UIUC Dept. of Mathematics Seminar Calendar

      March 2012             April 2012              May 2012      
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Tuesday, April 24, 2012

Joint Ergodic Theory/Number Theory Seminar
11:00 am in 347 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by jathreya.

Francesco Cellarosi (IAS/MSRI)
Ergodic Properties of Square-Free Numbers
Abstract: We study binary and multiple correlations for the set of square-free numbers and we construct a dynamical systems naturally associated to them. We prove that such dynamical system has pure point spectrum and it is therefore isomorphic to a translation on a compact abelian group. In particular, the system is ergodic but not weakly mixing, and it has zero metric entropy. The latter results were announced recently by Peter Sarnak and our approach provides an alternative approach. Joint work with Yakov Sinai.

Differential Geometry Seminar
1:00 pm in 243 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by sba.

Steven Rayan (U Toronto Math)
Combinatorics of the moduli space of L-twisted Higgs bundles at genus 0
Abstract: An L-twisted Higgs bundle on a compact Riemann surface is a vector bundle together E with an L-valued Higgs field, that is, an endomorphism taking values along a fixed line bundle L. (Ordinary Higgs bundles arise by choosing the canonical line bundle for L.) The Betti numbers of the moduli space of L-twisted Higgs bundles on P^1, with fixed numerical invariants, can be determined by Hitchin's localization calculation: the Poincar\'e series of the moduli space is the (weighted) sum of Poincar\'e series of certain subvarieties of the nilpotent cone. These subvarieties are precisely moduli spaces of holomorphic chains: these are chains of vector bundles where the maps are L-twisted Higgs fields. Some of the difficulty in classifying these chains is avoided in the case of P^1, over which the situation becomes very combinatorial. I will calculate Betti numbers for certain low values of the rank of E and degree of L, in order to verify some conjectural numbers coming from Mozgovoy's twisted version of Chuang, Diaconescu, and Pan's ADHM formula. I will also make some conjectures about properties of the Betti numbers, including in the case of arbitrary genus.

Harmonic Analysis and Differential Equations
1:00 pm in 347 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by laugesen.

Oscar Lopez-Pamies [email] (UIUC Civil and Environmental Engineering)
Cavitation instabilities in nonlinear elastic solids: a defect-growth formulation based on iterated homogenization
Abstract: I will introduce a new formulation to study cavitation instabilities in nonlinear elasticity. The basic idea is to first cast cavitation as a homogenization problem of nonlinear elastic solids containing random distributions of zero-volume cavities, or defects. This problem is then addressed by means of a novel iterated homogenization procedure. Ultimately, the relevant calculations amount to solving Hamilton-Jacobi equations, in which the initial size of the defects plays the role of ``time'' and the applied load plays the role of ``space''. When specialized to the case of isotropic loading conditions, isotropic solids, and vacuous defects, the proposed formulation recovers the classical result of John Ball (1982) for radially symmetric cavitation. I will discuss the nature and implications of this remarkable connection.

Logic Seminar
1:00 pm in Altgeld Hall, Tuesday, April 24, 2012

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Submitted by ssolecki.

Anush Tserunyan (UCLA)
Finite generators for countable group actions
Abstract: Consider a Borel action of a countable group $G$ on a standard Borel space $X$. A countable Borel partition $P$ of $X$ is called a generator if $GP=\{ gA: g \in G, A \in P\}$ generates the Borel $\sigma$-algebra of $X$. Existence of such $P$ of cardinality $n$ is equivalent to the existence of a $G$-embedding of $X$ into the shift $n^G$. For $G=Z$, the Kolmogorov-Sinai theorem implies that finite generators don't exist in the presence of an invariant probability measure with infinite entropy. It was asked by Weiss in the late 80s, whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator. We show that the answer is positive in case $X$ admits a $\sigma$-compact topological realization (e.g. if $X$ is a $\sigma$-compact Polish $G$-space). We also show that finite generators always exist in the context of Baire category thus answering a question of Kechris. In fact, we show that if $X$ is a Polish $G$-space having infinite orbits, then there is a 4-generator on an invariant comeager set.

Probability Seminar
2:00 pm in 347 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by kkirkpat.

Jose Blanchet (Columbia Engineering)
A Large Deviations Theory for Heavy-tailed Processes via Martingale Arguments and its Connections to Monte Carlo
Abstract: Modern large deviations theory centers around the work of Donsker and Varadhan in the setting of processes whose marginal distributions have a finite moment generating. The existence of a finite moment generating function turns out to be crucial for the development of the theory in that it allows to define suitable positive martingales which, in turn, generate changes of measure. In important areas of application such as risk theory and operations research, however, stochastic processes with heavy-tails arise naturally; so no moment generating function exists. Moreover, it is well known (as we shall discuss) that the qualitative features of most likely paths given rare events are completely different in light and heavy tailed settings. One of the most advantageous features of the Donsker-Varadhan theory is that it suggests, via changes of measure, efficient Monte Carlo simulation methods for sampling rare events. In this talk, we present an approach for the large deviations analysis of heavy-tailed processes that is based on changes of measure and thus it is parallel to the Donsker-Varadhan approach in that the analysis suggests efficient Monte Carlo simulation methods for rare events as well. (This talk builds on joint work with P. Glynn and J. C. Liu).

Group Theory (note unusual day)
2:00 pm in 241 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by jathreya.

Shubojoy Gupta (Yale)
Asymptoticity of grafting and Teichmuller rays
Abstract: We shall discuss a result showing that any grafting ray in Teichmuller space is (strongly) asymptotic to a Teichmuller geodesic ray. Our method involves constructing quasiconformal maps between the underlying Thurston metric of a complex projective surface on one hand, and the singular flat metric induced by a holomorphic quadratic differential on the other. As a consequence we can show that the set of points in Teichmuller space obtained by integer graftings on any hyperbolic surface projects to a dense set in moduli space.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by west.

Mohsen Jamaali (Sharif University (Iran))
On harmonious coloring of graphs
Abstract: Let $G$ be a simple graph, and let $\Delta(G)$ denote the maximum degree of $G$. A harmonious coloring of $G$ is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colors in such a coloring. In this talk, with a constraint on $\Delta(G)$, we determine the exact value of $h(G)$ when $G$ is a tree. Furthermore, some bounds on $h(G)$ are obtained in general for trees and for bipartite graphs. An analogous concept of harmonious edge coloring is introduced, and some results on the harmonious edge-chromatic number are proved.