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Braid Foliations in Low-Dimensional Topology
Offers a self-contained introduction to braid foliation techniques, which is a theory developed to study knots, links and surfaces in general 3-manifolds and more specifically in contact 3-manifolds. With style and content accessible to beginning students interested in geometric topology, each chapter centres around a key theorem or theorems.
The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations
The aim of this book is to provide beginning graduate students who completed the first two semesters of graduate-level analysis and PDE courses with a first exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces. Chapters 1 and 2 cover the fundamentals of the Euler theory: derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory for smooth solutions, and blowup criteria. Chapters 3, 4, and 5 cover ...
Hochschild Cohomology for Algebras
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.
Portfolio Theory and Arbitrage: A Course in Mathematical Finance
This book develops a mathematical theory for finance, based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability, starting with initial capital arbitrarily near zero. The principle is easy-to-test in specific models, as it is described in terms of the underlying market characteristics; it is shown to be equivalent to the existence of the so-called “Kelly” or growth-optimal portfolio, of the log-optimal portfolio, and of appropriate local martingale deflators. The resulting theory is powerful enough to treat in great generality the fundamental questions of hedging, valuation, and portfolio optimization. The bo...
Algebraic Geometry
This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course. The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and constructibility. $mathcal{O}$-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line. Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in $mathbb{P}^3$, and double planes, and it ends with applications of the Riemann-Roch Theorem.
Introduction to Complex Analysis
In this text, the reader will learn that all the basic functions that arise in calculus—such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet—are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed...
Lectures on Differential Topology
This book gives a comprehensive introduction to the theory of smooth manifolds, maps, and fundamental associated structures with an emphasis on “bare hands” approaches, combining differential-topological cut-and-paste procedures and applications of transversality. In particular, the smooth cobordism cup-product is defined from scratch and used as the main tool in a variety of settings. After establishing the fundamentals, the book proceeds to a broad range of more advanced topics in differential topology, including degree theory, the Poincaré-Hopf index theorem, bordism-characteristic numbers, and the Pontryagin-Thom construction. Cobordism intersection forms are used to classify compac...
Lectures on Poisson Geometry
This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way. —Alan Weinstein, University of California at Berkeley This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study. —Eckhard Meinrenken, University of Toronto
