Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 17, 2015.

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Tuesday, February 17, 2015

1:00 pm in Altgeld Hall 243,Tuesday, February 17, 2015

A mapping class group invariant parameterization of maximal $Sp(4,\mathbb{R})$ representations

Brian Collier (UIUC Math)

Abstract: Let $S$ be a closed surface of genus at least 2, and consider the moduli space of representations $\rho:\pi_1(S)\rightarrow Sp(4,\mathbb{R}).$ There is an invariant $\tau\in\mathbb{Z},$ called the Toledo invariant, which helps to distinguish connected components. The Toledo invariant satisfies a Milnor-Wood inequality $|\tau|\leq 2g-2.$ Representations with maximal Toledo invariant have many geometrically interesting properties, for instance, they are all discrete and faithful. In this talk, we will give a mapping class group invariant parameterization of all smooth connected components of the maximal $Sp(4,\mathbb{R})$ representations. Our main tool is Higgs bundles. However, to utilize Higgs bundle techniques, one has to fix a conformal structure of the surface $S,$ hence breaking the mapping class group symmetry. To restore the symmetry, we associate a unique `preferred' conformal structure to each such representation. This is done by exploiting the relationship between the associated Higgs bundles and minimal surfaces. View talk at http://youtu.be/lhJhEI1FGb8

1:00 pm in 345 Altgeld Hall,Tuesday, February 17, 2015

The model-theoretic content of a result of Junge and Pisier

Isaac Goldbring (UIC)

Abstract: An operator space is an operator-norm closed linear subspace of B(H), the C* algebra of bounded linear on a Hilbert space H. For reasons that will be explained in the talk, an operator space can be viewed as a noncommutative generalization of a Banach space. A fundamental result of Junge and Pisier states that there are many more operator spaces than there are Banach spaces. More precisely, for any n at least 3, the set of n-dimensional operator spaces, when equipped with the strong topology, is not separable. As a corollary, there is no universal separable operator space. In this talk, I will mention how the techniques of Junge and Pisier can be used to prove a model-theoretic result, namely that the class of so-called 1-exact operator spaces is not an omitting types class. I will mention a descriptive-set theoretic strategy for removing this dependence on the techniques of Junge and Pisier, which would be desirable as I will then show how the fact that the! 1-exact operator spaces are not an omitting types class, together with a model-theoretic analysis of the theory of the Gurarij operator space, can be used to give a purely model-theoretic proof of the fact that there is no universal separable operator space. This is joint work with Thomas Sinclair and separately with Martino Lupini.

1:00 pm in 347 Altgeld Hall,Tuesday, February 17, 2015

Two dimensional water waves in holomorphic coordinates

Mihaela Ifrim (UC Berkeley)

Abstract: This is joint work with Daniel Tataru, and in parts with John Hunter. My talk is concerned with the infinite depth water wave equation in two space dimensions, with either gravity or surface tension. Both cases will be discussed in parallel. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves case, we were the first to establish a global result (two months later, Ionescu-Pusateri also announced a related result). Our goal is improve the understanding of these problems by providing a single setting for both cases, and presenting simpler proofs. The talk will try to be self contained.

3:00 pm in 243 Altgeld Hall,Tuesday, February 17, 2015

On the punctual Hilbert scheme

Bill Haboush (UIUC Math)

Abstract: We consider simultaneous conjugacy classes of q nxn nilpotent matrices. We show that this is the set of representations of a certain nil algebra and that the representations satisfying a particular regularity condition are just the punctual Hilbert scheme of schemes of a certain co-length supported at a point of Projective n-space. We show that this is a homogeneous space and that it can be interpreted as the stable points of a certain action. We show that the scheme is an affine bundle over a projective space and that it is not a vector bundle. (Joint with D. H. Hyeon and H. P. Kraft.)

3:00 pm in 241 Altgeld Hall,Tuesday, February 17, 2015

Regular Graphs Into 4-Regular Graphs with Triangle-free Euler Tours

Michael Plantholt (Department of Mathematics - Illinois State University )

Abstract: Finding euler tours with higher girth is a standard technique for decomposing a graph into paths of given length. Heinrich, Liu and Yu used this approach to show that every 4-regular graph with size a multiple of 3 has a decomposition into paths of length 3. Oksimets generalized this by fully determining which graphs with all degrees even have a decomposition into paths of length 3. She also determined conditions for complete graphs to have decompositions into paths of given length by using the euler tour approach. We investigate the structure of regular graphs of even degree. We show that for k > 1, any connected 4k-regular graph has a decomposition into 4-regular graphs, each having a triangle-free euler tour, and that these tours can be re-connected in a certain way to get a triangle-free euler tour of the original. The key tool in the result above is the result by Heinrich et al that any 4-regular graph has a triangle-free euler tour if and only if no 5 vertices induce K5 or K5 - e. Thus our proof of the result above requires the 4-regular graphs in the decomposition to be free of these dense subgraphs. We discuss the approach to that problem, and possible generalizations.

4:00 pm in 243 Altgeld Hall,Tuesday, February 17, 2015

Bi-Lipschitz embedding of generalized Grushin spaces

Matthew Romney

Abstract: When can a metric space be embedded in Euclidean space under a bilipschitz mapping? We present a new sufficient condition for embeddability based on so-called Whitney curvature bounds. The motivation comes from the classical Grushin plane and its higher-dimensional generalizations. Embeddability of the Grushin plane was recently proven by Seo; our new condition provides an approach for proving embeddability of higher-dimensional spaces. This talk is in preparation for my prelim exam.