Model theory seminar

Archives of 2015-2016

April 27th: Zariski Geometries VI, Nick.

April 20th: Zariski Geometries V, Nick.

April 13th: What do graph polynomials tell us about their underlying graphs?, J.A. Makowsky.
Over hundred years ago, G. Birkhoff noted that counting the number of proper colorings of a graph $G$ with $k$ colors gives rise to a polynomial $\chi(G;k) \in \Z[k]$. Further developments brought us the Tutte polynomial and its many incarnations in statistical mechanics and knot theory. In the 1950ties chemists introduced the characteristic and matching polynomials to study chemical properties of molecules, which initiated the field of chemical graph theory. In 2005 B. Zilber and I discovered that one can associate with any finite structure a family of polynomial invariants, which led the discovery of many new graph polynomials. In this talk we give a gentle introduction to graph polynomials aimed at model theorists, and report about more recent developments.

April 6th: Zariski Geometries IV, Silvain.

March 30th: Zariski Geometries III, Silvain.

March 16th: Zariski Geometries II, continued, Michael.

March 16th: Algebraically closed fields with a generic multiplicative character, Chieu Minh Tran.
We consider a 2-sort structure $(F, K; \chi)$ where $F = \mathbb{F}_p^a$, $K = \mathbb{Q}^a$ and $\chi: F \to K$ is an injective, multiplicative preserving map. I will briefly summarize some earlier results on model theory of the structure: axiomatization, completeness, quantifier reduction, regular model completeness, $\omega$-stability. Then I will discuss some attempts to geometrically understand definable sets in connection to proving that the theory has the definability of multiplicity property. (This is a joint work with T. Hakobyan).

March 9th: Zariski Geometries II, Michael.

March 2nd: Zariski Geometries I: Definitions and Examples, Reid.
A gentle introduction to the concept of Zariski geometries, focused on proving that certain familiar objects give us Zariski geometries: smooth quasiprojective curves, special singular curves, and strongly minimal sets in DCF_0 (after taking away a few points). The main difficulty in verifying the Zariski axioms in this case come from verifying the dimension theorem (Z3). If there is any time left we will begin to discuss quantifier elimination for and strong minimality of Zariski geometries.

February 24th: Properly Ergodic Structures, Alex.
One natural notion of "random L-structure" is a probability measure on the space of L-structures with domain \omega which is invariant and ergodic for the natural action of S_\infty on this space. We call such measures "ergodic structures." Ergodic structures arise as limits of sequences of finite structures which are convergent in the appropriate sense, generalizing the graph limits (or "graphons") of Lovász and Szegedy. In this talk, I will address the properly ergodic case, in which no isomorphism class of countable structures is given measure 1. In joint work with Ackerman, Freer, and Patel, we give a characterization of those theories (in any countable fragment of L_{\omega_1,\omega}) which admit properly ergodic models. The main tools are a Morley-Scott analysis of an ergodic structure, the Aldous-Hoover representation theorem, and an "AFP construction" - a method of producing ergodic structures via inverse limits of discrete probability measures on finite structures.

February 17th: Nilpotent Galois groups, Massey products and the Section conjecture, continued, Adam.

February 10th: ACFA XI, continued, Tom.
We will finish looking at the new proofs regarding friendliness.

February 10th: Nilpotent Galois groups, Massey products and the Section conjecture, Adam.

February 3rd: ACFA XI, Tom.
We will consider the new version of the main theorem on friendliness.

January 29th: Decomposition of dependent structures, Pierre Simon.
A structure is a set equipped with a family of predicates and functions on it, for example a group or a field. A structure is said to be dependent if it is not as complicated as a random graph (in some precise sense). The fields of complex numbers, real numbers and p-adic numbers are examples of dependent structures. If a dependent structure contains no definable order, then it is stable: a much stronger property that is now very well understood. Stability can be thought of as an abstraction of algebraic geometry. At the other extreme, are dependent structures which are very much controlled by linear orders. We call such structures distal. Distality is meant to serve as a general model for semi-algebraic (or analytic) geometry. In this talk, I will explain those notions and state a recent theorem saying that any dependent structure can be locally decomposed into a stable-like and a distal-like part.

