Mathematics

Dense geodesic rays in the quotient of Outer space

Catherine Pfaff (Bielefeld)

Abstract: In 1981 Masur proved the existence of a dense Teichmueller geodesic in moduli space. As some form of analogue, we construct dense geodesic rays in certain subcomplexes of the $Out(F_r)$ quotient of outer space. This is joint work in progress with Yael Algom-Kfir.

Partial homogeneity of dual Fraisse limits and homogeneity of the pseudo-arc

Slawomir Solecki (UIUC)

Abstract: The pseudo-arc is the generic compact connected subset of the plane (or the Hilbert cube). By a fundamental result of Bing, it is homogeneous as a topological space. By work of Irwin and myself, the pseudo-arc is represented as a quotient of a dual Fraisse limit, which allows for a discretization of a continuous situation. (The limit is automatically "dually" homogeneous, but not "directly" homogeneous, so Bing's result does not follow.) In this joint work with Tsankov, we determine partial "direct" homogeneity of the limit, which involves combinatorial and basic "dual" model theoretic arguments (e.g., a notion of dual type). Further, we prove a transfer theorem, through which we recover Bing's result from our partial homogeneity. Time permitting, I will make comments on the possible generality of the method.

Regularity and long-time behavior of nonlocal heat flows

Stanley Snelson   [email] (U of Chicago Mathematics)

Abstract: We consider a nonlocal parabolic system with a singular target space. Caffarelli and Lin showed that a well-known optimal eigenvalue partition problem could be reformulated as a constrained harmonic mapping problem into a singular space. We show that the gradient flow corresponding to this problem is Lipschitz continuous in space, and study the regularity of a resulting free interface problem. We also show that the flow converges to a stationary solution of the constrained mapping problem as time approaches infinity. Time permitting, we will also discuss some related ongoing work involving more general non-smooth target spaces.

Arbitrary Orientations of Hamilton Cycles in Digraphs

Theodore Molla   [email] (UIUC Math)

Abstract: Let n be sufficiently large. Every digraph G on n vertices with minimum indegree and minimum outdegree at least n/2 contains every orientation of a Hamilton cycle except when n is even and G is isomorphic to one of two digraphs. Furthermore, both of these two exceptional digraphs have minimum indegree and minimum outdegree exactly n/2 and contain every orientation of a Hamilton cycle except the orientation in which every pair of consecutive edges alternate direction. Our result improves on an approximate result of Häggkvist and Thomason from 1995. Along with the proof of this result, we will discuss some of the innovative ideas employed in Häggkvist and Thomason's result and how these ideas can be used in a proof of the precise result. This is joint work with Louis DeBiasio, Daniela Kühn, Deryk Osthus and Amelia Taylor.

The Bergman kernel on some Reinhardt domains

Zhenghui Huo (UIUC Math)

Abstract: We provide a new method to compute the Bergman kernel on some Reinhardt domains. We express the kernel on certain domain in $\mathbb C^{n+1}$ in terms of an already known kernel on a domain in $\mathbb C^n$ and a first order differential operator. We find, for example, an exact formula for the kernel on $\{(z_1,z_2,w)\in \mathbb C^{3};e^{|w|^2}|z_1|^2+|z_2|^2<1\}$.