# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Informal Session

We will meet in an informal discussion to talk about recent work in model theory as well as any current problems we are interested in. Also we can use the opportunity to discuss plans for the seminar next semester. Bring any ideas or recent preprints you are interested in talking about. In particular see the most recent model theory preprints posted on the

ArXiv.

# First order expansions of the ordered group of real numbers

We discuss (part of) the classification of first order expansions of $(\mathbb{R},<,+)$ according to the geometry and topology of their definable sets. Joint work with Philipp Hieronymi, following work of Hieronymi, Fornasiero, Miller, and Tychonievich.

# Foundations of cologic

The existence of a robust categorical dual to first-order logic is hinted at in (at least) four independent bodies of work: (1) Projective Fraïssé theory [Solecki & coauthors, Panagiotopolous]. (2) The cologic of profinite groups (e.g. Galois groups), which plays an important role in the model theory of PAC fields [Cherlin – van den Dries – Macintyre, Chatzidakis]. (3) Ultracoproducts and coelementary classes of compact Hausdorff spaces [Bankston]. (4) Universal coalgebras and coalgebraic logic [Rutten, Kurz – Rosicky, Moss, others]. In this talk, I will propose a natural syntax and semantics for such a dual “cologic”, in which “coformulas” express properties of partitions of “costructures”, dually to the way in which formulas express properties of tuples from structures. I will show how the basic theorems and constructions of first-order logic (completeness, compactness, ultraproducts, Henkin constructions, Löwenheim-Skolem, etc.) can be dualized, and I will discuss some possible extensions of the framework.

# No seminars on Nov. 25

Friday, November 25 is the day after Thanksgiving, so there will be no seminars at the Graduate Center that day.

# Straw into gold: Turning a c.c.c. forcing construction of a model into a ZFC proof

We use $M$-normal ultrapowers to give a new proof of a theorem of Keisler that if one can construct a standard model of a sentence of $L_{\omega_1,\omega}(Q)$ using a c.c.c. forcing, then a standard model already exists in V.

We use this to investigate the class of atomic models of a countable, first order theory $T$. In particular, we show that various `unsuperstable-like’ behaviors imply the existence of many non-isomorphic atomic models of size $\aleph_1$.

This is joint work with John Baldwin and Saharon Shelah.

# Isomorphisms of ordered valued fields

A real closed field with a convex valuation is a nice model-theoretic object. In this talk, I will look at some examples of such structures to get a sense of why they are nice. I will discuss two theorems which illustrate the analogues with algebraically closed valued fields, which lead us to a notion of domination by the residue field. This is joint work with Clifton Ealy and Jana Marikova.

# Northeast Regional Model Theory Days

The Northeast Regional Model Theory Days will take place on Saturday the 21st of October and

Sunday the 22nd of October at the University of Pennsylvania in Philadelphia PA. See the Northeast Regional Model Theory Days website for full details.

# Imaginaries in valued differential fields II: Computing canonical bases

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in a algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon’s theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the second part of the proof that this result indeed holds. Our goal will be to explain a result, joint with Pierre Simon, on definable types in enrichments of NIP theories, which is crucial to prove elimination of imaginaries. We show that under certain hypothesis if a type in some NIP theory T is definable in an enrichment of T, then it h is already be definable in T.

# On Collapse of Generalized Indiscernibles.

I discuss the characterization of model theoretic dividing lines by collapse of generalized indiscernibles. For example, S. Shelah showed that a theory is stable if and only if all order indiscernibles are set indiscernibles. In her thesis, L. Scow showed that a theory has NIP if and only if all graph order indiscernibles are order indiscernibles. I explore new results characterizing dp-rank and rosiness using similar methods. I also talk about some attempts to create a general framework for such results. This work is joint with C. Hill and L. Scow.

# Complexity of classification problems for a class of discretely ordered rings

I will give a standard introduction to the theory of Borel reducibility, including some details for non-logicians. The rest of the talk will be about some results in classification problems for countable nonstandard models of arithmetic due to Samuel Coskey and myself.

# An introduction to Pillay’s differential Galois theory (part 2)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic *D*-groups.

# An introduction to Pillay’s differential Galois theory (part 1)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic *D*-groups.

