UIUC Dept. of Mathematics Seminar Calendar

     January 2010          February 2010            March 2010     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2       1  2  3  4  5  6       1  2  3  4  5  6
  3  4  5  6  7  8  9    7  8  9 10 11 12 13    7  8  9 10 11 12 13
 10 11 12 13 14 15 16   14 15 16 17 18 19 20   14 15 16 17 18 19 20
 17 18 19 20 21 22 23   21 22 23 24 25 26 27   21 22 23 24 25 26 27
 24 25 26 27 28 29 30   28                     28 29 30 31         
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Thursday, February 25, 2010

Group Theory Seminar
1:00 pm in 347 Altgeld Hall, Thursday, February 25, 2010

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Submitted by nmd.

Nathan Dunfield (University of Illinois)
Hyperbolic surfaces bundles of least volume
Abstract: Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. I will describe an explicit family of genus-g bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, I will show prove that this is indeed the case when g is large. The proof combines a soft geometric limit argument with a detailed Neumann-Zagier asymptotic formula for the volumes of Dehn fillings. The examples are all Dehn fillings on the sibling of the Whitehead manifold, and one can also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations. (Joint work with John W. Aaber.)

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, February 25, 2010

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Submitted by jarouse.

Atul Dixit (UIUC Math)
Transformation formulas associated with integrals involving the Riemann Xi function
Abstract: Page 220 of Ramanujan's Lost Notebook contains a beautiful transformation formula involving the digamma function which is also associated with an integral involving the Riemann Xi function. Here we discuss some new transformation formulas of this type, of which one generalizes the above-mentioned formula of Ramanujan. Also included are new extensions of formulas of N.S. Koshliakov, A.P. Guinand and W.L. Ferrar. The interesting history behind some of these formulas will also be discussed.

Probability Seminar
2:00 pm in 241 Altgeld Hall, Thursday, February 25, 2010

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Submitted by rsong.

Prof. Zhenqing Chen (University of Washington)
Sharp Heat Kernel Estimates for Fractional Laplacians in Unbounded Open Sets
Abstract: In this talk, I will present recent results on sharp heat kernel estimates for symmetric stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded $C^{1,1}$ open sets in Euclidean spaces: half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for the heat kernel for all time and all points in the open sets. As a consequence, sharp Green functions estimates will be given for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates will also be given for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets. Joint work with Joshua Tokle

Graph Theory and Combinatorics
3:00 pm in 1 Illini Hall (note change of room and day for this seminar), Thursday, February 25, 2010

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Submitted by west.

W. Tom Trotter (Georgia Institute of Technology)
Disjoint families of maximal chains and antichains
Abstract: Fix an integer k greater than 1. Duffus and Sands gave sufficient conditions for a poset to have k pairwise disjoint maximal antichains. The case k=2 is special (and straightforward). For k>2, they showed that if n≥k and n≤|C|≤n+(n-k)/(k-2) for every maximal chain C in P, then P has k pairwise disjoint maximal antichains. They showed that the result is sharp by constructing for n≥k a poset P(n,k) in which the upper bound on |C| is weaker by 1, yet P(n,k) has only k-1 pairwise disjoint maximal antichains. Since each P(n,k) is 2-dimensional, they noted that if the dual statement for families of pairwise disjoint maximal chains were to hold, then it would also be sharp. David Howard and the speaker then proved the dual version. Our approach actually yields a somewhat stronger result, which turns out also to hold for families of antichains. In this talk, we will outline the proofs.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, February 25, 2010

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Submitted by clein.

Zhen-Qing Chen (University of Washington)
Global Heat Kernel Estimates for Jump-Diffusions
Abstract: In this talk, I will describe recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric discontinuous processes (or equivalently, a class of symmetric integro-differential operators). A prototype of the Markov processes under consideration is the mixture of symmetric diffusion of uniformly elliptic divergence form operator and symmetric stable-like processes on $R^d$. I will focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, while briefly mention a priori Holder estimates and parabolic Harnack inequalities for their parabolic functions. To establish these results, we have employed methods from both probability theory and analysis. Based on joint work with Takashi Kumagai.