UIUC Dept. of Mathematics Seminar Calendar

    February 2011            March 2011             April 2011     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5          1  2  3  4  5                   1  2
  6  7  8  9 10 11 12    6  7  8  9 10 11 12    3  4  5  6  7  8  9
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 27 28                  27 28 29 30 31         24 25 26 27 28 29 30

Tuesday, March 1, 2011

Topology Seminar
11:00 am in 241 Altgeld Hall, Tuesday, March 1, 2011

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Submitted by mando.

HIsham Sati (University of Maryland)
M-theory and elliptic cohomology

Joint Ergodic Theory/Differential Geometry
1:00 pm in 243 Altgeld Hall, Tuesday, March 1, 2011

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Submitted by jathreya.

Steve Lalley (University of Chicago)
Self-Intersections of Closed Curves and Closed Geodesics on Negatively Curved Surfaces
Abstract: Fix a compact surface M with a Riemannian metric of constant curvature -1. From among all closed geodesics on M of length no greater than x, choose one at random; let L be its length, and let N be the number of its transversal self-intersections. A simple argument shows that as x gets large N / x^2 converges almost surely to a constant K. How big are the fluctuations of N around the mean? We will argue that they are of order x, and that [ N - K L^2 ]/x converges in distribution, most likely not to a Gaussian distribution. On the other hand, if the closed geodesic is instead chosen at random from among all closed geodesics on M represented by (conjugacy classes of) elements in the fundamental group with word length (in the natural generators) no larger than x, then fluctuations of N are of order x^(3/2), and the limit distribution is Gaussian. Part of this is joint work with Moira Chas.

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, March 1, 2011

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Submitted by sba.

Gregory Galperin (Eastern Illinois University)
Making invisible objects
Abstract: In my talk, I will show some examples of "invisible geometric objects". This means that light bends around an object, causing it to become as if it were invisible. When designing such objects, the billiard law of reflection will play a crucial role. The main example in my talk will concern the billiard reflection for ellipses and hyperbolas. To prove that the reflection in my design is indeed billiard, it is necessary to prove a tricky lemma about an angle bisector of a triangle. If time permits, I will provide a proof for this lemma. Everyone (especially students!) are invited to visit the talk. No prior knowledge is required; all necessary definitions and explanations will be given during the talk.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, March 1, 2011

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Submitted by west.

Ping Hu (UIUC Math)
Chromatic thresholds of hypergraphs
Abstract: Let F be a family of graphs. The chromatic threshold of F is the infimum of the values c such that the subfamily of F consisting of graphs G with minimum degree at least c|V(G)| has bounded chromatic number. This was defined by Andrásfai, Erdős and Sós in 1974 for K_l-free graphs. Łuczak and Thommassé showed how to use Vapnik-Chervonenkis dimension to give a new short proof that the chromatic threshold of triangle-free graphs is at most 1/3. They also generalized VC-dimension to paired VC-dimension and used it to give a new proof that C₅-free graphs have chromatic threshold 0. We generalize the concept for hypergraphs. Besides surveying results for graphs, this talk will focus on constructions providing lower bounds on chromatic threshold for various hypergraphs. Our constructions are based on an idea of Hajnal, using the Kneser graph. (Joint work with József Balogh, Jane Butterfield, John Lenz, and Dhruv Mubayi.)