Logic, Set Theory and Topology Ical Atom

Zilber’s trichotomy conjecture, asserting roughly, that a non-locally modular strongly minimal set is an algebraic curve over an algebraically closed field, was thoroughly refuted by Hrushovski in 1988. Nevertheless, the ideology underlying Zilber’s conjecture remains one of the most influential (and most successful) classifying principles in model theory. In the first talk I will explain Zilber’s trichotomy conjecture and its history, focusing on various restricted versions of the principle.

In the second talk I will discuss a long term project of classifying, along Zilber’s trichotomy, all structures interpretable in o-minimal theories. I will describe the main ideas in a recent result (joint with Eleftheriou and Peterzil) proving that if $(G,+)$ is a commutative complex Lie group and $G_S:=(G, +, S)$ is a strongly minimal non locally modular structure interpretable in an o-minimal expansion of the reals, then $G_S$ interprets an algebraically closed field. ​​

Zilber’s trichotomy conjecture, asserting roughly, that a non-locally modular strongly minimal set is an algebraic curve over an algebraically closed field, was thoroughly refuted by Hrushovski in 1988. Nevertheless, the ideology underlying Zilber’s conjecture remains one of the most influential (and most successful) classifying principles in model theory. In the first talk I will explain Zilber’s trichotomy conjecture and its history, focusing on various restricted versions of the principle.

In the second talk I will discuss a long term project of classifying, along Zilber’s trichotomy, all structures interpretable in o-minimal theories. I will describe the main ideas in a recent result (joint with Eleftheriou and Peterzil) proving that if $(G,+)$ is a commutative complex Lie group and $G_S:=(G, +, S)$ is a strongly minimal non locally modular structure interpretable in an o-minimal expansion of the reals, then $G_S$ interprets an algebraically closed field. ​​

(joint work with Elad Levi and Pierre Simon) Macpherson proved that omega-stable omega-categorical groups are nilpotent-by-finite. Krupinski and Krupinski with Dobrowolski (in two separate works, one with NIP, the other without) replaced the stability assumption by the much weaker assumption of being generically-stable. We go to the other direction, and try to generalize Krupinski’s first result (NIP omega-categorical groups with fsg are nilpotent-by-finite) to remove the fsg assumption. We succeed in the simplest NIP case, i.e., when the group is dp-minimal.

I will try to give a full proof of this result. All concepts will be defined during the talk, but some basic knowledge of model theory (e.g., omega categorical theories) might be helpful.

The argument “there are only countably many definitions in the language of set theory, therefore there are uncountably many undefinable reals” is fallacious. The same applies for the argument suggesting that the first undefinable (first order, no parameters) ordinal must be countable, and the two statements are closely linked. In the paper “Pointwise definable models of Set Theory” by Joel Hamkins et all, countable models of set theory are constructed in which every element is definable. We shall demonstrate a couple of these constructions, and then move on to consider models in which there are undefinable ordinals. We’ll show that in well-founded models of V=L there exists a dichotomy: Either the first undefinable ordinal is countable (according to the model), or it does not exist. We’ll construct well-founded models of ZFC (given very low-strength consistency hypotheses) in which the first undefinable ordinal is uncountable (according to the model). We’ll show that the consistency strength of “The first undefinable ordinal is a cardinal” is roughly that of an inaccessible cardinal, and that the consistency strength of “The first undefinable ordinal is a regular cardinal” is roughly that of a Mahlo cardinal.

Note: The lecture will be given in two sessions (this week, and the next). Basic knowledge of forcing is recommended, but not completely required - as notions and facts about the relevant forcings will be stated.

The argument “there are only countably many definitions in the language of set theory, therefore there are uncountably many undefinable reals” is fallacious. The same applies for the argument suggesting that the first undefinable (first order, no parameters) ordinal must be countable, and the two statements are closely linked. In the paper “Pointwise definable models of Set Theory” by Joel Hamkins et all, countable models of set theory are constructed in which every element is definable. We shall demonstrate a couple of these constructions, and then move on to consider models in which there are undefinable ordinals. We’ll show that in well-founded models of V=L there exists a dichotomy: Either the first undefinable ordinal is countable (according to the model), or it does not exist. We’ll construct well-founded models of ZFC (given very low-strength consistency hypotheses) in which the first undefinable ordinal is uncountable (according to the model). We’ll show that the consistency strength of “The first undefinable ordinal is a cardinal” is roughly that of an inaccessible cardinal, and that the consistency strength of “The first undefinable ordinal is a regular cardinal” is roughly that of a Mahlo cardinal.

