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In this volume the author presents, in great detail and with many examples, a basic collection of principles, techniques, and applications needed to conduct independent research in gauge theory and its use in geometry and topology. Complete and self-contained computations of the Seiberg-Witten invariants of most simply connected algebraic surfaces using only Witten's factorization method are included. Also given is a new approach to cutting and pasting Seiberg-Witten invariants, which is illustrated by examples such as the connected sum theorem, the blow-up formula, and a proof of a vanishing result of Fintushel and Stern. The book is a suitable textbook for advanced graduate courses in differential geometry, algebraic topology, basic PDEs and functional analysis.

The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi: \mathbb{R}\times X\rightarrow X$ on pfaffian set $X$ is tame if the graph of $\Phi$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. The author proves that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame.

This is a state-of-the-art introduction to the work of Franz Reidemeister, Meng Taubes, Turaev, and the author on the concept of torsion and its generalizations. Torsion is the oldest topological (but not with respect to homotopy) invariant that in its almost eight decades of existence has been at the center of many important and surprising discoveries. During the past decade, in the work of Vladimir Turaev, new points of view have emerged, which turned out to be the'right ones'as far as gauge theory is concerned. The book features mostly the new aspects of this venerable concept. The theoretical foundations of this subject are presented in a style accessible to those, who wish to learn and understand the main ideas of the theory. Particular emphasis is upon the many and rather diverse concrete examples and techniques which capture the subleties of the theory better than any abstract general result. Many of these examples and techniques never appeared in print before, and their choice is often justified by ongoing current research on the topology of surface singularities. The text is addressed to mathematicians with geometric interests who want to become comfortable users of this versatile invariant.

In this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family. All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eight symmetries in the real case and two in the complex case).