# Blog Archives

# Topic Archive: model theory

# Proaperiodic monoids and model theory

We begin with the observation that the free profinite aperiodic monoid over a finite set *A* is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite *A*-labelled linear orders (“*A*-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite *A*-words. From this perspective, the operations of multiplication and ω-power on proaperiodic monoids can be understood in a very concrete way. This point of view allows us to import methods from both topology and model theory, in particular saturated models, into the study of proaperiodic monoids. We use these methods to prove results about ω-terms in the free proaperiodic monoid and well-quasi-orders of factors in related proaperiodic monoids.

# 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.

Tuesday, October 4, 2016

# DART VII

DART VII, the seventh meeting of the conference series Differential Algebra and Related Topics, will take place at the CUNY Graduate Center (and other CUNY venues) from September 30 – October 4, 2016. Information about this conference is available here. It will include talks by Alexander Buium, Zoe Chatzidakis, Taylor Dupuy, James Freitag, Victor Kac, Joel Nagloo, Anand Pillay, and Thomas Scanlon, among others.

# DART VII

DART VII, the seventh meeting of the conference series Differential Algebra and Related Topics, will take place at the CUNY Graduate Center from September 30 – October 4, 2016. Information about this conference is available here.

# A model theory of affine n-space via differential algebra

Affine $n$-space $A_k^n$ and its algebraic equivalent, the polynomial ring $k[x_1,…,x_n]$, are basic and widely studied objects in geometry and algebra, about which we know a great deal. However, there remains a host of basic open problems (like the Jacobian conjecture, Zariski Cancellation Conjecture, Complement Problem, …) indicating that our knowledge is nonetheless quite limited. In fact, the greatest obstacle in solving the above conjectures is our inability to “pinpoint” affine space among all varieties (or $k[x]$ among all finitely generated $k$-algebras): this is the so-called Characterization Problem.

The most recent approach to these problems is via additive group actions on affine $n$-space, which corresponds on the algebraic side, to the theory of locally nilpotent derivations. Using this, for instance, N. Gupta recently showed the falsitude of the Zariski Cancellation Conjecture in positive characteristic.

From a model-theoretic point of view, the polynomial ring (in its natural ring language) is quite expressive: in characteristic zero, one can define the integers (as a subset), one can express in general that, say, Embedded Resolution of Singularities holds, etc. Of course, one of the peculiarities of model theory (and probably one of the reasons for its pariah status) is the unavoidable presence of non-standard models. In other words, a characterization problem is never solvable in model theory, unless one allows some non first-order conditions as well (e.g., cardinality in categorical theories–but most mainstream mathematicians would not be too happy about that either). But other, more intrinsic problems arise: there are elementary equivalent fields whose polynomial rings are not. So, can we find an expanded language plus a “natural” but non first-order condition, that pinpoints the standard model, i.e., $k[x]$ within the models of its theory. Or even better, since these complete theories will have unwieldy axiomatizations, can we find a (recursive?) theory, whose only model satisfying the extra non first-order condition is $k[x]$?

In view of the recent developments in algebra/geometry, to this end, I will propose in this talk some languages that include additional sorts, in particularly, a sort for derivations. This is different from the usual language of differential fields, where one only studies a fixed (or possibly finitely many) derivation: we need all of them! We also need a substitute for the notion of degree, and the corresponding group $Z$-action as power maps. To test our theories, we should verify which algebraic/geometric properties are reflected in this setup. For instance, affine $n$-space has no cohomology, which is equivalent to the exactness of the de Rham complex, and this latter statement is true in any of the proposed models. Nonetheless, this is only a preliminary analysis of the problem, and nothing too deep will yet be discussed in this talk.

# Differential fields — a model theorist’s view

In his book *Saturated Model Theory*, Gerald Sacks described differentially closed fields as “the least misleading” example of an Ω-stable theory. His remark was particularly prescient as many interesting model theoretic phenomena arise naturally in differential algebra. Model theory has been strangely effective in both solving and generating questions in differential algebraic geometry. I will survey some aspects of this interaction.

This talk is part of a weekend-long workshop in differential algebra. Details are available here.

# Trichotomy principle for partial differential fields

The Zilber trichotomy principle gives a precise sense in which the structure on a sufficiently well behaved one-dimensional set must have one of only three possible kinds: disintegrated (meaning that there may be some isolated correspondences, but nothing else), linear (basically coming from an abelian group with no extra structure), or algebro-geometric (essentially coming from an algebraically closed field). This principle is true in differentially closed fields when “one dimensional” is understood as “strongly minimal” (proven by Hrushovski and Sokolovic using the theory of Zariski geometries and then by Pillay and Ziegler using jet spaces).

