Department of

Mathematics


Seminar Calendar
for events the day of Monday, October 14, 2013.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, October 14, 2013

10:00 am in 145 Altgeld Hall,Monday, October 14, 2013

A generalization of the group of Hamiltonian homeomorphisms

Augustin Banyaga (Pennsylvania State University Math)

Abstract: The Eliashberg-Gromov rigidity theorem implies that Symplectic Geometry underlines a topology. This talk is about the automorphism groups of this "continuous" symplectic topology. The group of symplectic homeomorphisms (Sympeo) has a remarkable subgroup: the group of Hamiltonian homeomorphisms (Hameo), defined by Oh and Müller using the $L^{(1q,\infty)}$ Hofer norm. We introduce a generalization of Hameo, called the group of strong symplectic homeomorphisms (SSympeo), using a generalization of the Hofer norm from the group of Hamiltonian diffeomorphisms to the whole group of symplectic diffeomorphisms. Each group Hameo and SSympeo has also a $L^\infty$ version. The two versions coincide (Müller, Banyaga-Tchuiaga).

1:30 pm in 345 Altgeld Hall,Monday, October 14, 2013

The Configuration Space of Graphs

Sishen Zhou (UIUC Math)

Abstract: I will introduce the story of exploring the structure of configuration space of n points on a graph. This problem comes from the motion planning problem of robots in a factory with narrow pathways. Robert Ghrist discovered that this space is homotopy equivalent to a CW-complex of dimension determined by the structure of the graph (and surprisingly, independent of n). Farley and Sabalka applied Discrete Morse Theory to visualize those cells and to simplify the computation of the fundamental group and the (co)homology. We will go through the details of their work.

4:00 pm in Altgeld Hall 143,Monday, October 14, 2013

Relative Fatou theorem for $\Delta+\Delta^{\alpha/2}$ in $C^{1,1}$ open sets

Hyunchul Park (William and Mary Math)

Abstract: In 1906 Fatou showed that bounded (classical) harmonic functions in the open disk have nontangential limits almost everywhere on the unit circle. This theorem is called Fatou theorem and it is extended to some general open sets and for various diffusion processes. But when the processes have jumps, Fatou theorem does not hold even for very regular domains. Instead it is proved that \textit{relative} Fatou theorem holds for positive harmonic functions with respect to stable processes in various open sets. In this talk we will show that relative Fatou theorem is true for (independent) sum of Brownian motions and stable processes in $C^{1,1}$ open sets under minimal conditions. We will investigate the special case when the reference measure is given by the surface measure of the domain. Time permitting we will also discuss its connection to Hardy spaces of harmonic functions with respect to sum of Brownian motions and stable processes. This is a joint work with Y. Lee.

4:00 pm in 241 Altgeld Hall,Monday, October 14, 2013

Distribution of digits of rational power sequences

Yiannis Konstantoulas (Illinois)

Abstract: In this talk, we will describe recent history of and the author's ongoing work on the distribution of digits of the power sequence $r_n=\left(\frac{3}{2}\right)^n$ and other rational power sequences. Using algebraic and combinatorial tools, we will show that for the $n$-fold dyadic partition of the interval $[0,1]$, one can find among the $2^n$ subintervals at least $cn$ distinct intervals containing a limit point of the fractional parts of $r_n$ for an absolute constant $c>0$. Finally, we will introduce a topological space whose large scale structure reflects the metric theory of the map $\{(r_n)\xi\}$ and link the problem of distribution of digits of $r_n$ to properties of random walks on that topological space. The talk will contain many questions, a few answers and hopefully an interesting new perspective on rational power sequence maps.

5:00 pm in 241 Altgeld,Monday, October 14, 2013

Some unique group-measure space decomposition for type II_1 factors (after Popa-Vaes and Chifan-Peterson): Part II

Bogdan Udrea (UIUC Math)