Metrics, Connections and Gluing Theorems

Metrics, Connections and Gluing Theorems
Author: Clifford Taubes
Publisher: American Mathematical Soc.
Total Pages: 98
Release: 1996
Genre: Mathematics
ISBN: 0821803239
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In this book, the author's goal is to provide an introduction to some of the analytic underpinnings for the geometry of anti-self duality in 4-dimensions. Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics. The book reviews some basic geometry, but is is assumed that the reader has a general background in differential geometry (as would be obtained by reading a standard text on the subject). Some of the fundamental references include Atiyah, Hitchin and Singer, Freed and Uhlenbeck, Donaldson and Kronheimer, and Kronheimer and Mrowka. The last chapter contains open problems and conjectures.

Riemannian Holonomy Groups and Calibrated Geometry

Riemannian Holonomy Groups and Calibrated Geometry
Author: Dominic D. Joyce
Publisher: Oxford University Press
Total Pages: 314
Release: 2007
Genre: Mathematics
ISBN: 019921560X
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Riemannian Holonomy Groups and Calibrated Geometry covers an exciting and active area of research at the crossroads of several different fields in mathematics and physics. Drawing on the author's previous work the text has been written to explain the advanced mathematics involved simply and clearly to graduate students in both disciplines.

Compact Manifolds with Special Holonomy

Compact Manifolds with Special Holonomy
Author: Dominic D. Joyce
Publisher: OUP Oxford
Total Pages: 460
Release: 2000
Genre: Mathematics
ISBN: 9780198506010
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This is a combination of a graduate textbook on Reimannian holonomy groups, and a research monograph on compact manifolds with the exceptional holonomy groups G2 and Spin (7). It contains much new research and many new examples.

Grid Homology for Knots and Links

Grid Homology for Knots and Links
Author: Peter S. Ozsvath
Publisher: American Mathematical Soc.
Total Pages: 410
Release: 2017-01-19
Genre: Education
ISBN: 1470434423
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Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

Gauge Theory and the Topology of Four-Manifolds

Gauge Theory and the Topology of Four-Manifolds
Author: Robert Friedman
Publisher: American Mathematical Soc.
Total Pages: 233
Release: 1998
Genre: Mathematics
ISBN: 0821805916
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This text is part of the IAS/Park City Mathematics series and focuses on gauge theory and the topology of four-manifolds.

Geometric Analysis

Geometric Analysis
Author: Jingyi Chen
Publisher: Springer Nature
Total Pages: 615
Release: 2020-04-10
Genre: Mathematics
ISBN: 3030349535
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This edited volume has a two-fold purpose. First, comprehensive survey articles provide 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 of significant interest 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 will stimulate further 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, Sławomir Kołodziej, 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 Rivière, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger.

Families of Riemann Surfaces and Weil-Petersson Geometry

Families of Riemann Surfaces and Weil-Petersson Geometry
Author: Scott A. Wolpert
Publisher: American Mathematical Soc.
Total Pages: 130
Release: 2010
Genre: Mathematics
ISBN: 0821849867
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Provides a generally self-contained course for graduate students and postgraduates on deformations of hyperbolic surfaces and the geometry of the Weil-Petersson metric. It also offers an update for researchers; material not otherwise found in a single reference is included; and aunified approach is provided for an array of results.

Topology, $C^*$-Algebras, and String Duality

Topology, $C^*$-Algebras, and String Duality
Author: Jonathan R_osenberg
Publisher: American Mathematical Soc.
Total Pages: 122
Release: 2009-10-27
Genre: Mathematics
ISBN: 0821849220
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String theory is the leading candidate for a physical theory that combines all the fundamental forces of nature, as well as the principles of relativity and quantum mechanics, into a mathematically elegant whole. The mathematical tools used by string theorists are highly sophisticated, and cover many areas of mathematics. As with the birth of quantum theory in the early 20th century, the mathematics has benefited at least as much as the physics from the collaboration. In this book, based on CBMS lectures given at Texas Christian University, Rosenberg describes some of the most recent interplay between string dualities and topology and operator algebras. The book is an interdisciplinary approach to duality symmetries in string theory. It can be read by either mathematicians or theoretical physicists, and involves a more-or-less equal mixture of algebraic topology, operator algebras, and physics. There is also a bit of algebraic geometry, especially in the last chapter. The reader is assumed to be somewhat familiar with at least one of these four subjects, but not necessarily with all or even most of them. The main objective of the book is to show how several seemingly disparate subjects are closely linked with one another, and to give readers an overview of some areas of current research, even if this means that not everything is covered systematically.

Topics in the Homological Theory of Modules Over Commutative Rings

Topics in the Homological Theory of Modules Over Commutative Rings
Author: Melvin Hochster
Publisher: American Mathematical Soc.
Total Pages: 86
Release: 1975
Genre: Mathematics
ISBN: 0821816748
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Contains expository lectures from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. This book deals mainly with developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings.