Mathematics

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 $f_\mathcal{A}(n)$.

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.

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.

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.

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.

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.

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.

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.