Department of

# Mathematics

Seminar Calendar
for Logic Seminar events the year of Thursday, September 14, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2017           September 2017          October 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5                   1  2    1  2  3  4  5  6  7
6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
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Tuesday, January 24, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, January 24, 2017

#### Will the Nonstandard Analysis become the Analysis of Future?

###### Evgeny Gordon (Eastern Illinois Math)

Abstract: In 1973 Abraham Robinson gave a talk about the nonstandard analysis (NSA) at the Institute for Advanced Study. After his talk Kurt G\"odel made a comment, in which he predicted that "...there are good reasons to believe that Non-Standard Analysis in some version or other will be the analysis of the future". One has to admit that during the fifty years since this prediction, it did not come true. One of the reasons is that the most part of researchers in NSA considered it as a tool of obtaining new results in standard mathematics, instead of consider it as a more appropriate language, in which the "book of nature is written". Nowadays, the investigation of DE's that simulate processes in science and economy are based on computer (discrete) simulations of these DE's.
In this talk I will try to justify the point of view that the language of NSA is more appropriate for investigation of the interaction between continuous models and their discrete simulations (or maybe vise versa - between discrete models and their continuous simulation, according to a popular among applied mathematicians point of view). The reason is that not well defined properties ("very big", "very small", "far enough of the boundaries of computer memory", etc.) can be introduced in the language of NSA on the level of rigor of Cantor's Set Theory. I will discuss some NSA theorems in algebra, calculus, ergodic theorem and quantum mechanics) concerning this problem that have intuitively clear sense and agree with computer experiments, while their formulation in the language of standard mathematics looks irrelevant and sometimes even unreadable.

Tuesday, January 31, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, January 31, 2017

#### Uncountable Categoricity in Continuous Logic

###### Victoria Noquez (UIC Math)

Abstract: In recent years, some progress has been made towards understanding uncountable categoricity in the continuous setting, particularly in the context of classes of Banach spaces. Currently, it is unknown if the Baldwin-Lachlan characterization of uncountable categoricity holds in continuous logic. Namely, is it the case a continuous theory T is $\kappa$-categorical for some uncountable cardinal $\kappa$ if and only if T is $\omega$-stable and has no Vaughtian pairs?
In order to address this question, we provide the necessary continuous characterization of Vaughtian pairs, and in the process, prove Vaught's two-cardinal theorem, as well as a partial converse of the theorem in the continuous setting. This allows us to prove the forward direction of the Baldwin-Lachlan characterization.
Trying to prove the reverse direction leads us to an attempt to characterize strong minimality in continuous logic. We propose a notion of strong minimality, and show that it has many of the properties of its classical analogue. Unfortunately, we see that this does not provide the machinery required to show that $\omega$-stability and the absence of Vaughtian pairs are sufficient conditions for uncountable categoricity. We provide some examples towards understanding this failure.

Tuesday, February 7, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, February 7, 2017

#### Canceled

Tuesday, February 21, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, February 21, 2017

#### "Strong theories of ordered abelian groups" by A. Dolich and J. Goodrick (2nd talk)

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

Tuesday, February 28, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, February 28, 2017

#### Complexity in Dual Banach Spaces

###### Robert Kaufman (UIUC)

Abstract: X is a Banach space, X* is its dual space, composed of bounded linear functionals on X. The norm of a functional in X* is its supremum over the closed unit ball in X. NA is the set of functionals whose norm is attained there. S (for "sharp") is the set of functionals whose norm is attained at precisely one point in the closed ball. To obtain interesting conclusions about the complexity of these sets, the space X is re-normed. (This is not as scary as it sounds.)

Thursday, March 2, 2017

1:00 pm in 345 Altgeld Hall,Thursday, March 2, 2017

#### Henson's universal triangle-free graphs have finite big Ramsey degrees

###### Natasha Dobrinen (University of Denver)

Abstract: A triangle-free graph on countably many vertices is universal triangle-free if every countable triangle-free graph embeds into it. Universal triangle-free graphs were constructed by Henson in 1971, which we will denote as $\mathcal{H}_3$. Being an analogue of the random graph, its Ramsey properties are of interest. Henson proved that for any partition of the vertices in $\mathcal{H}_3$ into two colors, there is either a copy of $\mathcal{H}_3$ in one color (furthermore, only leaving out finitely many vertices in the first color), or else the other color contains all finite triangle-free graphs. In 1986, Komj\'{a}th and R\"{o}dl proved that the vertices in $\mathcal{H}_3$ have the Ramsey property: For any partition of the vertices into two colors, one of the colors contains a copy of $\mathcal{H}_3$. In 1998, Sauer showed that there is a partition of the edges in $\mathcal{H}_3$ into two colors such that every subcopy of $\mathcal{H}_3$ has edges with both colors. He also showed that for any coloring of the edges into finitely many colors, there is a subcopy of $\mathcal{H}_3$ in which all edges have at most two colors. Thus, we say that the big Ramsey degree for edges in $\mathcal{H}_3$ is two. It remained open whether all finite triangle-free graphs have finite big Ramsey degrees; that is, whether for each finite triangle-free graph G there is an integer n such that for any finitary coloring of all copies of G, there is a subcopy of $\mathcal{H}_3$ in which all copies of G take on no more than n colors. We prove that indeed this is the case.

