Graduate Studies in Mathematics, Volume: 204
2019; 264 pp; Hardcover
MSC: Primary 16;
Print ISBN: 978-1-4704-4931-5
This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work.
Graduate students and researchers interested in non-commutative geometry and representation theory.
Mathematical Surveys and Monographs, Volume: 244
2019; 297 pp; Hardcover
MSC: Primary 35; 37; 47; 81;
Print ISBN: 978-1-4704-4395-5
This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves.
The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrodinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.
Graduate students and researchers interested in the analysis of non-linear PDEs.
Contemporary Mathematics, Volume: 741
2020; 206 pp; Softcover
MSC: Primary 05; 60; 81; 82; 90;
Print ISBN: 978-1-4704-4841-7
This volume contains the proceedings of the Arizona School of Analysis and Mathematical Physics, held from March 5?9, 2018, at the University of Arizona, Tucson, Arizona.
A main goal of this school was to introduce graduate students and postdocs to exciting topics of current research that are both influenced by physical intuition and require the use of cutting-edge mathematics.
The articles in this volume reflect recent progress and innovative techniques developed within mathematical physics. Two works investigate spectral gaps of quantum spin systems. Specifically, Abdul-Rahman, Lemm, Lucia, Nachtergaele, and Young consider decorated AKLT models, and Lemm demonstrates a finite-size criterion for DD-dimensional models. Bachmann, De Roeck, and Fraas summarize a recent proof of the adiabatic theorem, while Bachmann, Bols, De Roeck, and Fraas discuss linear response for interacting Hall insulators. Models on general graphs are the topic of the articles by Fischbacher, on higher spin XXZ, and by Latushkin and Sukhtaiev, on an index theorem for Schrodinger operators. Probabilistic applications are the focus of the articles by DeMuse and Yin, on exponential random graphs, by Saenz, on KPZ universality, and by Stolz, on disordered quantum spin chains.
In all, the diversity represented here is a testament to the enthusiasm this rich field of mathematical physics generates.
Graduate students and researchers interested in analysis and mathematical physics.
Pure and Applied Undergraduate Texts,Volume: 43
2020; 336 pp; Hardcover
MSC: Primary 05;
Print ISBN: 978-1-4704-5342-8
Graph theory is a fascinating and inviting branch of mathematics. Many problems are easy to state and have natural visual representations, inviting exploration by new students and professional mathematicians. The goal of this textbook is to present the fundamentals of graph theory to a wide range of readers. The book contains many significant recent results in graph theory, presented using up-to-date notation. The author included the shortest, most elegant, most intuitive proofs for modern and classic results while frequently presenting them in new ways. Major topics are introduced with practical applications that motivate their development, and which are illustrated with examples that show how to apply major theorems in practice. This includes the process of finding a brute force solution (case-checking) when an elegant solution is not apparent. With over 1200 exercises, internet resources (e.g., the OEIS for counting problems), helpful appendices, and a detailed guide to different course outlines, this book provides a versatile and convenient tool for the needs of instructors at a large variety of institutions.
Undergraduate and graduate students interested in graph theory.
Due 2020-07-13
1st ed. 2020, Approx. 500 p.
Hardcover
ISBN 978-3-030-34952-3
Presents information on recent cutting edge progress in the field
Offers starters in the topic an introduction and paramount view of the subfields
Contributions from leading experts
This edited volume has a two-fold purpose.First, comprehensive survey articlesprovide a way for
beginners to ease into the corresponding sub-fields. These are then supplemented by original
works that give the more advanced readers a glimpse of the current research in geometric
analysis and related PDEs. The book is ofsignificantinterest for researchers, including advanced
Ph.D. students, working in geometric analysis. Readers who have a secondary interest in
geometric analysis will benefit from the survey articles. The results included in this book
willstimulatefurther advances in the subjects: geometric analysis, including complex differential
geometry, symplectic geometry, PDEs with a geometric origin, and geometry related to
topology. Contributions by Claudio Arezzo, Alberto Della Vedova, Werner Ballmann, Henrik
Matthiesen, Panagiotis Polymerakis, Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang, Tobias
Holck Colding, William P. Minicozzi II, Panagiotis Dimakis, Richard Melrose, Akito Futaki, Hajime
Ono, Jiyuan Han, Jeff A. Viaclovsky, Bruce Kleiner, John Lott, Sawomir Koodziej, Ngoc Cuong
Nguyen, Chi Li, Yuchen Liu, Chenyang Xu, YanYan Li, Luc Nguyen, Bo Wang, Shiguang Ma, Jie
Qing, Xiaonan Ma, Sean Timothy Paul, Kyriakos Sergiou, Tristan Riviere, Yanir A. Rubinstein,
Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua
Zhu and Aleksey Zinger.