UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, April 12, 2011

Topology Seminar
11:00 am in 241 Altgeld Hall, Tuesday, April 12, 2011

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Submitted by mando.

Marcy Robertson (U. Western Ontario)
Spaces of operad structures
Abstract: A multicategory, also known as a colored operad, is simply an operad with many objects. While operads encode families of algebras in a symmetric monoidal category of interest, multicategories encode more complex algebraic structures such as modules, bimodules, and morphisms between algebras. In this talk we focus on studying maps between multicategories enriched in simplicial sets or symmetric spectra. We show that the space of maps between any two multicategories can be computed as the moduli of a certain small category of bimodules. Time permitting, we will discuss applications of this theory to computing invariants of cosimplicial spaces.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, April 12, 2011

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Submitted by west.

Andrew Treglown (University of Birmingham, U.K.)
Matchings in 3-uniform hypergraphs
Abstract: A theorem of Tutte characterises the graphs that have a perfect matching. In contrast, a result of Garey and Johnson implies that the decision problem whether an r-uniform hypergraph contains a perfect matching is NP-complete for r>2. So it is natural to seek simple sufficient conditions that ensure a perfect matching. Given an r-uniform hypergraph H, the degree of a k-tuple (x₁,…,x_k) of vertices is the number of edges in H containing x₁,…,x_k. The minimum vertex degree of H is the minimum of these degrees over all 1-tuples. The minimum codegree of H is the minimum of all the degrees over all (r-1)-tuples of vertices in H.

In recent years there has been significant progress on this problem. In 2009, Rödl, Ruciński and Szemerédi characterised the minimum codegree that ensures a perfect matching in an r-uniform hypergraph. Much less is known about minimum vertex degree conditions for perfect matchings in r-uniform hypergraphs. In the case r=3, Hàn, Person, and Schacht asymptotically determined the minimum vertex degree that ensures a perfect matching. In this talk we discuss a result that determines this threshold exactly. (Joint work with Daniela Kühn and Deryk Osthus.)

Ergodic Theory
4:00 pm in 347 Altgeld Hall, Tuesday, April 12, 2011

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Submitted by jathreya.

Howard Masur (University of Chicago)
Ergodicity of the Weil-Petersson geodesic flow
Abstract: This is joint work with Keith Burns and Amie Wilkinson. Let $\Sigma$ be a surface of genus g with n punctures. We assume 3g-3+n>0. Associated to \Sigma is the Teichmuller space. This is the space of hyperbolic metrics one can put on \Sigma, up to isotopy. The mapping class group acts on the Teichmuller space with quotient, the Riemann moduli space \cal M(\Sigma). There are a number of interesting metrics on \cal M(\Sigma); one of which is the Weil-Petersson metric. It is a Riemannian metric of negative curvature and finite volume but it is not complete. In this talk I will discuss the background on this metric and the following theorem. Theorem: The Weil-Petersson geodesic flow is ergodic on \cal M(\Sigma).