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for events the day of Tuesday, August 29, 2017.

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Tuesday, August 29, 2017

10:00 am in 106B3 Engineering Hall,Tuesday, August 29, 2017

An introduction to HT-fields

Elliot Kaplan (UIUC Math)

Abstract: In this talk, I will introduce the class of $HT$-fields. Let $T$ be an o-minimal theory extending the theory of ordered fields and let $K$ be a model of $T$ which is also equipped with a nontrivial derivation $x \mapsto x'$, making it an $H$-field (a particularly nice type of ordered differential field). We require that this derivation interact nicely with the o-minimal structure on $K$. The class of $H$-fields has been thoroughly explored by Aschenbrenner, van den Dries, and van der Hoeven. I will establish some analogues of their results on $H$-fields for the class of $HT$-fields and discuss my ongoing work. This talk is part of my preliminary examination.

12:00 pm in 243 Altgeld Hall,Tuesday, August 29, 2017

The twisted rabbit problem via the arc complex.

Rebecca Winarski (UW Milwaukee)

Abstract: The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one? After remaining open for 25 years, this problem was solved by Bartholdi—Nekyrashevych using iterated monodromy groups. In joint work with Lanier and Margalit, we formulate the problem topologically and solve the problem using the arc complex.

1:00 pm in 345 Altgeld Hall,Tuesday, August 29, 2017

Invariantly universal equivalence relations

Filippo Calderoni (University of Torino Math)

Abstract: In this talk we analyze the phenomenon of invariant universality for analytic equivalence relations. Invariantly universality strengthens the notion of completeness, namely, the property of being on top of the hierarchy of analytic equivalence relations with respect to Borel reducibility. We survey recent results and discuss some open problems.

3:00 pm in 241 Altgeld Hall,Tuesday, August 29, 2017

On-line size Ramsey number for ordered tight paths

Douglas West (Illinois Math and Zhejiang Normal University)

Abstract: An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices. The appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number $R'_t(G)$ of an ordered hypergraph $G$ is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using $t$ colors to produce a monochromatic copy of $G$. The monotone tight path $P(r,k)$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices. We obtain good bounds on $R'_t(P(r,k))$. Letting $m=r-k+1$ (the number of edges in $P(r,k)$), we prove $m^{t-1}/(3\sqrt t) \le R'(P(r,2)) \le tm^{t+1}$. For general $k$, a trivial upper bound is $\left({N \atop k}\right)$, where $N$ is the vertex Ramsey number of $P(r,k)$ and is a tower of height $k-2$. We prove $N/(k\log N) \le R'_t(P(r,k)) \le N(\log N)^{2+c}$, where $c$ is any positive constant and $t(m-1)$ is sufficiently large. Our upper bounds improve prior results when $t$ grows faster than $m/\log m$, and our methods yield another derivation of the vertex Ramsey number. We also generalize our results to $l$-loose monotone paths, where each successive edge begins $l$ vertices after the previous edge (the tight path is 1-loose). This work is joint with Xavier Pérez Giménez and Pawel Pralat.