January 27th: Extra-amenability and amenability of automorphism group of generic structures, Hamed Khalilian.
In this talk I will focus on extra amenability and amenability of the automorphism group of some Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions. By modifying the Kechris, Pestov and Todorcevic correspondence and Justin Moore's amenability theorem to these cases we will see that this group doesn't have any of these properties.

January 20th: Simplicity without local character, Nick.
What are the meaningful dividing lines among unstable theories without the strict order property? Simplicity theory provides a paradigm case of one such dividing line, providing a characterization of the syntactic property of not having the tree property in terms of the structural tameness of forking. We will describe how similar characterizations can be given for the class of NSOP1 theories. In brief, we will attempt to make the case for the informal equation "NSOP1 = simple - local character".

December 2nd: ACFA X, Alex.
We continue our study of ACFA in characteristic p. In this talk, I will (at long last) define friendliness for pairs of coordinate rings and and prove that under the assumption of friendliness, the virtual loci of virtual ideals are closed under intersection. We will then discuss the main theorem of the section: the pair of coordinate rings associated to a semi-basic type is friendly. Instead of proving this theorem, I will present a counterexample.

November 18th: Banach lattice methods for proving axiomatizability of Banach spaces, Ward.
The context is first order logic for metric structures, applied to (unit balls of) Banach spaces and their expansions. Using this logic, many classes of Banach lattices are known to be axiomatizable. However, relatively few of the associated classes of Banach spaces have been similarly understood. With Yves Raynaud, we have developed some new techniques for transferring axiomatizability of suitable classes of Banach lattices to their Banach space reducts. The techniques involve facts about disjointness preserving linear isometries (between Banach lattices) from functional analysis, and facts about definability, especially definability of sets, from continuous model theory. These ideas also have a range of other applications to the model theory of familiar Banach spaces.

November 11th: ACFA IX,Reid.
We continue studying the model theory of difference fields in characteristic p, following the paper by Chatzidakis, Hrushovski, and Peterzil. In this talk, we will talk about the friendliness of virtual ideals.

November 4th: ACFA VIII, Alex.
We continue studying the model theory of difference fields in characteristic p, following the paper by Chatzidakis, Hrushovski, and Peterzil. In this talk, I will introduce the notion of a (semi-)basic type in a model of ACFA and show how to associate a pair of coordinate rings to such a type. "Virtual ideals" in these coordinate rings will eventually give us the closed sets of a Zariski geometry, which we will use to prove the Zil'ber trichotomy in all characteristics (but not in this talk!).

October 28th: Compact complex manifolds with a generic automorphism, Michael.
CCMA exists and eliminates geometric imaginaries, but it does not eliminate imaginaries. I’ll give definitions and then sketch proofs of these facts, relying heavily on black boxes from the model theory of generic automorphisms, as well as complex geometry.

October 21th: ACFA VII, Nick.
We will start the second paper.

October 14th: ACFA VI, Tom.
We will finish the proof of the dichotomy in characteristic zero.

October 7th: ACFA V, Tom.

September 30th: The Jacobi bound conjecture in differential algebra, Wei.
The Jacobi bound conjecture is a classical open problem in differential algebra. It conjectures that the order of a component of a differential polynomial equation system S=0 is bounded by the Jacobi number of S. First proposed by Jacobi heuristically, this conjecture was rediscovered and introduced to differential algebra by J.F. Ritt in 1930s. Later, it was shown by R. Cohn that the Jacobi bound conjecture is closely related to the differential dimension conjecture, another well-known conjecture in differential algebra. In this talk, I will give a survey on the contributions of differential algebraists made on these two conjectures and also introduce our recent new results in these directions.

September 23th: ACVF IV, Will.
We will talk about ranks.

September 16th: ACVF III, Will.
We will talk about imaginaries.

September 9th: ACVF II, Silvain.
This week, we will probably see the n-amalgamation theorem and the elimination of imaginaries.

September 2nd: ACFA I, Reid.
We, hopefully, will go through reduction of quantifiers and the independence theorem.