Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, February 26, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, February 26, 2015

11:00 am in 241 Altgeld Hall,Thursday, February 26, 2015

#### Mock theta functions and their ghosts

###### Robert Rhoades   [email] (Center for Communications Research)

Abstract: Ramanujan's last letter to Hardy has become celebrated, not only because of the tragic circumstances surrounding it, but because of the mysterious objects put forth in it. Ramanujan gave no formal definition of mock theta functions. He gave 17 examples and basic properties they satisfied.

I will explain why these properties were so mysterious. Thanks to the 2002 thesis of Sander Zwegers, we have a foundation for understanding these mysteries. I will present one of the remaining mysteries. The mystery is to place "ghost" terms associated to mock theta functions via a formal q-hypergeometric construction into a theoretical foundation.

1:00 pm in Altgeld Hall 243,Thursday, February 26, 2015

#### Higher rank geometric structures

###### Andy Zimmer (University of Chicago)

Abstract: Suppose G is a Lie group acting transitively on a manifold X. Then a (G,X)-structure on a manifold M is a collection of coordinate charts whose images are in X and whose transition functions are in G. Some classic examples include the locally symmetric spaces, real projective manifolds, complex projective manifolds, and affine manifolds. In this talk we will consider (G,X)-structures where X is a compact G-space having higher rank'' in the sense that G does not act two-transitively on X. For instance, the complete flag variety of a real vector space with dimension at least two is a higher rank $SL_d(\mathbb{R})$-space. We will show that a higher rank geometric structure on a manifold restricts the fundamental group. In particular, a manifold with Gromov hyperbolic fundamental group cannot have a proper (G,X)-structure when $G=SL_d(\mathbb{R})$ and X is a higher rank compact G-space.

2:00 pm in 140 Henry Administration Bldg,Thursday, February 26, 2015

#### Arithmetic of Apollonian Circle Packings, part II

###### Elena Fuchs (UIUC Math)

Abstract: We will explore some theorems about the arithmetic of ancient objects called Apollonian Circle Packings. The tools we will use have applications far beyond these packings, and we will give some idea of what such applications are, as well.

4:00 pm in 245 Altgeld Hall,Thursday, February 26, 2015

#### Bi-algebraic geometry and its application to Diophantine problems

###### Bruno Klinger (Jussieu/IAS)

Abstract: After recalling the classical Schanuel's conjecture, which encapsulates the transcendence properties of the exponential function, I will describe some functional analogues and their applications to Diophantine problems descending from the Mordell conjecture (theorem of Faltings), as developed by Pila and Zannier. The main tool will be o-minimal geometry. All the notions will be introduced from scratch.