Department of

Mathematics


Seminar Calendar
for events the day of Thursday, November 5, 2015.

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Thursday, November 5, 2015

11:00 am in 241 Altgeld Hall,Thursday, November 5, 2015

Indefinite theta functions, higher depth mock modular forms, and quantum modular forms

Larry Rolen (Pennsylvania State University)

Abstract: In this talk, I will describe several new results concerning the modularity of indefinite theta functions. From Zwegers' thesis, we know that special types of indefinite theta functions with prescribed signatures give rise to mock modular forms, which combined with important work of Andrews and others gives one road to understanding the mock theta functions of Ramanujan. Here, we will study several important examples of more general indefinite theta series inspired by physics and geometry and describe how to study the modularity properties of more complicated objects such as these, giving a glimpse into the general structure of indefinite theta functions. We will also study another class of indefinite theta functions, and we will discuss a new family of examples which give rise to quantum modular forms, and provide a family of canonical Maass waveforms whose Fourier coefficients are described by combinatorial functions with integer coefficients, placing the famous functions $\sigma$ and $\sigma^*$ of Andrews, Dyson, and Hickerson in a natural framework.

1:00 pm in Altgeld Hall 243,Thursday, November 5, 2015

Proper holomorphic mappings throughout mathematics

John P. D'Angelo (UIUC Math )

Abstract: After a brief review of the relationship between proper mappings and CR geometry, we consider in detail proper holomorphic mappings between balls. We discuss spherical equivalence, homotopy equivalence, and Hermitian analogues of Hilbert's $17$-th problem. If time permits we will introduce a complex variety associated with a rational proper map and state a sharp variational inequality for the volumes of images of proper polynomial mappings between balls.

2:00 pm in 347 Altgeld Hall,Thursday, November 5, 2015

A convoluted way to get representation theory out of cohomology theories

Joshua Wen (UIUC Math)

Abstract: The convolution algebra construction allows one to build algebras and their representations out of the (equivariant) Borel-Moore homology or (equivariant) K-theory of a suitable space. Iíll outline this construction for the Springer resolution, which yields the group algebra of the Weyl group and the affine Hecke algebra (and hopefully Iíll point out some cool/mind-boggling aspects of this). In the mid 90ís, speculative work of Ginzburg-Kapranov-Vasserot extended this construction to equivariant elliptic cohomology before such a cohomology theory even existed. The end goal of this talk is to give a rough picture of how the `loopsí hierarchies in representation theory and cohomology theories are related.

2:00 pm in 241 Altgeld Hall,Thursday, November 5, 2015

Recurrences for Eisenstein series

Larry Rolen (Pennsylvania State University)

Abstract: In this talk, we will learn about recursive formulas for Eisenstein series, some of which are classical, and some of which are surprisingly new. In particular, we will see that these important examples of modular forms can be recursively defined in many ways, which directly yields surprising identities between convolution sums of sums of divisor functions as well as relations among the classical Bernoulli numbers. Along the way, we will learn about important examples of doubly periodic, meromorphic functions, also known as elliptic functions, and their connections to modular forms. This talk will be self-contained, and no prior knowledge of modular forms or the related objects mentioned above will be assumed.

4:00 pm in 245 Altgeld Hall,Thursday, November 5, 2015

Some new rigidity results for smooth actions of large groups

Federico Rodriguez Hertz (Pennsylvania State University)

Abstract: In this talk I will survey some new results for smooth actions on manifolds by lattices on higher rank Lie groups. This will come in two flavors, measure theoretic and smooth rigidity. The main theme is that under reasonably general conditions a smooth action is either measurably conjugated, or even smoothly conjugated to an algebraic action. If time permits, I will also discuss some open problems with their approaches. This is based on joint work with A. Katok, A. Brown and Zhiren Wang in different combinations.