K-Theory Day
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Wed, 09/06/2010 09:00 |
Chris Douglas |
K-Theory Day  |
L3 |
| Ordinary homology is a geometrically defined invariant of spaces: the 0-th homology group counts the number of components; the n-th homology group counts n-cycles, which correspond to an intuitive notion of 'n-dimensional holes' in a space. K-theory, or more specifically the 0-th K-theory group, is defined in terms of vector bundles, and so also has an immediate relationship to geometry. By contrast, the n-th K-theory group is typically defined homotopy-theoretically using the black box of Bott periodicity. I will describe a more geometric perspective on K-theory, using Z/2-graded vector bundles and bundles of modules for Clifford algebras. Along the way I will explain Clifford algebras, 2-categories, and Morita equivalence, explicitly check the purely algebraic 8-fold periodicity of the Clifford algebras, and discuss how and why this periodicity implies Bott periodicity. The talk will not presume any prior knowledge of K-theory, Clifford algebras, Bott periodicity, or the like. Based on joint work with Arthur Bartels and Andre Henriques | |||
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Thu, 10/06/2010 09:00 |
Chris Douglas |
K-Theory Day |
L3 |
| Homology counts components and cycles, K-theory counts vector bundles and bundles of Clifford algebra modules. What about geometric models for other generalized cohomology theories? There is a vision, introduced by Segal, Stolz, and Teichner, that certain cohomology theories should be expressible in terms of topological field theories. I will describe how the 0-th K-theory group can be formulated in terms of equivalence classes of 1-dimensional topological field theories. Then I will discuss what it means to twist a topological field theory, and explain that the n-th K-theory group comes from twisted 1-dimensional topological field theories. The expectation is that 2-dimensional topological field theories should be analogously related to elliptic cohomology. I will take an extended digression to explain what elliptic cohomology is and why it is interesting. Then I will discuss 2-dimensional twisted field theory and explain how it leads us toward a notion of higher ("2-dimensional") algebra. Based on joint work with Arthur Bartels and Andre Henriques | |||
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Fri, 11/06/2010 09:00 |
Chris Douglas |
K-Theory Day |
L3 |
| Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory. Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory—that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners. I will explain the notion of a local or (0+1+1+1)-dimensionaltopological field theory, which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners. I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories. Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification. | |||
