UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, April 17, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, April 17, 2012

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Submitted by jathreya.

Slawomir Solecki (Department of Mathematics, University of Illinois at Urbana-Champaign)
Point realizations of Boolean actions
Abstract: We will look at measure preserving Boolean actions of Polish groups and consider the problem, going back to Mackey, of determining when such actions have point realizations. We will explore the boundary line between the groups whose all Boolean actions have point realizability and those that do not have this property. One result, joint with Kwiatkowska, states that Boolean action of Polish groups of isometries of locally compact separable metric spaces can always be point realized. On the other hand, a very recent result with Moore, states that the group of all continuous functions from an uncountable compact space to the circle does not have the point realizability property. In several respects, this last group is quite different from other groups that were shown earlier, by Vershik, Becker and Glasner-Weiss, not to have the point realizability property. Connections with the solution to Hilbert's 5-th problem, with the concentration of measure phenomena, and with the Cameron-Martin theorem will be mentioned.

Probability Seminar
2:00 pm in 347 Altgeld Hall, Tuesday, April 17, 2012

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Lee DeVille [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)
Noisy Perturbations of Kuramoto Oscillators
Abstract: We consider the Kuramoto model of coupled oscillators with various couplings and additive white noise. Synchronous solutions which are stable without the addition of noise become metastable and there are transitions amongst synchronous solutions on long timescales. We compute these timescales and, moreover, compute the most likely path in phase space that transitions will follow. We compute these timescales for various families of connection graphs and show that for several families, these timescales do not increase as the system size increases (i.e the action governing the large deviation timescale does not depend significantly on the number of oscillators).

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, April 17, 2012

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Submitted by sba.

Matthew Wright (Huntington University)
Hadwiger Integration and Applications
Abstract: Integration of integer-valued functionals with respect to Euler characteristic has surprising applications in problems arising in sensor networks, as Rob Ghrist demonstrated in the Trjitzinsky Lectures. This integration theory makes use of Euler characteristic as a topological invariant for counting objects detected by the network. However, Euler characteristic is only one of n+1 Euclidean-invariant valuations on "tame" subsets of n-dimensional Euclidean space. Integration of integer-valued functionals with respect to any of these valuations is straightforward. We can extend this integration theory to real-valued functionals to obtain what we call Hadwiger integrals. The Hadwiger integrals provide various notions of the size of a functional. This talk will explain the theory of Hadwiger integration and discuss some of its challenges and potential applications.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, April 17, 2012

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Submitted by west.

Frank R. Bernhart (visitor, Department of Mathematics, University of Illinois at Urbana-Champaign)
Enumeration of special classes of outerplanar graphs
Abstract: Arrange $n$ vertices equally spaced on a circle, labeled in counterclockwise order. Join some pairs by edges, drawn as chords without crossings. This general class has many interesting subfamilies, some of which can be counted using basic methods of enumeration, especially Lagrange Inversion. Catalan numbers and their generalizations arise frequently. We consider several such families:
(1) Polygons with $n$ sides and $r$ diagonals whose bounded regions have $k$ sides.
(2) Trees (no crossings allowed!).
(3) Graphs where every component is a triangle (requiring components to be edges yields a familiar family counted by the Catalan numbers).
(4) Every component is a vertex, an edge, or a cycle (plus subfamilies defined by forbidding isolated vertices or by forbidding edges joining consecutive points).
Finally, we also consider supplementing these techniques by Burnside's Lemma when drawings are considered to be equivalent under rotation and reflection, yielding for example an enumeration of maximal outerplanar graphs.