**CUNY Graduate Center**

**Room 6417**

**Fridays 12:30pm-2:00pm**

**Organized by Alf Dolich**

**Fall 2019**

**December 6**

**Rizos Sklinos**
Stevens Institute of Technology

**Fields definable in the theory of nonabelian free groups**

**Abstract**

In this talk I will show that only finite fields are definable in the theory of nonabelian free groups. This is joint work with Ayala Byron.

**November 29**

**Seminar cancelled**

Thanksgiving break.

**November 22**

**Alex Kruckman**
Wesleyan University

**A diversity of Kim's Lemmas**

**Abstract**

One of the most important steps in the development of simplicity theory by Kim and Pillay in the 1990s was a result now known as Kim's Lemma: In a simple theory, if a formula divides, then this dividing is witnessed by every Morley sequence in the appropriate type. More recently, variants on Kim's Lemma have been shown (by Chernikov, Kaplan, and Ramsey) to follow from, and in fact characterize, the combinatorial dividing lines NTP2 and NSOP1: two generalizations of simplicity in different directions. After surveying the Kim's Lemmas of the past, I will speculate about a new combinatorial dividing line, generalizing both NTP2 and NSOP1 and characterized by a new variant of Kim's Lemma. This is joint speculation with Nick Ramsey.

**November 15**

**Special time: 10:00-11:30am**

**Andrés Villaveces**
Universidad Nacional de Colombia

**Some New Infinitary Logics and a Canonical Tree**

**Abstract**

The main recent logic I will describe is Shelah's infinitary logic $L^1_\kappa$ (from 2012). I will describe some of the reasons for studying this logic (roughly, it is an infinitary logic that has interpolation and a weak form of compactness - therefore particularly well-adapted to model theory, as well as closure under chains) and some of the features lacking (mostly, a workable syntax). I will describe two other logics that have been created in order to capture better the syntax (one of these logics is my joint work with Väänänen, the other one is due originally to Karp and Cunningham and has recently been connected to $L^1_\kappa$ by Dzamonja and Väänänen. Finally I will connect these logics with the problem of axiomatizing abstract elementary classes. In particular, I will describe *canonical trees* of models that enables one to build a sentence to test models for membership into aecs. This last part is joint work with Shelah.

**November 8**

**Seminar cancelled**

MathFest at the Graduate Center.

**October 25**

**Alice Medvedev**
CUNY

**TBA**

**Abstract**

**October 18**

**Russell Miller**
CUNY

**Model Completeness and Relative Decidability**

**Abstract**

Model-completeness is a standard notion in model theory, and it is well known that a theory $T$ is model complete if and only if $T$ has quantifier elimination down to existential formulas. From the quantifier elimination, one quickly sees that every computable model of a computably enumerable, model-complete theory $T$ must be decidable. We call a structure *relatively decidable* if this holds more broadly: if for all its copies $\mathcal{A}$ with domain $\omega$, the elementary diagram of $\mathcal{A}$ is Turing-reducible to the atomic diagram of $\mathcal{A}$. In some cases, this reduction can be done uniformly by a single Turing functional for all copies of $\mathcal{A}$, or even for all models of a theory $T$.

We discuss connections between these notions. For a c.e. theory, model completeness is equivalent to uniform relative decidability of all countable models of the theory, but this fails if the condition of uniformity is excluded. On the other hand, for relatively decidable structures where the reduction is not uniform, it can be made uniform by expanding the language by finitely many constants to name certain specific elements. This is shown by a priority construction related to forcing. We had conjectured that a similar result might hold for theories $T$ such that every model of $T$ is relatively decidable, but in separate work, Matthew Harrison-Trainor has now shown relative decidability to be a $\Pi^1_1$-complete property of a theory, which is far more complicated than our conjectured equivalent property.

This is joint work with Jennifer Chubb and Reed Solomon.

**October 11**

**Alexander Van Abel**
CUNY

**On Pseudofinite Dimension and Measure II**

**Abstract**

A pseudofinite structure is (among many equivalent definitions) a structure which is elementarily equivalent to an infinite ultraproduct of finite structures. In this talk, we discuss how the natural counting measure on finite structures lifts to useful notions of dimension and measure on pseudofinite structures. We give a proof of Furstenburg's Correspondence Principle in combinatorial number theory, using pseudofinite measure. We sketch a proof, by Chernikov and Starchenko, of a special case ('stable' graphs) of the Erdös-Hajnal conjecture, using a particular notion of pseudofinite dimension. Finally, we discuss how a different notion of dimension leads to simplicity results. This talk is largely based on Darío García's lecture notes on 'Model theory of pseudofinite structures'.

**October 4**

**Alexander Van Abel**
CUNY

**On Pseudofinite Dimension and Measure**

**Abstract**

A pseudofinite structure is (among many equivalent definitions) a structure which is elementarily equivalent to an infinite ultraproduct of finite structures. In this talk, we discuss how the natural counting measure on finite structures lifts to useful notions of dimension and measure on pseudofinite structures. We give a proof of Furstenburg's Correspondence Principle in combinatorial number theory, using pseudofinite measure. We sketch a proof, by Chernikov and Starchenko, of a special case ('stable' graphs) of the Erdös-Hajnal conjecture, using a particular notion of pseudofinite dimension. Finally, we discuss how a different notion of dimension leads to simplicity results. This talk is largely based on Darío García's lecture notes on 'Model theory of pseudofinite structures'.

**September 27**

**Sam Braunfeld**
University of Maryland

**Monadic stability and growth rates of omega-categorical structures**

**Abstract**

Generalizing the classical combinatorial problem of counting the orbits of a group acting on a finite set, we consider the growth rate of an omega-categorical structure M, which counts the orbits of Aut(M) acting on n-sets. We show that for stable M, there is a gap from subexponential to superexponential growth, corresponding to whether M is monadically stable. This allows us to confirm some longstanding conjectures of Macpherson about the spectrum of possible growth rates.

**September 20**

**Matěj Konečný**
Charles University

**Extending partial automorphisms of structures**

**Abstract**

This is based on joint work with David Evans, Jan Hubička and Jaroslav Nešetřil. The extension property for partial automorphisms (EPPA), also called the Hrushovski property is a property of classes of finite structures stating that for every A there is B containing A as a substructure such that every isomorphism of substructures of A extends to an automorphism of B. Every class with EPPA is an amalgamation class, in fact, EPPA is equivalent to some properties of the automorphism group of the Fraisse limit of the class. In particular, EPPA is a key ingredient in proving ample generics, the small index property etc. In this talk, we show a new easy way of proving EPPA for the class of all finite graphs and then explain how to extend these techniques to get EPPA for two-graph and also the strongest sufficient condition for EPPA so far. This talk should be self-contained.

**August 30**

**Seminar cancelled**