Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 31, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 31, 2015

1:00 pm in 347 Altgeld Hall,Tuesday, March 31, 2015

Gap Eigenvalues and the Asymptotic Behavior of Geometric Wave Equations on Hyperbolic Space

Sohrab Shahshahani (U Michigan Math)

Abstract: (joint work with A. Lawrie and S.-J. Oh) We consider the equivairant wave maps problem from the hyperbolic plane into two model rotationally symmetric targets, namely the two sphere and the hyperbolic plane itself. Due to the non-Euclidean geometry of the domain, these problem exhibits markedly different phenomena compared to its Euclidean counterpart. For instance, there exist a one parameter family of finite energy stationary solutions (harmonic maps) even in the case where the target of the wave map is negatively curved (i.e. $H^2$). Moreover, when the target is the sphere the spectrum of the linearized operator about certain stationary solutions possesses a gap eigenvalue, i.e., a simple eigenvalue in the gap $(0, 1/4)$ between $0$ and the essential spectrum of the Laplacian on hyperbolic plane. The existence of such a gap eigenvalue raises interesting questions about the asymptotic dynamics of the solutions to the corresponding wave equations. If time permits, we will discuss similar phenomena for an equivariant Yang-Mills problem on hyperbolic space.

1:00 pm in 243 Altgeld Hall,Tuesday, March 31, 2015

Approximating codimension one foliations of 3-manifolds

Rachel Roberts (Washington University)

Abstract: Eliashberg and Thurston proved that any smooth taut co-oriented foliation of a closed, orientable 3-manifold can be approximated by a pair of contact structures, one positive and one negative. These contact structures are weakly symplectically fillable and universally tight. This result has been used both to establish the fillability of certain contact structures and to prove the nonexistence of taut foliations in certain 3-manifolds. Sometimes this has been done without consideration of the smoothness of the foliations under consideration, resulting in proof gaps. I'll discuss 3-manifolds and 2-plane fields on 3-manifolds. I'll then focus in on the statement of the Elisahberg-Thurston theorem, defining all terms, and give an overview of its proof. As time permits, I will describe how to generalize the Eliashberg-Thurston theorem to continuous foliations. This work is joint with Will Kazez.

2:00 pm in 347 Altgeld Hall,Tuesday, March 31, 2015

Analysis on fractals and differential forms on Dirichlet spaces

Daniel Kelleher (Purdue)

Abstract: Analysis on fractals is a subject which lies at the intersection of probability, analysis and geometry. I will begin by talking about some of the advances and problems in the area, such as convergence of Laplacians/central limit theorems for random walks on approximating graphs, or calculating the spectrum and heat kernel estimates on the limiting fractal. I will also talk about the development of geometric objects on spaces with Dirichlet forms, such as intrinsic metrics, differential forms and Dirac operators.

3:00 pm in 241 Altgeld Hall,Tuesday, March 31, 2015

Stability of extremal graphs, Simonovits' stability from Szemeredi's regularity

Zoltan Furedi   [email] (Renyi Institute of Mathematics and UIUC Math)

Abstract: The following sharpening of Turan's theorem is proved. Let $T_{n,p}$ denote the complete $p$-partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there exists an (at most) $p$-chromatic subgraph $H_0$ such that $e(H_0)\geq e(G)-t$. Using this result we present a concise, contemporary proof (i.e., one applying Szemeredi's regularity lemma) for the classical stability result of Simonovits.

4:00 pm in 243 Altgeld Hall,Tuesday, March 31, 2015

Tauberian theory

Byron Heersink (UIUC Math)

Abstract: We will give an introductory overview of some of the results and methods in Tauberian theory, giving special attention to Freud's method of obtaining an error term for Karamata's Tauberian theorem.