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1. Instantons and Four-Manifolds
2. Biography - Uhlenbeck, Karen (Keskulla)
3. Surveys in Differential Geometry,
4. Geometry and Quantum Field Theory:
5. Regularity of Minimizing Harmonic

1. Instantons and Four-Manifolds (Mathematical Sciences Research Institute Publications)
by Daniel S. Freed, Karen K. Uhlenbeck
 Hardcover: 194 Pages (1990-12-03)
list price: US$43.95
Isbn: 038797377X
Average Customer Review: 5.0 out of 5 stars
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From the reviews of the first edition: "This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must." #Science#1 "I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book." #Bulletin of the American Mathematical Society#2 ... Read more

Customer Reviews (1)

5-0 out of 5 stars the best introduction to Donaldson's gauge theory, even if it is outdated
Gauge theory as a field for study by mathematicians, and in particular, topologists, was initiated by Donaldson's proof, which is the content of this book, that the only simply connected smooth closed 4-manifolds with definite intersection form are connected sums of complex projective planes (with the factors having either orientation). This was the first major advance in 4-manifold smooth topology beyond Rohlin's theorem on the signature of a smooth spin 4-manifold and the Kirby-Siebenmann obstruction. Donaldson's theorem was followed by many other related results, in particular, the discovery of the Donaldson polynomial invariants and various gluing and vanishing theorems, none of which appear in this early book, so this work is not suitable as a comprehensive reference for the theory. But it is still essential for a student wishing to learn gauge theory because of the extensive coverage of the requisite analysis, topology, and geometry that one needs to work in this field (or even read any later works). The authors endeavor to explain intuitively the motivation behind the different steps in this long proof (it takes more than 100 pages to prove the theorem), with some effective figures included as well.

Donaldson's proof is based on studying the moduli space of certain (anti-self-dual) solutions (mod gauge equivalences) of the Yang-Mills equations, the most fundamental equation in particle physics, which includes the vacuum Maxwell equations of electromagnetics, as well as those of QED and QCD, as special cases. For the SU(2) case over a smooth simply connected compact 4-manifold, the moduli space can be shown to be nonempty, orientable, smooth except for a finite number of points, near which the moduli space is a cone on a projective space, and having one noncompact end that is diffeomorphic to a collar on the original 4-manifold. This establishes a cobordism between the original manifold and a disjoint union of copies of CP^2, which proves the theorem. The various chapters of the book prove each of the aforementioned properties of the moduli spaces, mainly through tedious technical analytic results. Along the way the necessary material concerning bundles, connections, Sobolev spaces, Fredholm operators, index theory, classifying spaces, Bochner-Weitzenboeck formulas, etc., is presented, making this the most accessible gauge-theoretic work for (2nd-year) graduate students.

The second author, Karen Uhlenbeck, was one of the pioneers in the field, deriving several critical analytic results about sequences of solutions to the YM equations that are ubiquitous in gauge theory proofs; the book includes her proof of the famed removable singularities theorem as an appendix. There's also an early section on the construction of fake R^4s, one of the strangest results of 4-manifold topology (although I think Lawson's Theory of Gauge Fields in Four Dimensions does a better job of explaining this, and orientability as well). It is preferable to read the second edition, which includes a brief introduction that sumarizes advances in the field between 1984 and 1991, in addition to having been re-typeset.

A demerit of the book is that it is filled with mathematical typos and many of the technical proofs require some modification in the choice of constants owing to some errors in the inequalities. However, the theorems themselves remain true, as the reader can verify with some effort.

This book will provide the reader the tools to proceed to Donaldson and Kronheimer's The Geometry of Four-Manifolds and Morgan's Gauge Theory and the Topology of Four-Manifolds and The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. ... Read more

2. Biography - Uhlenbeck, Karen (Keskulla) (1942-): An article from: Contemporary Authors
by Gale Reference Team
Digital: 5 Pages (2002-01-01)
list price: US$9.95 -- used & new: US$9.95
(price subject to change: see help)
Asin: B0007SHQ7A
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This digital document, covering the life and work of Karen (Keskulla) Uhlenbeck, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1274 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

  • Place and date of birth and death (if deceased)
  • Family members
  • Education
  • Professional associations and honors
  • Employment
  • Writings, including books and periodicals
  • A description of the author's work
  • References to further readings about the author
... Read more

3. Surveys in Differential Geometry, Vol. 4: Integrable Systems
by Karen Uhlenbeck
Hardcover: 519 Pages (1998-04-01)
list price: US$65.00 -- used & new: US$61.00
(price subject to change: see help)
Asin: 1571460667
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The Surveys in Differential Geometry are supplements to the Journal of Differential Geometry, which are published by International Press. They include significant invited papers combining original research and overviews of the most current research in specific areas of interest to the growing Journal of Differential Geometry community. The survey volumes serve as continuing references, inspirations for new research, and introductions to the variety of topics of interest to differential geometers. These supplements are published annually since 1999.

The theory of integrable systems has been an active area of mathematics for the past thirty years. Different aspects of the subject have fundamental relations with mechanics and dynamics, applied mathematics, algebraic structures, theoretical physics, analysis and spectral theory and geometry. This volume (Vol. 4) is a collection of papers intended to introduce and explain some of the more geometric aspects of integrable systems to the community. ... Read more

4. Geometry and Quantum Field Theory: June 22-July 20, 1991, Park City, Utah (Ias/Park City Mathematics, Vol 1)
by Daniel S. Freed
Paperback: 459 Pages (1995-03-24)
list price: US$54.00 -- used & new: US$54.00
(price subject to change: see help)
Asin: 0821804006
Average Customer Review: 4.0 out of 5 stars
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Editorial Review

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Exploring topics from classical and quantum mechanics and field theory, this book is based on lectures presented in the Graduate Summer School at the Regional Geometry Institute in Park City, Utah, in 1991. The chapter by Bryant treats Lie groups and symplectic geometry, examining not only the connection with mechanics but also the application to differential equations and the recent work of the Gromov school. Rabin's discussion of quantum mechanics and field theory is specifically aimed at mathematicians. Alvarez describes the application of supersymmetry to prove the Atiyah-Singer index theorem, touching on ideas that also underlie more complicated applications of supersymmetry. Quinn's account of the topological quantum field theory captures the formal aspects of the path integral and shows how these ideas can influence branches of mathematics which at first glance may not seem connected. Presenting material at a level between that of textbooks and research papers, much of the book would provide excellent material for graduate courses. The book provides an entree into a field that promises to remain exciting and important for years to come. ... Read more

Customer Reviews (1)

4-0 out of 5 stars Decent survey on "quantum mathematics"
This book has four articles on mathematics in quantum field theory. Some of it is well known and "classical", particularly the article by Bryant on symplectic geometry (however, Bryan does discuss the modernmodern developments by Gromov in the last section). The article by Rabinattempts to do the impossible: try to teach mathematicians the physicsbehind quantum field theory via path integrals. The article by Alvarez onindex theorems summarizes nicely the connections between supersymmetry andindex theory. The last article by Quinn on topological field theory iswritten for the mathematician in mind. It is fascinating mathematics andgives the reader the impression that research in topological field theorywill no doubt continue to grow in importance. Several advances have takenplace since the book was published (1995), so peruse the literature beforebuying it. ... Read more

5. Regularity of Minimizing Harmonic Maps Into the Sphere: MSRI 039-83
by Richard; Karen Uhlenbeck Schoen
 Paperback: Pages (1983)

Asin: B000KGOV90
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