Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, November 6, 2014.

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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, November 6, 2014

10:00 am in 143 Altgeld Hall,Thursday, November 6, 2014

#### Bilipschitz Embedding of the Grushin Plane via Curvature Growth Bounds

###### Matthew Romney (UIUC Math)

Abstract: A common problem in metric space geometry is to embed a given space in a well-known model space under a sufficiently nice mapping. In the case of this talk, we consider bilipschitz embeddings into Euclidean space. We prove a theorem giving sufficient conditions for such an embedding in terms of the sectional curvature of the space. As an application, we offer a new proof of Seo's recent result that the Grushin plane is bilipschitzly embeddable in some Euclidean space. This talk should be accessible to all graduate students.

11:00 am in 241 Altgeld Hall,Thursday, November 6, 2014

#### Singular invariants, mock modular forms of weight 5/2, and partitions

###### Nickolas Andersen   [email] (UIUC Math)

Abstract: We study the coefficients of a natural basis for the space of mock modular forms of weight 5/2 on the full modular group. The "shadow" of the first element of this infinite basis encodes the values of the partition function p(n). We show that the coefficients of these forms are given by traces of singular invariants. These are values of modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoglu, and Toth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

12:30 pm in 345 Altgeld Hall,Thursday, November 6, 2014

#### Revisiting the p - q duality of 2D quantum gravity

###### Martin Luu (UIUC Math)

Abstract: I will explain how one can view the p - q duality of 2D gravity as a kind of Fourier duality. More precisely, it will be the so-called local Fourier transform that plays a key role. It was introduced in the late 80’s by Laumon in the context of p-adic sheaves and later adapted to the complex setting by Bloch and Esnault. This viewpoint on the p - q duality also reveals very close connections with certain Langlands dualities and I will briefly explain this as well.

1:00 pm in 243 Altgeld Hall,Thursday, November 6, 2014

#### Local finiteness of the curve complex and tight geodesics.

###### Yohsuke Watanabe (Utah Math)

Abstract: The curve complex is locally infinite. In this talk, we recover this obstacle in the following sense. First, we observe the property which any locally finite graph with a uniformly bounded valency satisfies. Then, we show that the curve complex also satisfies this property after passing to the curve complex of some subsurface by subsurface projections. As a corollary of this, we give a computable and uniform bound for the cardinality of the Bowditch’s slices on tight geodesics. View talk at http://youtu.be/oOZcjhBdhUA

2:00 pm in 243 Altgeld Hall,Thursday, November 6, 2014

#### Polynomial hulls without analytic structure

###### Alexander Izzo (Bowling Green State University and Indiana University )

Abstract: It was once hoped that whenever a compact set in complex Euclidean space has a nontrivial polynomially convex hull, there must be analytic structure in the hull. This hope was dashed by a counterexample given by Stolzenberg in 1963. I will present recent joint work with Samuelsson Kalm and Wold showing that every smooth manifold of dimension at least three can be smoothly embedded in some complex Euclidean space so as to have hull without analytic structure and present current joint work with Stout extending this to two dimensional manifolds. (It is well known that a smoothly embedded one dimensional manifold never has hull without analytic structure.)

3:00 pm in 243 Altgeld Hall,Thursday, November 6, 2014

#### A Brief History of Lech’s Conjecture

###### Matthew Mastroeni (UIUC Math)

Abstract: Lech’s Conjecture is a very simple statement about the behavior of Hilbert-Samuel multiplicity under a flat local extension of Noetherian local rings. And yet, in the fifty years since Lech first made his conjecture, not much can be said about its validity in general. In this talk, we will attempt to summarize as much as possible what is currently known about Lech’s Conjecture, to indicate various simplifications that can be made to the problem, and to describe the main techniques that have been used to attack it.

4:00 pm in 245 Altgeld Hall,Thursday, November 6, 2014

#### Partition Regular Equations.

###### Imre Leader (University of Cambridge, UK)

Abstract: A finite or infinite matrix M is called `partition regular' if whenever the natural numbers are finitely coloured there exists a vector x, with all of its entries the same colour, such that Mx=0. Many of the classical results of Ramsey theory, such as van der Waerden's theorem or Schur's theorem, may be naturally rephrased as assertions that certain matrices are partition regular. While the structure of finite partition regular matrices is well understood, little is known in the infinite case. In this talk we will review some known results and then proceed to some recent developments. No knowledge of anything at all will be assumed.