Past K-Theory Day

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.

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.  

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.

Martin Bridson will give a "repeat" performance of his Abel Lecture which he delivered a few weeks ago in Oslo as part of the scientific programme in honour of Abel Prize laureate Mikhail Gromov.

Abstract:

Gromov has illuminated great swathes of mathematics with the bright light of geometry. By means of example, I hope to convey the sense of wonder that his work engenders and something of the profound influence he has had on the way my generation thinks about mathematics.

I shall focus particularly on Geometric Group Theory. Gromov's ideas turned the study of discrete groups on its head, infusing it with an array of revolutionary ideas and unveiling deep connections to many other branches of mathematics.

We generalize rings, Banach algebras and C*-algebras to ringoids, Banach algebroids and C*-algebroids. We construct algebraic and topological K-theory of these objects. As an application we can formulate Farrell-Jones Conjecture in algebraic K-theory, Bost- and Baum-Connes-Conjecture in topological K-theory