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for events the day of Thursday, December 3, 2015.

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    November 2015          December 2015           January 2016    
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Thursday, December 3, 2015

11:00 am in 241 Altgeld Hall,Thursday, December 3, 2015

The weight part of Serre's conjecture

Brandon Levin (University of Chicago)

Abstract: Serre's modularity conjecture (now a Theorem due to Khare-Wintenberger and Kisin) states that every odd irreducible two dimensional mod p representation of the absolute Galois group of Q comes from a modular form. I will begin with an overview of the Serre's original conjecture on modular forms focusing on the weight part of the conjecture. Herzig gave a generalization of the conjecture for n-dimensional Galois representations which predicts the modularity of so-called shadow weights. After briefly describing Herzig's conjecture, I will discuss joint work with D. Le, B. Le Hung, and S. Morra where we prove instances of this conjecture in dimension three.

12:30 pm in 464 Loomis Laboratroy,Thursday, December 3, 2015

To Be Announced

Albion Lawrence (Brandeis Physics)

1:00 pm in Altgeld Hall 243,Thursday, December 3, 2015

Complex deformations of Anosov representations

Andy Sanders (University of Illinois at Chicago)

Abstract: An Anosov representation of a hyperbolic surface group is a homomorphism from the surface group into a semi-simple Lie group which satisfies certain dynamical properties: from these properties one deduces that Anosov representations are discrete, faithful and the set of all Anosov representations is an open subset of the space of all homomorphisms. In recent years, Guichard-Wienhard produced examples of co-compact domains of discontinuity for Anosov representations, which lie in various homogeneous spaces, thus giving an answer to the question of whether or not Anosov representations appear as monodromies of locally homogeneous geometric structures on compact manifolds. In this talk, which comprises joint work with David Dumas, I will discuss some of the complex analytic features of these locally homogeneous geometric manifolds in the case the relevant homogeneous space is a generalized complex flag variety. In particular, we give sufficient conditions to compute the space of all infinitesimal deformations of the complex manifold underlying these locally homogeneous manifolds.

2:00 pm in 243 Altgeld Hall,Thursday, December 3, 2015

The geometry of Radon-Nikodym Lipschitz differentiability spaces

David Bate (University of Chicago)

Abstract: We give a purely geometric characterization of those metric measure spaces that satisfy the differentiability theory of Cheeger for Lipschitz functions taking value in Banach spaces with the Radon-Nikodym property. This characterization is centered on a notion of connecting points in the metric space by certain Lipschitz curves that form partial derivatives of any Lipschitz function at almost every point. This allows us to form the total Cheeger derivative from partial derivatives analogously to the Euclidean case. Further, Cheeger and Kleiner showed that doubling spaces that satisfy a Poincare inequality are RNP Lipschitz differentiability spaces. We show that the existence of such curves give a characterization of RNP Lipschitz differentiability spaces in terms of a Poincare type inequality. This is joint work with Sean Li.

2:00 pm in 347 Altgeld Hall,Thursday, December 3, 2015

Power Operations on Ring Spectra

Peter Nelson   [email] (UIUC Math)

Abstract: Unlike in classical algebra, being "commutative" in homotopy theory is more structure, rather than a property. An algebraic topologist might wonder: how does this extra structure manifest on homotopy groups, and what does it buy us? I hope to indicate some answers through a few examples. I might even say some things about connections to arithmetic.

4:00 pm in 245 Altgeld Hall,Thursday, December 3, 2015

Supersymmetry and tensor categories

Vera Serganova (UC Berkeley)

Abstract: The goal of the lecture is to show the interplay between supersymmetry and tensor categories. Original motivation of supersymmetry comes from physics and topology (a chain complex can be considered as a supercommutative ring and differential as a superderivation.) In supersymmetry we work with Z_2-graded objects and modify usual identities by the sign rule. A systematic approach to this involves the language of tensor categories.
After introduction to supersymmetry and tensor categories I illustrate how both theories enrich each other on the following examples:
(1) Theorem of Deligne about supertannakian category.
(2) Mixed Schur -Weyl duality and Deligne's category Rep GL(t).
(3) Universal tensor categories via representations of supergroups.