Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, September 10, 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 10, 2013

11:00 am in 243 Altgeld Hall,Tuesday, September 10, 2013

#### Power operations in height-2 Morava E-theory and its K(1)-localization

###### Yifei Zhu (Northwestern )

Abstract: For E a Morava E-theory spectrum of height 2, its ring of power operations has the structure of a graded twisted bialgebra satisfying a Frobenius congruence. For the K(1)-localization of E, its ring of power operations has a single generator over the coefficient ring. We illustrate this structure and provide explicit formulas by doing calculations, at the prime 3, with moduli of elliptic curves.

11:00 am in Altgeld Hall,Tuesday, September 10, 2013

#### Properties of digital representations

###### Katie Anders (UIUC Math)

Abstract: Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We will discuss patterns and properties of sequences and polynomials arising from these digital representations. We will also consider asymptotics of the summatory function of $f_\mathcal{A}(n)$.

11:00 am in 347 Altgeld Hall,Tuesday, September 10, 2013

#### Chanillo-Chiu-Yang embeddability for 3-dim CR structures

###### Gabe La Nave (UIUC)

Abstract: Chanillo-Chiu-Yang show that the eigenvalues of Kohn's Laplacian on a 3D CR manifold M are bounded away from zero if the CR Paneitz operator is non-negative and the Webster curvature is positive. This implies that M is embeddable if the CR-Yamabe constant is positive and the CR-Paneitz operator is nonnegative.

1:00 pm in 345 Altgeld Hall,Tuesday, September 10, 2013

#### Extendability of automorphisms of generic substructures

###### Aristotelis Panagiotopoulos (UIUC Math)

Abstract: Let $\boldsymbol{M}$ be any Fraisse limit which has no algebraicity such as the Rado graph(random graph), the rational numbers with their ordering, the rational Urysohn space, etc. We will provide necessary and sufficient condition for the following to hold:For a generic substructure $\boldsymbol{A}$ of $\boldsymbol{M}$, every automorphism $f\in\mathrm{Aut}(\boldsymbol{A})$ extends to a full automorphism $\tilde{f}\in\mathrm{Aut}(\boldsymbol{M})$''. We will also discuss extensions of the main result to uncountable structures such as the Urysohn metric space.

1:00 pm in 347 Altgeld Hall,Tuesday, September 10, 2013

#### Interaction functions and Boundary conditions

###### Ikemefuna Agbanusi (UIUC Math)

Abstract: In this talk, I'll summarize my work comparing the use of interaction functions and boundary conditions to model (stochastic) reaction diffusion. The focus is on quantifying how close the two methods can be in practice. The main example will be the so-called large coupling limits of Schroedinger equations. I'll end with an interesting set of problems and a possible line of attack.

2:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2013

#### On elliptesque and hyperbolesque curves

###### Bruce Reznick (UIUC Math)

Abstract: Any five points in the plane (no four on a line) determine a unique conic section. What can be said about a curve $C$ with the property that any five points chosen from $C$ either always determine an ellipse (or circle) or always determine a hyperbola? Such a curve is "elliptesque" or "hyperbolesque". Non-trivial examples include $y = x^3, x \ge 0$, which is hyperbolesque and $y = x^{3/2}, 1 \le x \le 1.3$, which is elliptesque. We show that if a smooth closed curve $C$ satisfies either condition, then it must be elliptesque and bound a convex region; no unbounded smooth curve can be elliptesque. Proofs are elementary.

The opening act of this talk is a discussion of the remarkable differential equation $((y'')^{-2/3})'''=0$; Sylvester observed in 1886 that the solutions to this equation are precisely the non-degenerate conic sections, simplifying a result originally proved by Monge in 1809. Two proofs of this will be given, and both are readily accessible to undergraduate math majors who have had calculus as well as linear algebra.

Departmental veterans will recognize this as a "Potpourri" talk.

3:00 pm in 245 Altgeld Hall,Tuesday, September 10, 2013

#### Symplectic Galois groups and Springer theory

###### Kevin McGerty (University of Oxford)

Abstract: One of the fundamental phenomena in geometric representation theory is Springer's action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. Recently there has been much interest in the geometry of symplectic resolutions, of which the Springer resolution is an example. We will discuss how Springer theory can be generalized to this setting.

3:00 pm in 347 Altgeld Hall,Tuesday, September 10, 2013

#### Dynamical systems on networks with noise

###### Lee DeVille   [email] (UIUC Math)

Abstract: We consider stochastic dynamical systems defined on networks . We will concentrate on the case of SDE with small white noise for concreteness. We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators. Along the way, we show that there is a non-standard spectral problem that appears in all of these calculations, and that the important features of this spectral problem is related to a certain homology group.

3:00 pm in 241 Altgeld Hall,Tuesday, September 10, 2013

#### List Coloring Graph Powers

###### Benjamin Reiniger   [email] (UIUC Math)

Abstract: I will give some background on conjectures concerning chromatic-choosability, then present a family of graphs that proves that graph powers are not generally chromatic-choosable. This result answers a question of Zhu and is joint work with Kosar, Petrickova, and Yeager.