Thu, 04 Jun 2009
16:00
L3

Structure of some integral Galois representations

Alex Bartel
(Cambridge)
Abstract

Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

Thu, 28 May 2009
16:00
L3

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture

Werner Bley
(Kassel)
Abstract

In the first part of the talk we briefly describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson.

Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an analytic and an algebraic element in a relative algebraic K-group. As a first application we give some numerical evidence for ETNC in the case of the base change of an elliptic curve defined over the rational numbers. In this special case ETNC is an equivariant version of the Birch and Swinnerton-Dyer conjecture

Thu, 26 Mar 2009
14:00
L3

Representation of Quantum Groups and new invariants of links

Chen QingTao
(UC Berkeley)
Abstract

The colored HOMFLY polynomial is a quantum invariant of oriented links in S³ associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure, which is the part of Labastida-Marino-Ooguri-Vafa conjecture. The new integer invariants are also predicted by the LMOV conjecture and recently has been proved. LMOV also give the application of Licherish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial could also be developed in a same fashion by using more complicated algebra method.

In a joint work with Lin Chen and Nicolai Reshetikhin, we rigorously formulate the orthogonal quantum group version of LMOV conjecture in mathematics by using the representation of Brauer centralizer algebra. We also obtain formulae of Lichorish-Millet type which could be viewed as the application in knot theory and topology. By using the cabling technique, we obtain a uniform formula of colored Kauffman polynomial for all torus links with all partitions. Combined these together, we are able to prove many interesting cases of this orthogonal LMOV conjecture.

Thu, 26 Mar 2009
11:00
L3

Applications of the Cobordism Hypothesis

Jacob Lurie
(MIT)
Abstract

In this lecture, I will illustrate the cobordism hypothesis by presenting some examples. Exact content to be determined, depending on the interests of the audience.

Wed, 25 Mar 2009
11:00
L3

The Cobordism Hypothesis

Jacob Lurie
(MIT)
Abstract

In this lecture, I will give a more precise statement of the Baez-Dolan cobordism hypothesis, which gives a description of framed bordism (higher) categories by a universal mapping property. I'll also describe some generalizations of the cobordism hypothesis, which take into account the structure of diffeomorphism groups of manifolds and which apply to manifolds which are not necessarily framed.

Tue, 24 Mar 2009
11:00
L3

An Overview of Higher Category Theory

Jacob Lurie
(MIT)
Abstract

In this lecture, I'll give an overview of some ideas from higher category theory which are needed to make sense of the Baez-Dolan cobordism hypothesis. If time permits, I'll present Rezk's theory of complete Segal spaces (a model for the theory of higher categories in which most morphisms are assumed to be invertible) and explain how bordism categories can be realized in this framework.

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