Department of

Mathematics


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for events the day of Tuesday, February 27, 2018.

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Tuesday, February 27, 2018

1:00 pm in 214 Ceramics Bldg,Tuesday, February 27, 2018

Strong conceptual completeness for ℵ0-categorical theories

Jesse Han (McMaster Math)

Abstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is "definable", i.e. really just the points in all models of some imaginary sort of our theory?
In the '80s, Michael Makkai provided the following answer to this question: a functor $\mathrm{Mod}(T) \to \mathrm{Set}$ is definable if and only if it preserves all ultraproducts and all "formal comparison maps" between them, called ultramorphisms (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness.
Any general framework which reconstructs theories from their categories of models should be considerably simplified for $\aleph_0$-categorical theories. Indeed, we show:
If $T$ is $\aleph_0$-categorical, then $X : \mathrm{Mod}(T) \to \mathrm{Set}$ is definable, i.e. isomorphic to ($M \mapsto \psi(M)$) for some formula $\psi \in T$, if and only if $X$ preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when $T$ is $\aleph_0$-categorical. We show this definability criterion fails if we remove the $\aleph_0$-categoricity assumption, by constructing examples of theories and non-definable functors $\mathrm{Mod}(T) \to \mathrm{Set}$ which exhibit this.

3:00 pm in 110 Speech and Hearing Building,Tuesday, February 27, 2018

Extending edge-colorings of complete hypergraphs into regular colorings

Amin Bahmanian (Illinois State Math)

Abstract: Let $({X \atop h})$ be the collection of all $h$-subsets of an $n$-set $X\supseteq Y$. Given a coloring (partition) of a set $S\subseteq ({X \atop h})$, we are interested in finding conditions under which this coloring is extendible to a coloring of $({X \atop h})$ so that the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. The case $S=\emptyset, r=1$ was studied by Sylvester in the 18th century, and remained open until the 1970s. The case $h=2,r=1$ is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. An $r$-factorization is a coloring of $({[n] \atop h})$ so that the number of times each element of $[n]$ appears in each color class is $r$. Let $\chi (m,h,r)$ be the smallest $n$ such that any "partial" $r$-factorization of $({[m] \atop h})$ satisfying $r \mid ({{n-1} \atop {h-1}})$, $h \mid rn$ can be extended to an $r$-factorization of $({[n] \atop h})$. We show that $2m\leq \chi (m,4,r)\leq 4.847323m$, and $2m\leq \chi (m,5,r)\leq 6.285214m$.

4:00 pm in 1 Illini Hall,Tuesday, February 27, 2018

Introduction to GIT

Itziar Ochoa de Alaiza Gracia (UIUC)

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.