Mathematics

Finding paths in graphs of triangulations

Mark Bell (UIUC Math)

Abstract: There are many different ways of triangulating a surface using n arcs. As some triangulations are more similar than others, we get a natural topology on the space of triangulations which can be seen as an infinite graph; where two triangulations are connected if and only if they share (n-1) edges. This graph provides a combinatorial model for the surfaces mapping class group as it acts geometrically. We will look at some techniques for efficiently finding paths through this graph, allowing us to efficiently represent and compute with mapping classes.

Analytic sets defined by the metric in Polish/Banach spaces

Robert Kaufman   [email] (UIUC Math)

Abstract: We study analytic sets whose definition reflects exact choice of a metric, e.g., sets defined by best approximation. A pair $(a,b)$ is "balanced (bisected? split? dyadic?) if some element bisects the pair. The set B of balanced pairs in a Polish space is sometimes non-Borel; the proof yields a much more precise statement.
     A set $S$ in a Banach space $M$ is "centered" if $S$ admits a best approximation by singletons in the Hausdorff metric. The remaining results depend on the notions of re-norming, dual space, reflexive space [in order of difficulty]. In dual spaces, all sets of finite diameter are centered. (Converse false) The following result is proved for the space $\ell^1$ by elementary methods: if $M$ is non-reflexive (NR) then it can be re-normed so that some tripleton isn't centered. When $M$ is separable, the set of centered tripletons is analytic; if $M$ is also NR, then that set becomes non-Borel after a re-norming.

The Bergman kernel on some Reinhardt domains

Zhenghui Huo (UIUC Math)

Trace and Weyl asymptotics for non-local operators

Selma Yildirim Yolcu   [email] (Bradley University)

Abstract: I will present a broad overview of exciting problems and conjectures which lie at the interface of probability and the spectral theory. In particular, I will discuss a certain problem of computing the coefficients in the two-term asymptotic formula for the trace of some Schrodinger operators. After recalling some of the well-known results, I will discuss recent (and not so recent) progress in understanding the analogous picture when the Laplacian, which "goes" with the Brownian motion, is replaced by non-local operators which "go" with rotationally invariant symmetric stable processes and other closely related Levy processes.

Ordered Ramsey Theory and Track Representations of Graphs

Douglas B. West   [email] (Zhejiang Normal University and UIUC Math)

Abstract: We study an ordered version of hypergraph Ramsey numbers for linearly ordered vertex sets, due to Fox, Pach, Sudakov, and Suk. In the $k$-uniform ordered path, the edges are the sets of $k$ consecutive vertices in a linear order. Moshkovitz and Shapira described its ordered Ramsey number in terms of an enumerative problem: it equals $1$ plus the number of elements in the poset obtained by starting with a certain disjoint union of chains and repeatedly taking the poset of down-sets, $k-1$ times. We will describe a proof of this and apply the bounds to study the minimum number of interval graphs whose union is the line graph of the $n$-vertex complete graph. We prove the conjecture of Heldt, Knauer, and Ueckerdt that this value grows with $n$. In fact, the growth rate is between $\Omega({\log\log n}/{\log\log\log n})$ and $O(\log\log n)$. This work is joint with Kevin Milans and Derrick Stolee.

Life After Comps

Abstract: The panelists will be math graduate students who are near or past the prelim, and the event is meant to benefit graduate students who are at the stage of choosing a thesis adviser and thesis topic, or earlier. Come and listen, and ask whatever questions you’d like, or share your experience and advice, whether you are a panelist or not.