Tuesday, March 7, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, March 7, 2017

#### Relatively Random First-Order Structures

###### Henry Towsner (UPenn Math)

Abstract: The Aldous–Hoover Theorem gives a characterization of those random processes which generate "exchangeable" first-order structures. A random first-order structure on the natural numbers is exchangeable if, after any permutation of the natural numbers, it has the same distribution. The original proof of the full Aldous–Hoover Theorem used ultraproducts, and the topic remains intimately tied to the way probability measures behave in ultraproducts.
For some purposes, full exchangeability is too strong. We investigate "relative exchangeability", where we only require that the distribution be preserved by automorphisms of a fixed first-order structure $M$. A full Aldous–Hoover theorem is not always possible in this setting, and how much we recover turns out to depend on the amalgamation properties of $M$.

Tuesday, March 14, 2017

1:00 pm in Altgeld Hall,Tuesday, March 14, 2017

#### Model theory and Painleve equations

###### James Freitag (UIC Math)

Abstract: We will discuss how to use model theory to prove some transcendence results for solutions of Painleve equations.

Thursday, March 16, 2017

1:00 pm in 243 Altgeld Hall,Thursday, March 16, 2017

#### On Grothendieck's proof of the Fundamental Theorem of Stability Theory

###### Sergei Starchenko (Notre Dame)

Abstract: In the paper "Model theoretic stability and definability of types, after A. Grothendieck" (2014) Itai Ben Yaacov observed that the Fundamental Theorem of Stability Theory (also known as the Definability of Types Theorem) follows from Grothendieck's paper "Critères de compacité dans les espaces fonctionnels généraux" (1952) on what can be called "double limits property". In this talk we discuss this connection and provide an elementary proof of a version of Grothendieck's theorem equivalent to the Fundamental Theorem of Stability Theory.

Tuesday, March 28, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, March 28, 2017

#### Simultaneous stationary reflection and failure of SCH

###### Dima Sinapova (UIC Math)

Abstract: We will show that it is consistent to have finite simultaneous stationary reflection at $\kappa^+$ with not SCH at $\kappa$. This extends a result of Assaf Sharon. We will also present an abstract approach of iterating Prikry type forcing and use it to bring our construction down to $\aleph_\omega$. This is joint work with Assaf Rinot.

Tuesday, April 4, 2017

1:00 pm in UIC SEO 636,Tuesday, April 4, 2017

#### MidWest Model Theory Day at UIC

Tuesday, April 25, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, April 25, 2017

#### Differential-henselian extensions

###### Nigel Pynn-Coates (UIUC)

Abstract: The general motivating question is: What aspects of valuation theory can be adapted to the setting of valued differential fields, and under what assumptions? In valuation theory, henselian fields play an important role. I will concentrate on differential-henselian fields, introduced by Scanlon and developed in a more general setting by Aschenbrenner, van den Dries, and van der Hoeven. What do we know about uniqueness of differential-henselian extensions? Do differential-henselizations exist? After reviewing what is known, I will discuss my ongoing work towards answering these questions, and sketch a proof of the answers when the value group has finite archimedean rank. This talk is part of my preliminary examination.

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.

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.

Tuesday, September 5, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, September 5, 2017

#### Word and conjugacy problems in finitely generated groups

###### Arman Darbinyan (Vanderbilt Math)

Abstract: Word and conjugacy problems are two of the main decision problems associated with finitely generated groups. In particular, there are deep results which bridge some of the main concepts of the theories of computability and computational complexity with group theoretical invariants through the word problem in groups. In this talk we will recall some of the well-known facts about word and conjugacy problems in groups as well as discuss new results concerning the relationship between these decision problems.

1:00 pm in 345 Altgeld Hall,Tuesday, September 5, 2017

#### To Be Announced

###### Arman Darbinyan (Vanderbilt Math)

Tuesday, September 12, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, September 12, 2017

#### The complexity of the classification problem in ergodic theory

###### Martino Lupini (Caltech Math)

Abstract: Classical results in ergodic theory due to Dye and Ornstein–Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic. This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable amenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. This builds on previous work of Epstein, Tornquist, and Popa, and answers a question of Kechris.

Tuesday, September 19, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, September 19, 2017

#### No seminar

Tuesday, September 26, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, September 26, 2017

#### Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets

###### Tobias Kaiser (University of Passau)

Abstract: (Joint work with Julia Ruppert.) Understanding integration in the o-minimal setting is an important and difficult task. By the work of Comte, Lion and Rolin, succeeded by the work of Cluckers and Miller, parameterized integrals of globally subanalytic functions are very well analyzed. But very little is known when the exponential function comes into the game. We consider certain parameterized exponential integrals which come from considering the Brownian motion on globally subanalytic sets. We are able to show nice asymptotic expansions of these integrals.

