geometry, functional analysis, or the theory of Lie groups and have numerous applications in the applied sciences. A wealth of methods has been devised for their analysis. Over the past decades, operator algebras in connection with ideas and structures from geometry, topology, and theoretical physics have contributed a large variety of particularly useful tools. One typical example is the analysis on singular configurations, where elliptic equations have been studied successfully within the framework of operator algebras with symbolic structures adapted to ...
Geometric, Algebraic and Topological Methods for Quantum Field Theory
Based on lectures held at the 8th edition of the series of summer schools in Villa de Leyva since 1999, this book presents an introduction to topics of current interest at the interface of geometry, algebra, analysis, topology and theoretical physics. It is aimed at graduate students and researchers in physics or mathematics, and offers an introduction to the topics discussed in the two weeks of the summer school: operator algebras, conformal field theory, black holes, relativistic fluids, Lie groupoids and Lie algebroids, renormalization methods, spectral geometry and index theory for pseudo-differential operators.
Semigroups, Boundary Value Problems and Markov Processes
A careful and accessible exposition of functional analytic methods in stochastic analysis is provided in this book. It focuses on the interrelationship between three subjects in analysis: Markov processes, semi groups and elliptic boundary value problems. The author studies a general class of elliptic boundary value problems for second-order, Waldenfels integro-differential operators in partial differential equations and proves that this class of elliptic boundary value problems provides a general class of Feller semigroups in functional analysis. As an application, the author constructs a general class of Markov processes in probability in which a Markovian particle moves both by jumps and ...
Nonlinear Dispersive Waves and Fluids
This volume contains the proceedings of the AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations and the AMS Special Session on PDE Analysis on Fluid Flows, which were held in January 2017 in Atlanta, Georgia. These two sessions shared the underlying theme of the analysis aspect of evolutionary PDEs and mathematical physics. The articles address the latest trends and perspectives in the area of nonlinear dispersive equations and fluid flows. The topics mainly focus on using state-of-the-art methods and techniques to investigate problems of depth and richness arising in quantum mechanics, general relativity, and fluid dynamics.
Elliptic and Parabolic Equations
The international workshop on which this proceedings volume is based on brought together leading researchers in the field of elliptic and parabolic equations. Particular emphasis was put on the interaction between well-established scientists and emerging young mathematicians, as well as on exploring new connections between pure and applied mathematics. The volume contains material derived after the workshop taking up the impetus to continue collaboration and to incorporate additional new results and insights.
Approaches to Singular Analysis
This collection presents various approaches to analytic problems that arise in the context of singular spaces. It contains articles offering introductions to various pseudodifferential calculi and discussions of relations between them, plus invited papers from mathematicians who have made significant contributions to this field
Functional Analytic Techniques for Diffusion Processes
This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems. The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readabl...
