Past Advanced Class

6 March 2014
10:00
Abstract
<p>&nbsp;The theory of derivators is an approach to homotopical algebra<br />that focuses on the existence of homotopy Kan extensions. Homotopy<br />theories (e.g. model categories) typically give rise to derivators by<br />considering the homotopy categories of all diagrams categories<br />simultaneously. A general problem is to understand how faithfully the<br />derivator actually represents the homotopy theory. In this talk, I will<br />discuss this problem in connection with algebraic K-theory, and give a<br />survey of the results around the problem of recovering the K-theory of a<br />good Waldhausen category from the structure of the associated derivator.</p>
5 March 2012
11:00
James Griffin
Abstract
<p>A cactus product is much like a wedge product of pointed spaces, but instead of being uniquely defined there is a moduli space of possible cactus products. I will discuss how this space can be interpreted geometrically and how its combinatorics calculates the homology of the automorphism group of a free product with no free group factors. Then I will reinterpret the moduli space with Outer space in mind: the lobes of the cacti now behave like boundaries and our free products can now include free group factors.</p>
30 January 2012
11:00
Andre Henriques
Abstract
<p>The idea of three-tier conformal field theory (CFT) was first proposed by Greame Segal. It is an extension of the functorial approach to CFT, where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. The Baez-Dolan cobordism hypothesis is a meta-mathematical principle. It claims that functorial quantum field theory (i.e. quantum field theory expressed as a functor from some bordism category) becomes simper once "you go all the way down to points", i.e., once you replace the bordism category by a higher category. Three-tier CFT is an example of "going all the way down to points". We will apply the cobordism hypothesis to the case of three-tier CFT, and show how the modular invariance of the partition function can be derived as a consequence of the formalism, even if one only starts with genus-zero data.<br /><br /></p>

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