Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 11, 2014.

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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, March 11, 2014

1:00 pm in 347 Altgeld,Tuesday, March 11, 2014

#### Tire tracks, the stationary Schrodinger's equation and forced vibrations.

###### Mark Levi   [email] (Penn State Math)

Abstract: I will describe a newly discovered equivalence between the first two objects mentioned in the title. The stationary Schrodinger's equation, a.k.a. Hill’s equation, is ubiquitous in mathematics, physics, engineering and chemistry. Just to mention one application, the main idea of the Paul trap (for which W. Paul earned the 1989 Nobel Prize in physics) amounts to a certain property of Hill's equation. As it turns out, Hill's equation is equivalent to a seemingly completely unrelated problem of “tire tracks”. In addition to this equivalence, I will describe a yet another connection between the tire tracks” problem and the high frequency forced vibrations.

1:00 pm in 345 Altgeld Hall,Tuesday, March 11, 2014

#### Expansions of the ordered additive group of real numbers by two discrete subgroups

###### Philipp Hieronymi (UIUC Math)

Abstract: The theory of $(\mathbb{R},<,+,\mathbb{Z},a\mathbb{Z})$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},a\mathbb{Z})$ defines multiplication by $a$. The results are established by using the Ostrowski representation of a real number to define the above structures in monadic second order logic of one successor.

1:00 pm in 243 Altgeld Hall,Tuesday, March 11, 2014

#### A new proof of Bowen's theorem on Hausdorff dimension of quasi-circles

###### Andy Sanders (UIC Math)

Abstract: A quasi-Fuchsian group is a discrete group of Mobius transformations of the Riemann sphere which is isomorphic to the fundamental group of a compact surface and acts properly on the complement of a Jordan curve: the limit set. In 1979, Bowen proved a remarkable rigidity theorem on the Hausdorff dimension of the limit set of a quasi-Fuchsian group: it is equal to 1 if and only if the limit set is a round circle. This theorem now has many generalizations. We will present a new proof of Bowen's result as a by-product of a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. This lower bound is in terms of the differential geometric data of an immersed, incompressible minimal surface in the quotient manifold. If time permits, generalizations of this result to other convex-co-compact surface groups will be presented.

2:00 pm in Altgeld Hall 347,Tuesday, March 11, 2014

#### Small-time asymptotics for fast mean-reverting stochastic volatility models

###### Rohini Kumar (Wayne State Math)

Abstract: We use stochastic volatility models, with fast mean-reverting volatility, to price out-of-the-money (OTM) European call options near maturity. The regime of interest is when time to maturity is small, but large compared to the mean-reversion time of the stochastic volatility. This leads to a problem of large deviations for a fast-slow system.

2:00 pm in 241 Altgeld Hall,Tuesday, March 11, 2014

#### Generic representations of Abelian groups and extreme amenability (part 3)

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

Abstract: We finish the proof of the set $\{\phi \in \text{Hom}(\Gamma, L^0(\mathbb{T})) : \phi(\Gamma) \text{ dense in } L^0(\mathbb{T})\}$ being dense $G_\delta$. Then we show how to embed the unitary group of a separable Hilbert space into $\text{Aut}(X,\mu)$ using the Gauss measure construction, thus finishing the proof that the set $\{\phi \in \text{Hom}(\Gamma, \text{Aut}(X,\mu)) : \overline{\phi(\Gamma)} \cong L^0(\mathbb{T})\}$ is dense, and hence the set $\{\phi \in \text{Hom}(\Gamma, \text{Aut}(X,\mu)) : \overline{\phi(\Gamma)} \text{ extremely amenable}\}$ is dense $G_\delta$.

3:00 pm in 243 Altgeld Hall,Tuesday, March 11, 2014

#### Birational geometry of the moduli space of one-dimensional sheaves

###### Jinwon Choi (KIAS)

Abstract: We study the birational geometry of the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. We determine the effective/nef cone in terms of natural geometric divisors. We also present the birational model constructed from the locally free resolutions of the general sheaves. The two spaces are related by the Bridgeland-type wall-crossing. As corollaries, we compute the Betti numbers of the moduli spaces when $d \leq 6$. The results confirm the prediction from physics. This is joint work with Kiryong Chung.

3:00 pm in 241 Altgeld Hall,Tuesday, March 11, 2014

#### Characterization of $(2m,m)$-paintable graphs

###### Tom Mahoney   [email] (UIUC Math)

Abstract: A graph $G$ is $(a,b)$-choosable if for any list assignment giving $a$ colors to every vertex admits a coloring assigning each vertex $b$ colors from its list so that the color sets assigned to adjacent vertices are disjoint. Paintability is a generalization of list coloring where list elements are presented in an online fashion. Given $m \ge 1$, a graph $G$ is $(2m,m)$-paintable if and only if it is $2$-paintable. In 2009, Zhu conjectured that $k$-paintable graphs are $(km,m)$-paintable for all $m \ge 1$. Our results prove this conjecture for $k=2$.

4:00 pm in 245 Altgeld Hall,Tuesday, March 11, 2014

#### Diffusion Limit of a Limit-Order Book

###### Steven Shreve, Orion Hoch and University Professor (Department of Mathematical Sciences, Carnegie Mellon University)

Abstract: With the wholesale movement of the trading of stocks, currencies, and commodities futures to electronic exchanges, the need for models that capture the operation of these exchanges has become paramount. Simple questions such as whether high frequency traders contribute or remove liquidity from the market cannot be studied in the absences of such models. The construction of such models is complicated by their inherent high dimensionality and the strategic play involved. Adapting ideas from queueing theory, we present a simplified model for the limit-order book of an electronic exchange that is driven by Poisson processes. We then describe the limit obtained by diffusion scaling. This is joint work with Christopher Almost and John Lehoczky.

4:00 pm in 243 Altgeld Hall,Tuesday, March 11, 2014

#### Infinitesimal Algebraic Geometry and Infinitesimal Infinitesimal Algebraic Geometry

###### Peter Nelson (UIUC Math)

Abstract: Sometimes the more classical infinitesimal objects attached to a "smooth" group don't contain as much information as one would like, especially in an algebraic setting. I'll discuss one or two (still pretty classical) improvements on the situation. Since I like thinking about universal things, I'll try to say a few things about the moduli spaces of these improvements, and maybe even how they relate to the moduli of the original groups.