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for events the day of Tuesday, August 27, 2019.

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Tuesday, August 27, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

Views of the Scenery Flow

Albert Fisher (University of São Paulo)

Abstract: I will give an overview of work carried out together with T. Bedford, M. Urbanski, P. Arnoux, and M. Talet. We consider the result of zooming toward a point of a geometric object embedded in an ambient Euclidean space. For a smooth submanifold this just converges to the tangent space at the point, a fixed point for the flow. But for irregular objects like fractal sets the scenery at small scales keeps changing. The collection of the asymptotic limiting objects can be thought of as the tangent bundle of the fractal, these ``scenes'' are related to Furstenberg's ``microsets''. This collection of sets is acted upon by the flow of zooming exponentially fast toward small scales. A mathematical challenge, then, is to precisely define this ``scenery flow'' and to study its dynamical properties. As we sketch, for the limit set of a Kleinian group, the scenery flow is isomorphic to the geodesic frame flow of the associated hyperbolic three-manifold. Now as Sullivan proved, this flow entropy equals the Hausdorff dimension of the limit set. On the other hand, Bowen showed the Hausdorff dimension of the limit set is given in terms of a zero of the pressure function for log of the derivative. This formula for dimension was extended by Ruelle to hyperbolic Julia sets. With Bedford and Urbanski, we constructed the scenery flow for hyperbolic Julia sets, leading to a unification of Bowen's formula with Sullivan's: in both cases, "Hausdorff dimension equals scenery flow entropy". In this sense the scenery flow thus provides an analogue for Julia sets of the geodesic flow. These ideas extend to other situations. In work with M. Talet and P. Arnoux we have studied the scenery flow for Brownian motion paths, and for the nested tilings associated with circle rotations and interval exchange transformations and defined by renormalization. We sketch these ideas, indicating the methods of proof.

1:00 pm in Altgeld Hall,Tuesday, August 27, 2019

Deflated Continuation: A bifurcation analysis tool for Nonlinear Schrodinger (NLS) Systems

Stathis Charalampidis (Mathematics Department, California Polytechnic State University)

Abstract: Continuation methods are numerical algorithmic procedures for tracing out branches of fixed points/roots to nonlinear equations as one (or more) of the free parameters of the underlying system is varied. On top of standard continuation techniques such as the sequential and pseudo-arclength continuation, we will present a new and powerful continuation technique called the deflated continuation method which tries to find/construct undiscovered/disconnected branches of solutions by eliminating known branches. In this talk we will employ this method and apply it to the one-component Nonlinear Schrodinger (NLS) equation in two spatial dimensions. We will present novel nonlinear steady states that have not been reported before and discuss bifurcations involving such states. Next, we will focus on a two-component NLS system and discuss about recent developments by using the deflated continuation method where the landscape of solutions of such a system is far richer. A discussion about the challenges in the two-component setting will be offered and a summary of open problems will be emphasized.

2:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

The Alon-Tarsi number of subgraphs of a planar graph

Seog-Jin Kim (Konkuk University Math)

Abstract: This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable. This is joint work with Ringi Kim and Xuding Zhu.