When working with differentially closed fields with finitely many, but more than one, distinguished commuting derivations, there are sets which from a certain model theoretic point of view (having to do with the notion of a regular type) are one dimensional even though they are infinite dimensional from the point of view of differential dimension. Moosa, Pillay and Scanlon showed that a weakening of the trichotomy principle is true for these sets: if there is a counter example to the trichotomy principle, then one can be found for a set defined by linear PDEs.

In this lecture, I will explain in detail what the trichotomy principle means in differential algebra, how the reduction to the linear case works, and then how one might approach the open problems.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# Classification of strongly normal extensions of a differential field, and related issues

The material is taken from a joint paper with M. Kamensky, “Interpretations and differential Galois extensions.” Given a differential field *K* with field of constants *k*, and a logarithmic differential equation over *K*, the strongly normal extensions of *K* for the equation correspond (up to isomorphism over *K*) with the connected components of *G(k)* where *G* is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory,….), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# Studying profinite monoids via logic

This talk is about my ongoing joint research project with Benjamin Steinberg (CCNY). We begin with the observation that the free profinite aperiodic monoid over a finite set *A* is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite *A*-labelled linear orders (“*A*-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite *A*-words. We exploit this view of the free profinite aperiodic monoid to prove both old and new things about it using methods from model theory, in particular (weakly) saturated models.

The talk is aimed at anyone with a basic knowledge of model theory, not necessarily of profinite monoids; in particular I will take care to review some background on profinite monoids and on how they relate to logic and regular languages.

# The model theory of some j-functions

The model theory of j-functions (and more generally of modular and Shimura curves) has been studied by Adam Harris and Christopher Daw. They connected in an intriguing way the categoricity of (an infinitary theory of) the j-function with results in arithmetic geometry (a version of the Mumford-Tate conjecture). I will discuss some of these connections and the questions they raise for model theory, especially in connection with the quest for new versions of j-functions (e.g. on real fields).

# Zero-one laws for discrete metric spaces

Fix an integer $r \geq 3$. Given an integer $n$, we define $M_r(n)$ to be the set of metric spaces with underlying set ${1,\ldots,n}$ such that the distance between any two points lies in ${1,\ldots,r}$. We present results describing the approximate structure of these metric spaces when $n$ is large. As a consequence of these structural results in the case when $r$ is even, we obtain a first-order labeled $0$-$1$ law. This is joint work with Dhruv Mubayi.

# Loftiness

Loftiness is a weak notion of saturation. It was defined and studied in detail by Matt Kaufmann and Jim Schmerl in two substantial papers published in 1984 and 1987. Kaufman and Schmerl discovered that there are many shades of loftiness. I will give an overview of model theory of lofty models of arithmetic and I will talk about constructions of lofty models that are not recursively saturated.

# Scott ranks of models of a theory

I will talk about a few different results about the Scott ranks of models of a theory. (By a theory, I mean a sentence of *L _{ω1ω}*.) These results are all related in that they all follow from the same general construction; this construction takes a pseudo-elementary class

**C**of linear orders and produces a theory

*T*such that the Scott ranks of models of

*T*are related to the well-founded parts of linear orders in

**C**.

The main result is a descriptive-set-theoretic classification of the collections of ordinals which are the Scott spectrum of a theory. We also answer some open questions. First, we show that for each ordinal *β*, there is a *Π _{2}^{0}* theory which has no models of Scott rank less than

*β*. Second, we find the Scott height of computable infinitary sentences. Third, we construct a computable model of Scott rank

*ω*which is not approximated by models of low Scott rank.

_{1}^{CK}+1# Categories turned models: taming the finite

Whereas a category theorist sees mathematics as objects interacting with each other via maps, a model theorist looks instead at their internal structure. So we may think of the former as the sociologues of mathematics and the latter as their psychologues. It is well-known that to a first-order theory we can associate the category of its models, but this produces often a non-natural category, as the maps need to be elementary, and maps rarely are! I will discuss the opposite (Jungian?) perspective: viewing a category as a first-order structure. This yields some unexpected rewards: it allows us to define certain second-order concepts, like finiteness, in a first-order way. I will illustrate this with some examples: sets, modules, topologies, …