Department of

# Mathematics

Seminar Calendar
for Logic Seminar events the year of Tuesday, March 13, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 16, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, January 16, 2018

#### Turing and Machines

###### Kay Kirkpatrick (Illinois Math)

Abstract: We will discuss newly defined machines that out-perform Turing machines. In his unpublished 1948 paper, Intelligent Machinery, Alan Turing identified several types of machines, with one dichotomy that is false, between active and controlling machines. This mistake became an assumption in his famous 1950 paper and has probably been the source of the confusion about the Turing Test and the Chinese Room Argument. I'll introduce a new type of machine and define a subtype that cannot be simulated by a deterministic Turing machine.

Tuesday, January 23, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, January 23, 2018

#### First order expansions of the ordered real additive group

###### Erik Walsberg (UIUC Math)

Abstract: I will discuss joint work with Philipp Hieronymi on first order expansions of $(\mathbb{R},<,+)$. The definable subsets of $\mathbb{R}^n$ in the known natural model-theoretically tame expansions are geometrically tame objects. Many of these examples satisfy the essentially strongest geometric tameness assumption, o-minimality. In this talk I will discuss type A expansions. The assumption of being type A appears to be the broadest possible generalization of o-minimality for expansions of $(\mathbb{R},<,+)$ as some of the main o-minimal tools extend to this setting, and do not appear to extend further.

Tuesday, January 30, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, January 30, 2018

#### Canceled

Tuesday, February 13, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, February 13, 2018

#### Topological dimension through the lens of Baire category

###### Anush Tserunyan (Illinois Math)

Abstract: In 1913, Brouwer proved that the topological dimension of $\mathbb{R}^n$ is $n$, which implies that there is no continuous injection of $\mathbb{R}^{n+1}$ into $\mathbb{R}^n$. More recently, Izzo and Li wondered if the last statement survives when the requirement of injectivity is relaxed to being injective on a large set. They showed in 2013 that the answer is negative when the largeness is measure-theoretic, i.e. when the continuous function is required to be injective only on a conull set. However, they conjectured that the answer should be positive for the notion of largeness provided by Baire category, namely: there does not exist a continuous function $\mathbb{R}^{n+1} \to \mathbb{R}^n$ that is injective on a comeager set. We will discuss this conjecture and its dramatic (for the speaker) resolution.

Tuesday, February 20, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, February 20, 2018

#### A Descriptive Set in Topological Dynamics

###### Robert Kaufman (Illinois Math)

Abstract: Let $X$ be a compact metric space and $H(X)$ the group of homeomorphisms of $X$, a Polish group. Then the orbit of $x \in X$ under $h \in H(X)$ (the two-sided orbit) has an obvious definition; its closure is called the orbit-closure of $x$. When all the orbit-closures are minimal then $h$ is called "sharp". (Every orbit-closure contains a minimal one). There are two main theorems.
A. The set $S(X)$ of sharp homeomorphisms is always co-analytic.
B. For a certain $X$, $S(X)$ is not Borel.
In the proof of B we need a variant of Hurewicz' theorem (1930) on the class of uncountable compact sets. This variant should be (but so far isn't) a consequence of Hurewicz' theorem. I'll say a very few words about a new method of proof.

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.

Tuesday, March 6, 2018

1:00 pm in 113 Speech & Hearing Bldg at Daniel and 6th,Tuesday, March 6, 2018

#### Some aspects of simple proofs informed by history, ancient and modern

###### Victor Pambuccian (Arizona State University Math)

Abstract: We'll look at (i) the existence of a direct proof (even within intuitionistic logic) of the Steiner-Lehmus theorem, stating that a triangle with two congruent angle bisectors must be isosceles, (ii) the most primitive arithmetic capable of proving the irrationality of $\sqrt{2}$, going back to the early Pythagoreans, and why it cannot prove the irrationality of $\sqrt{17}$, the case Theodorus of Cyrene apparently stumbled upon, and (iii) the simplest proof for the fact that 30 is the greatest number all of whose totitives are prime.

Tuesday, March 13, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, March 13, 2018

#### Weak containment in ergodic theory and representation theory

###### Peter Burton (U Texas at Austin)

Abstract: The relation of weak containment for unitary representations of locally compact groups is a very useful tool in comparing such representations. Recently Kechris introduced an analogous definition of weak containment for measure-preserving actions of countable discrete groups. We will discuss the relationship between these concepts, and present a result showing that weak containment of measure-preserving actions is an essentially stronger notion than weak containment of the corresponding Koopman representations.

Tuesday, April 3, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, April 3, 2018