Note: The lecture will be given in two sessions. Basic knowledge of forcing is recommended, but not completely required - as notions and facts about the relevant forcings will be stated.

Generically stable groups and more specifically algebraic groups with generically stable generics were studied by Hrushovski in “Valued fields, Metastable Groups”. We’ll talk about a class of group schemes over the valuation ring definable in ACVF and show that they are generically stable. In fact, we define a functor between the category of algebraic varieties with a Zariski dense generically stable type concentrated on it to generically stable definable schemes over the valuation ring. If everything is over a model, algebraic groups get sent to group schemes. If time permits we may say a few words about strongly stably dominated types.

Do not come. We will not be there.

Do not come. We will not be there.

My plan is to remind you, what I told you last semester about the recovery of the action of semigroups and clones from their algebraic structure.

Then I’ll prove a transfer theorem from semigroups to clones. It is short and simple.

I shall present a bunch of open problems which are quite elementary (in their formulation).

Finally I shall try to give excerpt arguments from one of the theorems that we (Yonah and me) proved in this topic.

TBA

We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible.

What is the relationship between a Souslin tree and its reduced powers? and is there any difference between, say, the reduced w-power and the reduced w1-power of the same tree? In this talk, we shall present tools recently developed to answer these sort of questions. For instance, these tools allow to construct an w6-Souslin tree whose reduced w_n-power is Aronszajn iff n is not a prime number.

This is joint work with Ari Brodsky.

We will define the “lexicographic product” of two structures and show that if both structures admit quantifier elimination, then so does their product. As a corollary we get that nice (model theoretic) properties such as (ultra)homogeneity, stability, NIP and more are preserved under taking products.

It is clear how to iterate the product finitely many times, but we will introduce a new infinite product construction which, while not preserving quantifier elimination, does preserve (ultra)homogeneity. As time allows, we will use this to give a negative answer to the last open question from a paper by A. Hasson, M. Kojman and A. Onshuus who asked ``Is there a rigid elementarily indivisible* structure?’’

As time allows, we will introduce an approach for using the lexicographic product to generalize a result by Lachlan and Shelah to the following: given a finite relational language L, denote by H(L) the class of countable ultrahomogeneous stable L-structures. For M in H(L), define the rank of M to be the maximum value of CR(p,2) where p is a complete 1-type and CR(p,2) is the Shelah’s complete rank. There is a uniform finite bound on the rank of M, where M ranges over H(L). The result was proven by Lachlan and Shelah for L binary and proven in general by Lachlan using the Classification Theorem for finite simple groups.

• A structure M is said to be elementarily indivisible structure if for every colouring of its universe in two colours, there is a monochromatic elementary substructure N of M such that N is isomorphic to M.

We will define the “lexicographic product” of two structures and show that if both structures admit quantifier elimination, then so does their product. As a corollary we get that nice (model theoretic) properties such as (ultra)homogeneity, stability, NIP and more are preserved under taking products.

It is clear how to iterate the product finitely many times, but we will introduce a new infinite product construction which, while not preserving quantifier elimination, does preserve (ultra)homogeneity. As time allows, we will use this to give a negative answer to the last open question from a paper by A. Hasson, M. Kojman and A. Onshuus who asked ``Is there a rigid elementarily indivisible* structure?’’

As time allows, we will introduce an approach for using the lexicographic product to generalize a result by Lachlan and Shelah to the following: given a finite relational language L, denote by H(L) the class of countable ultrahomogeneous stable L-structures. For M in H(L), define the rank of M to be the maximum value of CR(p,2) where p is a complete 1-type and CR(p,2) is the Shelah’s complete rank. There is a uniform finite bound on the rank of M, where M ranges over H(L). The result was proven by Lachlan and Shelah for L binary and proven in general by Lachlan using the Classification Theorem for finite simple groups.

• A structure M is said to be elementarily indivisible structure if for every colouring of its universe in two colours, there is a monochromatic elementary substructure N of M such that N is isomorphic to M.

An w2-Suslin tree is a tree of height omega-2 which has no branches nor anti-chains of size omega-2. I will show that if we assume (less than) GCH and Square principle on omega-1 we can build an w2-Suslin Tree.