UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, October 11, 2011

Topology Seminar
11:00 am in 241 Altgeld Hall, Tuesday, October 11, 2011

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Submitted by franklan.

Olga Stroilova (MIT)
The dual of L(n)
Abstract: Fix a prime p, and work in the p-complete setting. The spectrum L(n) can be presented as a summand of the Steinberg factor in the classifying spectrum of B(Z/p)ⁿ. Studying the action of the general linear group GL_n(F_p) on the decomposition of the functional dual of B(Z/p)ⁿ given by the Segal conjecture allows us to show that F(L(n), S) is a wedge of just two indecomposables - a copy of L(n) and an L(n-1). This result was also previously independently obtained by Alan Cathcart in his thesis under the guidance of J. Frank Adams.

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, October 11, 2011

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Submitted by jathreya.

Robert Niemeyer (UC-Riverside)
On the nature of periodic orbits of fractal billiard tables
Abstract: First, we discuss various results on the Koch snowflake fractal billiard. We give a construction of Cantor orbits and hybrid periodic orbits of a prefractal approximation of the Koch snowflake fractal billiard. We show that there exist countably infinitely many directions for which a compatible sequence of periodic orbits exists. We discuss properties of certain compatible sequences of orbits and describe other self-similar fractal billiard tables for which a number of our results hold. We then speculate on the necessary criterion for our results holding in a more general setting. We then provide a number of open problems and outline a number of avenues of research in fractal billiards. We finish by describing an analogy between a region with varying indecies of refraction and cells of a prefractal approximation to the Koch snowflake fractal billiard.

Harmonic Analysis and Differential Equations
1:00 pm in 347 Altgeld Hall, Tuesday, October 11, 2011

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Submitted by ekirr.

Eduard Kirr [email] (UIUC Math)
On the global bifurcation picture in NLS
Abstract: I will present recent results, obtained in collaboration with Vivek Natarajan (UIUC), on the properties of the ground state branches of the nonlinear Schroedinger equations. Motivated by applications in optics, fluids, quantum chemistry and condensed matter physics we would like to know all these branches, together with the excited state branches, and the way they connect to each other, i.e. the way they bifurcate. While the results I will present make a significant progress in this direction, the complete description of all the nonlinear bound states of the Schroedinger equation is still an open problem.

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, October 11, 2011

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Submitted by sba.

Gregory Galperin (EIU)
Billiard bouncing in gravitational field
Abstract: There is a collection of semi-circles of diameter 1 in the upper half plane centered at the integer points (n, 0) on the x-axis. A released billiard ball falls down under the vertical constant gravitational force g. The ball bounces off the semi-circles according to the billiard law and describes a tra jectory γ . Record the indices of the semi-circles the ball hits as a sequence ω = (ω1 , ω2 , ...), which we call the one-sided itinerary of the tra jectory γ . What one-sided itineraries are realizable, if one can change (a) the initial height of the bal l; (b) both the initial height and the initial velocity of the bal l? For example, can the following itinerary ω = (ω1 , ω2 , ...) be realized by some billiard tra jectory γ ? Draw the digits of π = 3.14159265... on an one-sided infinite strip and cut this strip into pieces of positive integers, each of which has an arbitrary length not exceeding one billion. Reversing time, we can also consider billiard tra jectories in the gravitational field g with two-sided itineraries. What two-sided itineraries are realizable? For example, can the following two-sided itinerary ω = (..., ω −2 , ω−1 , ω0 , ω1 , ω2 , ...) be realized by some billiard tra jectory γ ? Draw the digits of π = 3.14159265... in the forward direction and the digits of e = 2.718281828... in the backward direction on an infinite two-sided strip and cut this strip into pieces of positive integers, each of which has an arbitrary length not exceeding one billion. The speaker will describe the sets of all realizable one- and two-sided itineraries of billiard tra jectories.

Probability Seminar
2:00 pm in 347 Altgeld Hall, Tuesday, October 11, 2011

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Submitted by rsong.

Prof. Robin Pemantle (University of Pennsylvania)
Concentration inequalities under negative dependence
Abstract: Let X_1 , ... , X_n be a collection of binary valued random variables and let f be a Lipschitz function on {0,1}^n, with S := f(X_1 , ... , X_n). Concentration inequalities are known when {X_j} are independent or when f = sum_j X_j, and {X_j} are negatively associated, but it is unknown whether such an inequaolity holds for general Lipschitz functions of negatively associated variables. We show that the Strong Rayleigh condition is enough to imply good concentration. Examples include determinantal measures, generalized negative binomials and exclusion measures. We also prove a continuous version for determinantal point processes, examples of which include the zero set of a random Gaussian function. This is joint work with Yuval Peres.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall 245, Tuesday, October 11, 2011

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Submitted by kapovich.

Robin Pemantle (University of Pennsylvania)
On the coefficients of a bivariate rational function
Abstract: Problem: describe the asymptotic behavior of the coefficients a_{ij} of the Taylor series for 1/Q(x,y) where Q is a polynomial. This problem is the simplest of a number of such problems arising in analytic combinatorics whose answer was not until recently known. In joint work with J. van der Hoeven and T. DeVries, we give a solution that is completely effective and requires only assumptions that are met in the generic case. Symbolic algebraic computation and homotopy continuation tools are required for implementation.