Department of

Mathematics

Seminar Calendar
for Model Theory events the year of Tuesday, January 1, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 18, 2019

4:00 pm in 345 Altgeld Hall,Friday, January 18, 2019

Generic flat pregeometries

Omer Mermelstein (University of Wisconsin, Madison.)

Abstract: The property of "flatness" of a pregeometry (matroid) is best known in model theory as the device with which Hrushovski showed that his example refuting Zilber's conjecture does not interpret an infinite group. I will dedicate the first part of this talk to explaining what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. In the second part, I will conjecture that the family of flat pregeometries associated to strongly minimal sets is model theoretically nice, and share some intermediate results.

Friday, February 8, 2019

4:00 pm in 345 Altgeld Hall,Friday, February 8, 2019

"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 1)

Anush Tserunyan (UIUC)

Abstract: This will be the introductory talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups.

Friday, February 15, 2019

3:00 pm in 341 Altgeld Hall ,Friday, February 15, 2019

Note the time and room change!"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 2)

Jenna Zomback (UIUC)

Abstract: This will be the second talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will fill in some proofs left out last time and prove this general criterion.

Friday, February 22, 2019

4:00 pm in 345 Altgeld Hall ,Friday, February 22, 2019

Cancelled

Friday, March 1, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 1, 2019

"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 3)

Mary Angelica Gramcko-Tursi (UIUC)

Abstract: This will be the third talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will show that one or two of the aforementioned classes satisfy this criterion.

Friday, March 8, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 8, 2019

Organizational meeting

Friday, March 15, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 15, 2019

The theory of addition with predicates for the powers of 2 and 3

Christian Schulz (UIUC Math)

Abstract: This talk concerns the intricate boundary between decidable and undecidable of expansions of Presburger artithmetic, i.e., the structure $(\mathbb{N}, +)$. For a natural number $p \ge 2$, let $p^{\mathbb{N}}$ denote the set of powers of $p$, and let $V_p$ be a predicate that allows us to access the full base-$p$ expansion of a natural number. It is known that the expansion $(\mathbb{N}, +, V_p)$ of Presburger arithmetic retains decidability, but $(\mathbb{N}, +, V_p, q^{\mathbb{N}})$, for $q$ multiplicatively independent from $p$, has an undecidable theory. In this talk, I present a proof that the reduct $(\mathbb{N}, +, p^{\mathbb{N}}, q^{\mathbb{N}})$ also has an undecidable theory, specifically in the case $p = 2$, $q = 3$. I conclude with a note on how the proof extends to other structures, as well as some discussion of directions for further research.

Friday, March 29, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 29, 2019

Generalized sum-product phenomenon for polynomials

Souktik Roy (UIUC Math)

Abstract: Suppose $P(x,y)$ and $Q(x,y)$ are real polynomials with non-trivial dependence on $x$ and $y$, and $\epsilon$ is any positive constant. If, for a sufficiently large $n$-element set $A$ of real numbers, both $|P(A,A)|$ and $|Q(A,A)|$ are simultaneously smaller than $n^{5/4-\epsilon}$, then we shall prove that either $P(x,y) = f(u(x)+Cu(y)) \text{ and } Q(x,y) = g(u(x)+Du(y)),$ or $P(x,y) = f(u(x)u^{c}(y)) \text{ and } Q(x,y) = g(u(x)u^{d}(y)),$ where $f,g,u$ are polynomials and $C,D,c,d$ are constants. As a corollary, we obtain a strengthening of a classic result of Elekes and Rónyai in a symmetric setting of natural interest. The proof combines ideas from incidence geometry and o-minimality in model theory. This is joint work with Yifan Jing (UIUC) and Minh Chieu Tran (UIUC).

Friday, April 5, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 5, 2019

Generic derivations on o-minimal structures

Elliot Kaplan (UIUC Math)

Abstract: We study derivations $\delta$ on o-minimal fields $K$. We introduce the notion of a $T$-derivation, which is a derivation which cooperates with the 0-definable $\mathcal{C}^1$-functions on $K$. For example, if $K$ is an elementarily equivalent to the real exponential field, we require that $\delta \exp(a) = \exp(a)\delta a$ for all $a \in K$. Let $T$ be the theory of $K$ in an appropriate language $L$ and let $T^\delta$ be the $L\cup \{\delta\}$ theory stating that $\delta$ is a $T$-derivation. We show that if $T$ has quantifier elimination, then $T^\delta$ has a model completion $T^\delta_G$. The derivation in models $K$ of $T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $K$. If $T$ is the theory of real closed ordered fields, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core and that $T^\delta_G$ is distal. This is joint work with Antongiulio Fornasiero.

Friday, April 12, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 12, 2019

Ultraproducts as a tool in the model theory of metric structures

Ward Henson (UIUC)

Abstract: L is a signature of continuous first order logic for metric structures and we have a class C of L-structures which we want to investigate from the point of view of model theory. In general, this involves letting T be the L-theory of C, and working to understand the models of T as fully as possible. This means not only knowing which L-structures are models of T, but also understanding the definable predicates and (especially important) the definable sets in models of T. (A valuable byproduct might be an explicit axiomatization of T.) In this talk we will lay out how understanding ultraproducts of members of C can be an important practical tool for understanding the full class of models of T. As much as time permits, we will discuss examples that have been successfully treated in this way, including some new ones, focusing on Banach spaces and Banach lattices. (Most of this work on examples is part of a collaboration with Yves Raynaud.)

