Department of

Mathematics


Seminar Calendar
for events the day of Monday, February 17, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, February 17, 2014

2:00 pm in 147 Altgeld Hall,Monday, February 17, 2014

Phase Transitions in Random Čech complexes

Omer Bobrowski (Duke Math)

Abstract: In manifold learning, one often wishes to infer geometric and topological features of an unknown manifold embedded in a d-dimensional Euclidean space from a finite (random) point cloud. One topological invariant of a considerable interest is the homology of the underlying space. A common method for recovering the homology of a manifold from a set of random samples is to cover each point with a d-dimensional ball and study the union of these balls. By the Nerve Lemma, this method is equivalent to study the homology of the Čech complex generated from the random point cloud. In this talk we discuss the limiting behavior of random Čech complexes as the sample size goes to infinity and the radius of the balls goes to zero. We show that the limiting behavior exhibits multiple phase transitions at different levels, depending on the rate at which the radius of the balls goes to zero. We present the different regimes and phase transitions discovered so far, and observe the nicely ordered fashion in which homology groups of different dimensions appear and vanish. One interesting consequence of this analysis is a sufficient condition for the random Čech complex to successfully recover the homology of the original manifold.

3:00 pm in 145 AH,Monday, February 17, 2014

Upper bounds for the Gromov width of coadjoint orbits of compact Lie groups

Alexander Caviedes Castro (University of Toronto)

Abstract: I will show how to find an upper bound for the Gromov width of coadjoint orbits with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated Non-squeezing theorem.

4:00 pm in 241 Altgeld Hall,Monday, February 17, 2014

Straus's examples and ergodic theory

Donald Robertson (Ohio State)

Abstract: A result due to Hindman states that, no matter how the positive integers are finitely partitioned, one cell of the partition contains a sequence and all its sums without repetition. Straus, answering a question of Erdos, later gave an example showing that a density version of Hindman's result does not hold. He exhibited sets of positive integers with arbitrarily large density, each having the property that no shift contains a sums set of the above kind. In this talk I will present recent joint work with V. Bergelson, C. Christopherson and P. Zorin-Kranich in which we generalize Straus's example to a class of locally compact, second countable, amenable groups and show, using ergodic theory techniques recently developed by Host and Austin, that positive density subsets of groups outside this class must contains sets with strong combinatorial properties. In particular, this allows us to give a combinatorial characterization of minimally almost periodic, amenable groups.

4:00 pm in 143 Altgeld Hall,Monday, February 17, 2014

Super Yang-Mills and spinors

Sheldon Katz (Illinois Math)

Abstract: I will finish the overview that I started last week by describing N=1 super Yang-Mills theories. Then I will go back and add expository material about spinors, from the viewpoints of both physics and mathematics. The presentation of this expository material will be primarily organized to benefit mathematicians who are not familiar with this circle of ideas.

5:00 pm in 241 Altgeld,Monday, February 17, 2014

On the Kadison-Singer Problem, Part III

Qiang Zeng (UIUC Math)