# Structure and enumeration theorems in hereditary properties of L-structures

The study of structure and enumeration for hereditary graph properties has been a major area of research in extremal combinatorics. Over the years such results have been extended to many combinatorial structures other than graphs. This line of research has developed an informal strategy for how to prove these results in various settings. In this talk we formalize this strategy. In particular, we generalize certain definitions, tools, and theorems which appear commonly in approximate structure and enumeration theorems in extremal combinatorics. Our results apply to classes of finite L-structures which are closed under isomorphism and model-theoretic substructure, where L is any finite relational language.

# The logical complexity of Schanuel’s Conjecture

In its most natural form Schanuel’s Conjecture is a $\Pi_1^1$-statement. We will show that there is an equivalent $\Pi^0_3$-statement. They key idea is a result of Jonathan Kirby showing that, if Schanuel’s Conjecture is false, then there are canonical counterexamples. Most of my lecture will describe Kirby’s work.

# No logic seminars on October 14

Since CUNY will follow a Tuesday schedule on Friday, October 14, we will not have any of the usual Friday logic seminars that day. However, there will be a talk by George Metcalfe at 4 pm, described below.

# $\omega$-stability and Morley rank of bilinear maps, rings and nilpotent groups

In this talk I will present our recent results on the algebraic structure of $\omega$-stable bilinear maps, arbitrary rings and nilpotent groups. I will also discuss some rather complete structure theorems for the above structures in the finite Morley rank case. The main technique in this work is associating a canonical scalar (commutative associative unitary) ring to a bilinear map. This canonical scalar ring happens to preserve many of the logical and algebraic properties of the bilinear map. This talk is based on the joint work with Alexei Miasnikov.

# Measuring definable sets in nonarchimedean o-minimal fields and an application to the reals

We introduce a measure on the definable sets in an o-minimal expansion of a real closed field which takes values in an ordered semiring and assigns a positive value to a definable set iff the interior of the set is non-empty (joint work with M. Shiota). We then discuss an application to Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field (joint work with E. Walsberg).

# The Dixmier-Moeglin problem for D-varieties

By a D-variety we mean, following Buium, an algebraic variety V over an algebraically closed field k equipped with a regular section s: V–> TV to the tangent bundle of V. (This is equivalent to the category of finite dimensional differential-algebraic varieties over the constants.) There are natural notions of D-rational map and D-subvariety. Motivated by problems in noncommutative algebra we are lead to ask under what conditions (V,s) has a maximum proper D-subvariety over k. (Model-theoretically this asks when the generic type is isolated.) A necessary condition is that (V,s) does not admit a nonconstant D-constant, that is, a D-rational map from (V,s) to the affine line equipped with the

zero section. When is this condition sufficient? I will discuss this rather open-ended problem, including some known cases.

# Imaginaries in valued fields of positive characteristic

I will discuss the basic properties of the theory SCVF of separably closed valued fields, and the closely related theory SCVH of separably closed valued fields equipped with Hasse derivations. The main new result is elimination of imaginaries (in a suitable language). This is joint work with Martin Hils and Silvain Rideau.

# On a question of Gaifman concerning invariant measures

In his 1964 paper “Concerning measures in first order calculi” Gaifman introduces the notion of a symmetric measure-model: a measure on the formulas of a first order calculus that is invariant under permutations of the elements instantiating the free variables of each formula, where these elements come from some fixed domain. To each symmetric measure-model there is associated a measure on sentences, which we can think of as a random (consistent) theory that the measure-model satisfies. Gaifman shows that every such random theory has a symmetric measure-model satisfying it. However, the symmetric measure-models that he constructs sometimes, necessarily, assign positive measure to instantiations of the formula “x=y” by unequal elements. Gaifman goes on to pose the question of characterizing those classical theories that admit symmetric measure-models without this pathology — those with so-called `strict equality’. In this talk I will show that when the instantiating domain is the set of natural numbers, a symmetric measure-model with strict equality is essentially a probability measure on a space of structures, with underlying set the natural numbers, that is invariant under the logic action. I will then give necessary and sufficient conditions for a classical theory to admit such an invariant measure, thereby providing an answer to the question posed by Gaifman. This is joint work with Nathanael Ackerman and Cameron Freer.