Tuesday, April 16, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, April 16, 2019

Positive model theory and sober spaces

Levon Haykazyan (University of Waterloo)

Abstract: I will talk about positive model theory (also known as coherent logic) where formulas are not closed under negation. This setting is in fact more general that full first-order logic, since negation can be expressed by changing the language. The result is that we can have as much negation as necessary, however no extra negation is forced by the framework.
We can associate to a positive theory a natural spaces of types, which will no longer be Hausdorff, but (quasi-)compact and sober. I will show that these spaces play the role of the Stone spaces in the full first-order logic. In particular I will show how classical results (due to Vaught) connecting the structure of countable models to Stone spaces carry over to the positive setting, provided we find the appropriate formulations of topological properties for non-Hausdorff spaces.

Friday, April 19, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 19, 2019

Cancelled

(UIUC Math)

Friday, September 6, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 6, 2019

"On the nonexistence of Følner sets" by Isaac Goldbring

Elliot Kaplan (UIUC Math)

Abstract: This will be the first (and possibly only) talk on the preprint "On the nonexistence of Følner sets" by Isaac Goldbring (https://arxiv.org/abs/1901.02445). I will introduce all of the necessary model-theoretic and group-theoretic background. Time permitting, I may get to the proof of the main result.

Friday, September 20, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 20, 2019

A Logician's Introduction to the Problem of P vs. NP

Alexi Block Gorman (UIUC Math)

Abstract: Central to much of computer science, and some areas of mathematics, are questions about various problems' computability and complexity (whether the problem can be solved "algorithmically," and how "hard" it is to do so). In this talk, I will first give an overview of the complexity hierarchy for machines (from finite automata to Turing machines) and the mathematical properties of the space of languages that we associate with them. Next, I will discuss the relationship of deterministic and non-deterministic machines, which will allow us to segue from questions of computability to that of complexity. Finally, I will give a precise formulation of the problem of P vs. NP, and try to illustrate why the problem remains rather elusive. This talk does not require any background in logic or computer science, and should be accessible to all graduate students.

Friday, September 27, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 27, 2019

O-minimal complex analysis according to Peterzil–Starchenko (Part 1)

Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Friday, October 4, 2019

4:00 pm in 345 Altgeld Hall,Friday, October 4, 2019

O-minimal complex analysis according to Peterzil–Starchenko (Part 2)

Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Friday, October 11, 2019

4:00 pm in 345 Altgeld Hall,Friday, October 11, 2019

"Complex-like" analysis in o-minimal structures (Part 3)

Neer Bhardwaj (UIUC)

Abstract: Analogues of many of the basic results in complex analysis can be established over an arbitrary algebraically closed field $K$ of characteristic zero, in the context of an o-minimal expansion of a real closed field $R$, with $K=R[i]$. I will show in particular how one can define winding numbers, and how differentiability begets infinite differentiability in this setting. This expository talk follows the survey given by Lou in the last two weeks and will be based mostly on: "Expansions of algebraically closed fields in o-minimal structures" by Starchenko–Peterzil (https://link.springer.com/article/10.1007/PL00001405)

Friday, November 1, 2019

4:00 pm in 345 Altgeld Hall,Friday, November 1, 2019

O-minimal complex analysis (Part 4)

Elliot Kaplan (UIUC)

Abstract: I will continue discussing Peterzil and Starchenko's treatment of definable functions on the algebraic closure of an o-minimal field.

Friday, November 8, 2019

4:00 pm in 345 Altgeld Hall,Friday, November 8, 2019

O-minimal complex analysis (Part 5)

Elliot Kaplan (UIUC)

Abstract: I will continue discussing Peterzil and Starchenko's treatment of definable functions on the algebraic closure of an o-minimal field.

Tuesday, November 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, November 12, 2019

Model theory of $\mathbb{R}$-trees and of ultrametric spaces

Ward Henson (UIUC Math)

Abstract: First, we consider the class of metric spaces $(M,d)$ that are $\mathbb{R}$-trees with a convex metric. To treat this class using continuous first order logic, we fix a base point $p$ in $M$ and require that $M$ have radius at most $r$ with respect to $p (r>0)$. The class of these structures $(M,d,p)$ is axiomatizable. Moreover, the theory of this class has a model companion $T$, whose models we describe precisely. This theory is a well behaved continuous theory. For example, $T$ has QE and is complete; it is stable (but not superstable) and has the maximum possible number of models in each infinite cardinal.
Second, given a model $M = (M,d,p)$ of $T$, we consider the closed subset $E_r(M) := \{x \in M | d(p,x)=r\}$. This is a definable set for T, and the entire structure $M$ can be reconstructed from $(E_r(M),d)$. The metric $d$ on $E_r(M)$ is an ultrametric; further, at every $x \in E_r(M)$, the set of distances $\{d(x,y) | y \in E_r(M)\}$ is dense in the interval $[0,2r]$. These properties are easily seen to be axiomatizable in continuous logic, and we let $T^*$ denote the resulting theory. We show that $T^*$ has QE, so it is complete; further, $T$ and $T^*$ are bi-interpretable.
This is joint work with Sylvia Carlisle.