Department of

Mathematics


Seminar Calendar
for events the day of Thursday, October 31, 2013.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 31, 2013

1:00 pm in Altgeld Hall 347,Thursday, October 31, 2013

Generalized north-south dynamics on the space of geodesic currents

Caglar Uyanik (UIUC Math)

Abstract: The study of outer automorphism group of a free group $Out(F_N)$ is closely related to study of Mapping Class Groups. The $Out(F_N)$ analog of a pseudo-Anosov homeomorphism is called a fully irreducible or iwip (short for irreducible with irreducible powers). We will talk about types of iwips, geometric and hyperbolic, and prove that any iwip $\phi$ acts on the space of geodesic currents with a certain kind of north-south dynamics. As an application, we will give a criterion for subgroups of $Out(F_N)$ to contain a hyperbolic iwip.

2:00 pm in 243 Altgeld Hall,Thursday, October 31, 2013

Current results and future directions regarding fractal billiard tables.

Robert Niemeyer (University of New Mexico)

Abstract: In this talk, we will discuss recent results on three different fractal billiard tables: the Koch snowflake, a self-similar Sierpinski carpet and (for lack of a better name and none ever given in the literature) the T-fractal. Initially, an investigation of the flow on a fractal billiard table was made on the Koch snowflake. Results on the Koch snowflake motivated us to investigate the other two billiard tables. Using a result of J. Tyson and E. Durand-Cartagena, we show that there are periodic orbits of a self-similar Sierpinski carpet billiard table. J. P. Chen and I have begun investigating whether or not it makes sense to discuss the existence of a dense orbit of a self-similar Sierpinski carpet billiard table. Substantial progress has been made in determining periodic orbits of the T-fractal billiard table. We detail some of the recent results concerning periodic orbits, determined in collaboration with M. L. Lapidus and R. L. Miller. Less has been done to determine what may constitute a dense orbit of the T-fractal billiard. We provide substantial experimental and theoretical evidence in support of the existence of an orbit that is dense in the T-fractal billiard table but is not a space-filling curve. We briefly touch on a long-term goal of determining a topological dichotomy for the flow on a fractal billiard table, namely that, in a fixed direction, the flow is either closed or minimal. Parts of this talk will be suitable for an advanced undergraduate/beginning graduate student audience.

4:00 pm in 245 Altgeld Hall,Thursday, October 31, 2013

Lecture 3: Optimal bounds for singular integrals, Fourier multipliers, connections to rank-one convexity, quasi convexity and weighted norm inequalities.

Rodrigo Banuelos (Purdue University)

Abstract: In 1966, D. L. Burkholder published his seminal paper on the boundedness of martingale trans- form. While the main results were of considerable importance, the ideas introduced in this paper became the building blocks for many subsequent results in martingales, leading to the celebrated Burkholder-Davis-Gundy inequalities (the bread and butter of modern stochastic analysis) and to numerous applications in analysis, including the solution by Burkholder, Gundy and Silver- stein of a longstanding open problem concerning a real variables characterization of the Hardy spaces. In 1984, motivated in part by applications to the geometry of Banach spaces, Burkholder gave a new proof of his 1966 inequality which bypassed L2 theory, introducing a novel and rev- olutionary new method for proving optimal inequalities in probability and harmonic analysis. This method, commonly referred to now days as the "Burkholder method" (and also as "the Bellman function method"), has had many applications in recent years. The aim of these lectures is to present some of these martingale inequalities and applications, emphasizing ideas rather than technicalities, taking a historical point of view but discussing applications to problems of current interest. 
Lecture 1: The 1966 martingale inequalities, the good-principle, some applications. 
Lecture 2: Sharp inequalities. The "why" and the "how".
Lecture 3: Optimal bounds for singular integrals, Fourier multipliers, connections to rank-one convexity, quasi convexity and weighted norm inequalities.