In the Fall of 1975 we started a joint project with the ultimate goal of topo logically classifying real algebraic sets. This has been a long happy collaboration (c.f., [K2)). In 1985 while visiting M.S.R.1. we organized and presented our classification results up to that point in the M.S.R.1. preprint series [AK14] -[AK17]. Since these results are interdependent and require some prerequisites as well as familiarity with real algebraic geometry, we decided to make them self contained by presenting them as a part of a book in real algebraic geometry. Even though we have not arrived to our final goal yet we feel that it is time to introduce them in a self contained coherent version and demonstrate their use by giving some applications. Chapter I gives the overview of the classification program. Chapter II has all the necessary background for the rest of the book, which therefore can be used as a course in real algebraic geometry. It starts with the elementary properties of real algebraic sets and ends with the recent solution of the Nash Conjecture. Chapter III and Chapter IV develop the theory of resolution towers. Resolution towers are basic topologically defined objects generalizing the notion of manifold.

Lively and engaging articles from the lectures and the participants of the 21st Goekova Geometry- Topology Conference, held on the shores of Goekova Bay, Turkey, in May of 2014.

In the spring of 1985, A. Casson announced an interesting invariant of homology 3-spheres via constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives surprising corollaries in low-dimensional topology. In the fall of that same year, Selman Akbulut and John McCarthy held a seminar on this invariant. These notes grew out of that seminar. The authors have tried to remain close to Casson's original outline and proceed by giving needed details, including an exposition of Newstead's results. They have often chosen classical concrete approaches over general methods. For example, they did not attempt to give gauge theory explanations for the results of Newstead; instead they followed his original techniques.

Originally published in 1990.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In the spring of 1985, A. Casson announced an interesting invariant of homology 3-spheres via constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives surprising corollaries in low-dimensional topology. In the fall of that same year, Selman Akbulut and John McCarthy held a seminar on this invariant. These notes grew out of that seminar. The authors have tried to remain close to Casson's original outline and proceed by giving needed details, including an exposition of Newstead's results. They have often chosen classical concrete approaches over general methods. For example, they did not attempt to give gauge theory explanations for the results of Newstead; instead they followed his original techniques. Originally published in 1990. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

This volume presents lively and engaging articles from the lecturers and the participants of the 25th and 26th Goekova Geometry-Topology Conferences, held on the shores of Goekova Bay, Turkey, in May/June of 2018 and May/June of 2019. The 25th and 26th Goekova Geometry-Topology Conferences were sponsored by the National Science Foundation, by the Turkish Mathematical Society, and by the European Research Council.

This volume presents lively and engaging articles from the lecturers and the participants of the 23rd Goekova Geometry-Topology Conference, held on the shores of Goekova Bay, Turkey, in May/June of 2016. Topics include manifolds, special holonomy, symplectic topology, gauge theory, Lie groups, hyperkahler manifolds, and more. The 23rd Goekova Geometry-Topology Conference was sponsored by the National Science Foundation, by the Turkish Mathematical Society, and by the European Research Council.

This volume presents lively and engaging articles from the lecturers and the participants of the 22nd Goekova Geometry-Topology Conference, held on the shores of Goekova Bay, Turkey, in May of 2015. Topics include high-dimensional geometric, symplectic and contact geometry, and more. The 22nd Goekova Geometry-Topology Conference was sponsored by the National Science Foundation, by the Turkish Mathematical Society, and by the European Research Council.

Lively and engaging articles from the lecturers and the participants of the 20th Goekova Geometry-Topology Conference, held on the shores of Goekova Bay, Turkey, in May of 2013.

Lively and engaging articles from the lecturers and the participants of the 19th Goekova Geometry-Topology Conference, held on the shores of Goekova Bay, Turkey, in May of 2012.

The Goekova Geometry-Topology Conferences were inaugurated in 1992 with the support of TUBITAK (The Scientific and Technological Research Council of Turkey) and are held annually on the shores of the scenic Goekova Bay on the southwestern coast of Turkey. These conferences have been supported by TUBITAK since their inception and since 2005 they have received partial funding from NSF. Over the years the topics of these conferences were chosen from the exciting subjects of geometry and topology, usually with the most recent developments taking the front stage.

Lively and engaging articles from the lecturers and the participants of the 17th Gokova Geometry-Topology Conference, held on the shores of Gokova Bay, Turkey, in May of 2010. This volume contains Borgeois' notes from his mini-course on contact homology, and an expository article on the solution of the celebrated Arf-Kerviere invariant problem by Hill, Hopkins, and Ravenel, as well as some new research articles.

Lively and engaging articles from the lecturers and the participants of the 18th Goekova Geometry-Topology Conference, held on the shores of Goekova Bay, Turkey, in May of 2011.

Features the proceedings of the 14th Gokova Geometry-Topology Conference, held on the shores of Gokova Bay, Turkey, in May of 2007.

Features the proceedings of the 13th Gokova Geometry-Topology Conference, held on the shores of Gokova Bay, Turkey, in May of 2006.

Dedicated to the memory of Raoul Bott, a great mathematician of the 20th Century, this volume contains articles from both eleventh and twelfth Gokova Conferences, held in Gokova, Turkey.

Since its inception in 1992, the Gokova conference has been a premiere mathematics event sponsored by the Scientific and Technical Research Council of Turkey. The participants include distinguished mathematicians, who came from different parts of the world. This conference proceedings presents their inspiring talks and follow-up articles, which reflect the latest developments in Geometry and Topology. The proceedings of previous years, by the same editors, are also available.