UIUC Dept. of Mathematics Seminar Calendar

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                        30 31

Thursday, October 20, 2011

Department of Mathematics Retiree's Luncheon
11:30 am in Kennedy's Restaurant, Thursday, October 20, 2011

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Submitted by seminar.

Number Theory
1:00 pm in 241 Altgeld Hall, Thursday, October 20, 2011

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Submitted by jathreya.

Yitwah Cheung (San Francisco State University)
Some applications of diagonal flows to simultaneous Diophantine approximation
Abstract: In this talk I will discuss an approach to the study of diagonal flows that has recently led to some new results in Diophantine approximation, particularly in higher dimensions. These include a generalization of the fundamental inequalities satisfied by convergents of continued fractions (Ann. of Math., 173, (2011), 127-167), determination of Hausdorff dimension of the set of singular vectors (joint with Nicolas Chevallier), and a solution to Schmidt-Starkov type questions regarding the behavior of successive minima of lattices (joint with Barak Weiss).

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, October 20, 2011

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Submitted by kapovich.

Andrey Nikolaev (Stevens Institute of Technology)
Verbal subgroups of hyperbolic groups have infinite width
Abstract: Let $G$ be a non-elementary hyperbolic group. Let $w$ be a group word such that the set $w[G]$ of all its values in $G$ does not coincide with $G$ or $1$. We show that the width of verbal subgroup $w(G)=\langle w[G]\rangle$ is infinite. That is, there is no such $l\in\mathbb Z$ that any $g\in w(G)$ can be represented as a product of $\le l$ values of $w$ and their inverses. As a consequence, we obtain the same result for a wide class of relatively hyperbolic groups. This is a joint work with Alexei Myasnikov.

Graduate Geometry and Topology Seminar
2:00 pm in 241 Altgeld Hall, Thursday, October 20, 2011

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Submitted by lukyane2.

Chih-Chung Liu (Department of Mathematics, University of Illinois)
Evolution of Metrics on Vector(Line) Bundles
Abstract: I will briefly introduce/summarize the properties of Hermitian metric $H$ on a vector bundle $E$ and connections and curvatures it defines. A brief introduction of Chern classes, which is a topological invariants derived from curvatures but independent of connections, will be presented. With these preliminaries, I will introduce PDE's of connection and sections defined on a vector bundle, which are often minimizing equations of some functionals carrying physical information. The space of solutions to these PDE's is linked tightly to some algebraic properties of the vector bundles called "stability". If time permitted, I will introduce dynamics of solutions, i.e. metrics on the moduli space of solutions.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 20, 2011

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Submitted by aimo.

Ron Ji (Indiana University and Purdue University at Indianapolis)
Relative Property A and relative amenability for countable groups
Abstract: We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if $G$ has property A relative to a family of subgroups ${\mathcal H}$ and if each $H\in {\mathcal H}$ has property A, then $G$ has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with stabilizers of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, October 20, 2011

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Submitted by kapovich.

Yitwah Cheung (San Francisco State University)
Ergodic properties of the BCZ map
Abstract: The "BCZ map" was introduced by Boca, Cobeli and Zaharescu in 2001 as a powerful tool to study the statistical properties of Farey sequences. Recognizing its importance as an object of study in its own right, they later raised questions about the ergodic properties of this map. In this talk, we will show that the BCZ map is in fact ergodic and has entropy zero. These results essentially follow from the fact that the BCZ map can be viewed as a Poincare section of the horocycle flow on the modular surface. Using the well-known equidistribution principle for closed horocycle orbits, we are also able to give new proofs of some of the earlier results on Farey statistics. If time permits, I will also discuss an application of the BCZ map to analysis of the behavior of horocycle orbits, which was the original motivation for our investigations. This is joint work with Jayadev Athreya.