Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, September 3, 2013.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, September 3, 2013

11:00 am in 347 Altgeld Hall,Tuesday, September 3, 2013

#### Nazarov's estimate of Mahler's volume product.

###### Alex Tumanov (UIUC)

Abstract: Let K be a centrally symmetric convex body in R^n, and let K* be its dual, that is, K* consists of all y in R^n such that for all x in K the inner product xy<1. The quantity M(K)=vol(K)vol(K*) is called Maler's volume product. Mahler (1939) conjectured that the minimum of M(K) is attained on the cube Q. Mahler proved it only for n=2; for n>2 it is still open. Bourgain and Milman (1987) proved the estimate M(K) > c^n M(Q), in which the best constant was obtained by Kuperberg (2008). I present the amazing proof by Nazarov (2012) based on Bergman kernel and Hormander's estimates of d-bar equations.

1:00 pm in 345 Altgeld Hall,Tuesday, September 3, 2013

#### Generic finite generators

###### Anush Tserunyan (UIUC Math)

Abstract: Consider a continuous action of a countable group G on a Polish space X. A countable Borel partition P of X is called a generator if GP={gA: g in G, A in P} generates the Borel sigma-algebra of X. If n=|P|, we say that P is an n-generator. We will discuss some results related to finite generators, and focus on one that states that aperiodic actions always admit a 4-generator on a comeager set, answering a question of Kechris from the mid 90s.

1:00 pm in 243 Altgeld Hall,Tuesday, September 3, 2013

#### Positively curved Alexandrov spaces with many symmetries

###### John Harvey (Notre Dame)

Abstract: I will introduce two new tools -- the ramified orientable double cover and the slice theorem -- for Alexandrov geometry. These will be used to classify positively curved Alexandrov spaces under certain symmetry conditions, shedding new light on similar Riemannian results. This is joint work with Catherine Searle.

2:00 pm in Altgeld Hall 347,Tuesday, September 3, 2013

#### Non-normal asymptotics of the mean-field Heisenberg model

###### Kay Kirkpatrick (UIUC Math)

Abstract: I will discuss spin models of magnets and superconductors, models with particles that have spins in the set $\{\pm1\}$ (Ising model), in the circle (XY model), or in the sphere (Heisenberg model). Spin models have interesting phase transitions and are the most challenging in the most realistic cases, so the first step is often understanding the mean-field case. I will present results joint with Elizabeth Meckes on the mean-field Heisenberg model and its complex behavior at the phase transition. There is much that is still unclear about these models; I'll mention work in progress and what could be done next.

3:00 pm in 241 Altgeld Hall,Tuesday, September 3, 2013

#### The Ramsey number of the clique and the hypercube

###### Jozef Skokan   [email] (London School of Economics)

Abstract: The Ramsey number $r(K_s,Q_n)$ is the smallest integer $N$ such that every red-blue colouring of the edges of the complete graph on $N$ vertices contains either a red $n$-dimensional hypercube $Q_n$, or a blue clique on $s$ vertices. In 1983, Burr and Erdős conjectured that $r(K_s,Q_n) = (s-1)(2^n - 1)+1$ for every positive integer $s$ and sufficiently large $n$. In this talk we shall sketch the proof of this conjecture and discuss some related problems. Joint work with Gonzalo Fiz Pontiveros, Simon Griffiths, Rob Morris and David Saxton.

3:00 pm in 243 Altgeld Hall,Tuesday, September 3, 2013

#### The Hitchin fibration and real forms through spectral data

###### Laura Schaposnik (UIUC)

Abstract: The talk will be dedicated to the study of the moduli space of G-Higgs bundles and the Hitchin fibration through spectral data, where G is a real form of a complex Lie group. Through some examples we shall see applications of this new geometric way of understanding the moduli space and, time permitting, we will mention how the data approach relates to Langlands duality and (A,B,A)-branes.

4:00 pm in 243 Altgeld Hall,Tuesday, September 3, 2013

#### CANCELLED

5:00 pm in Altgeld Hall Common Room,Tuesday, September 3, 2013

#### Organizational Meeting

Abstract: In this learning seminar students and faculty will discuss recent and less recent results on Rigidity/deformation and other topics in Operator algebra theory