Proceedings of Symposia in Pure Mathematics Introduction Jonathan Rosenberg Abstract. The papers in this volume are the outgrowth of an NSF-CBMS Regional Conference in the Mathematical Sciences, May 18–22, 2009, orga- nized by Robert Doran and Greg Friedman at Texas Christian University. This introduction explains the scientific rationale for the conference and some of the common themes in the papers. During the week of May 18–22, 2009, Robert Doran and Greg Friedman orga- nized a wonderfully successful NSF-CBMS Regional Research Conference at Texas Christian University. I was the primary lecturer, and my lectures have now been published in [29]. However, Doran and Friedman also invited many other mathe- maticians and physicists to speak on topics related to my lectures. The papers in this volume are the outgrowth of their talks. The subject of my lectures, and the general theme of the conference, was highly interdisciplinary, and had to do with the confluence of superstring theory, algebraic topology, and C∗-algebras. While with “20/20 hindsight” it seems clear that these subjects fit together in a natural way, the connections between them developed almost by accident. Part of the history of these connections is explained in the introductions to [11] and [17]. The authors of [11] begin as follows: Until recently the interplay between physics and mathematics fol- lowed a familiar pattern: physics provides problems and mathe- matics provides solutions to these problems. Of course at times this relationship has led to the development of new mathematics. . . . But physicists did not traditionally attack problems of pure mathematics. This situation has drastically changed during the last 15 years. Physicists have formulated a number of striking conjectures (such as the existence of mirror symmetry) . . . . The basis of the physi- cists’ intuition is their belief that underlying quantum field theory 2010 Mathematics Subject Classification. Primary 81-06 Secondary 55-06, 46-06, 46L87, 81T30. Key words and phrases. string theory, supersymmetry, D-brane, C∗-algebra, crossed product, topological K-theory, twisted K-theory, classifying space, noncommutative geometry, Landau- Ginzburg theory, Yang-Mills theory. Partially supported by NSF grant DMS-0805003. c 0000 (copyright holder) 1 Volume 81, 2010 c 2010 American Mathematical

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