Tuesday, October 3, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, October 3, 2017

#### Strong conceptual completeness for $\mathcal L_{\omega_1\omega}$

###### Ronnie Chen (Caltech)

Abstract: A strong conceptual completeness (SCC) theorem for a logic allows the syntax of a theory to be canonically recovered from its space of models equipped with suitable structure. SCC theorems are known for finitary first-order logic (Makkai, who introduced the name SCC) and fragments thereof (Gabriel–Ulmer, Lawvere, and others). In this talk, I will present a SCC theorem for $\mathcal L_{\omega_1\omega}$: a countable $\mathcal L_{\omega_1\omega}$-theory can be recovered from its standard Borel groupoid of countable models. As a consequence, we obtain a generalization of a recent result of Harrison-Trainor, Miller, and Montalbán.

Tuesday, October 17, 2017

1:00 pmTuesday, October 17, 2017

#### No seminar

Tuesday, October 24, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, October 24, 2017

#### Cancelled

Tuesday, October 31, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, October 31, 2017

#### Presburger Arithmetic and its computational complexity

###### Danny Nguyen (UCLA)

Abstract: Presburger Arithmetic (PA) is a classical topic in logic, with numerous connection to computer science and combinatorics. Formally, is the first order structure on the integers with only additions and inequalities. Despite its long history, many problems in PA have remained unsolved until recently. We study the complexity of decision problems in PA, and classify them according to hierarchy levels. Along the way, connections to Integer Programming and Optimization will be explained. The talk will be self contained and assumes no prior knowledge of the subject. Joint work with Igor Pak.

Tuesday, November 7, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, November 7, 2017

#### A direct solution to the Generic Point Problem

###### Andrew Zucker (Carnegie Mellon)

Abstract: We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If $G$ is a Polish group and X is a minimal, metrizable $G$-flow with all orbits meager, then the universal minimal flow $M(G)$ is non-metrizable. In particular, we show that given $X$ as above, the universal highly proximal extension of $X$ is non-metrizable.

Tuesday, November 14, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, November 14, 2017

#### Ax–Schanuel and Strong Minimality

###### Vahagn Aslanyan (Carnegie Mellon)

Abstract: Schanuel's conjecture is a transcendence conjecture on the complex exponential function. The Ax–Schanuel theorem is a differential analogue of that conjecture proven by J. Ax in 1971. It establishes a transcendence result for the solutions of the exponential differential equation $y' = yx'$ in a differential field. I will discuss Ax–Schanuel type results for other differential equations/functions, most importantly, for the third order non-linear differential equation of the modular $j$-invariant, due to J. Pila and J. Tsimerman. I will show how that kind of results can be used to show that certain sets in differentially closed fields are strongly minimal and geometrically trivial. As an application I will give a new proof for a theorem of J. Freitag and T. Scanlon establishing strong minimality and geometric triviality of the differential equation of the $j$-function (which is the first example of a strongly minimal set in $\mathrm{DCF}_0$ with trivial geometry which is not $\aleph_0$-categorical). If time permits, I will discuss Ax–Schanuel type conjectures for the Painleve equations based on the results of J. Nagloo and A. Pillay on strong minimality and geometric triviality of those equations.

Tuesday, November 28, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, November 28, 2017

#### Differential-henselizations and maximality

###### Nigel Pynn-Coates (Illinois Math)

Abstract: Henselian fields play an important role in valuation theory and the model theory of valued fields. Two useful results are that every valued field has a henselization, and that henselianity is equivalent to algebraic maximality (at least in equicharacteristic 0). I will state my analogous results for asymptotic valued differential fields (where "differential" is put in the right places) and hope to say something about the proof and possible applications in ongoing work.

Monday, December 4, 2017

4:00 pm in 343 Altgeld Hall,Monday, December 4, 2017

#### Amenability of discrete groups and their actions

###### Kate Juschenko (Northwestern Math)

Abstract: The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on reals as well as on other spaces. In particular the study of isometry-invariant measure led to the Banach-Tarski decomposition theorem in 1924. The class of amenable groups was introduced by von Neumann in 1929, who explained why the paradox appeared only in dimensions greater or equal to three, and does not happen when we would like to decompose the two-dimensional ball. In 1940's, M. Day formally defined a class of elementary amenable groups as the largest class of groups amenability of which was known to von Naumann. He asked whether there are other groups then that. Currently there are many groups that answer von Neumann-Day's question. However, in each particular case it is algebraically difficult to show that the group is not elementary amenable, and analytically difficult to show that it is amenable. The talk is aimed to discuss recent developments and approaches in the field. In particular, it will be shown how to prove amenability of all known non-elementary amenable groups using only one single approach. We will also discuss techniques coming from random walks of groups.

Tuesday, December 12, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, December 12, 2017

#### When does the union of two model complete theories have a model companion

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

Abstract: Suppose $T_1,T_2$ are consistent model complete theories in languages $L_1,L_2$. Suppose that $T_1 \cap T_2$ is a complete consistent $(L_1 \cap L_2)$-theory. Then the Robinson joint consistency theorem implies $T_1 \cup T_2$ is consistent. We describe general conditions which ensure that $T_1 \cup T_2$ has a model companion. These conditions cover a number of important examples of model complete theories. Joint with Minh Tran.