# Past talks

**April 14th**, Itay
Kaplan.

**April 7th: About ***Model theoretic dynamics in a
more Galois fashion*, Daniel Hoffmann.

I
will present the "nicest" results from my last draft, which
investigates a theory of models equipped which an action of a
fixed group. In the draft, I show that, under the assumption
of stability of the basic theory and the assumptions of having
bounded models, the arising model companion of the theory of
models equipped with an action of our fixed group is simple
and eliminates quantifiers up to some existential formulas -
similar to ACFA. Moreover, it codes finite sets and allows a
geometric elimination of imaginaries, but not always a weak
elimination of imaginaries. At some point, I will try to
explain my motivation for this research.

**March 24th: On the structure of certain valued
fields**, Jung-Uk Lee.

We show that for finitely ramified valued fields with
perfect residue fields, for large enough n, any homomorphism
of the n-th residue rings can be lifted to a homomorphism of
valuation rings quite naturally. Here, the n-th residue ring
is the quotient of the valuation ring by the n-th power of
the maximal idea. Thus we have a lifting map for finitely
ramified valued fields like as the lifting map of
homomorphisms of residue fields to homomorphisms of
valuation rings for the unramified case. Moreover this
lifting map is compatible with composition of homomorphisms
of n-th residue rings. This provides a functor from a
category of certain principal Artinian local rings of length
n to a category of certain valuation ring of fixed
ramification index, which naturally generalizes the
functorial property of Witt ring. The result also
strengthens Basarab's result on the AKE-principle for
finitely ramified henselian valued fields, which solve a
question raised by Basarab, when residue fields are
perfect. This is joint work with Wan Lee.

**March 17th: Reducts of algebraic curves
III**, Dmitry
Sustretov.

**March 15th: Stability and sparsity in sets of
natural numbers**, Gabriel
Conant.

The additive group of integers is a
well-studied example of a stable group, whose definable
sets can be easily and explicitly described. However,
until recently, very little has been known about stable
expansions of this group. In this talk, we examine the
relationship between model-theoretic stability of
expansions of the form (Z,+,0,A), where A is a subset of
the natural numbers, and the number theoretic behavior of
A with respect to sumsets, asymptotic density, and
arithmetic progressions.

**March 15th at 9am in 891 Evans: Reducts of
algebraic curves II**, Dmitry
Sustretov.

**March 13th at 9am in 891 Evans: Reducts of
algebraic curves I**, Dmitry
Sustretov.

**March 10th: Dreaming about NIP
fields**, Franziska
Jahnke.

We discuss the main conjectures about
NIP fields, the implications between them and what their
consequences are. In particular, we show that the two most
common versions of the conjecture that an infinite NIP
field is separably closed, real closed or 'p-adic like'
are equivalent. This is joint work in progress with Sylvy
Anscombe.

**March 8th: FSWFT VII**, Caroline Terry

**March 3rd: Decidability and definability results
for theories of modules over a Bézout
domain**, Françoise
Point.

We prove a Feferman-Vaught type theorem
for a class of modules over a Bézout domain B. We analyse
the definable sets in terms on, one hand of the definable
sets in the classes of modules over the localizations of B
at maximal ideals, and on the other hand of the
constructible subsets of the maximal spectrum of B. We
discuss two applications: the ring of algebraic integers
and the ring of holomorphic functions over the complex
numbers.

This is joint work with Sonia L'Innocente
(Camerino). (No prerequisite on the theory of modules (or
on Bézout domains) will be assumed.)

**March 1st: FSWFT VI - Envelopes**, Silvain Rideau

**February 24th: Residue field domination in real closed valued fields**, Clifton Healy.

In an algebraically closed valued field, as shown by Haskell, Hrushovski, and Macpherson, the residue field and the value group control the rest of the structure: Assume $C$ is maximal and algebraically closed. Then $tp(L/Ck(L)\Gamma(L)$ will have a unique extension to $M\supseteq C$, as long as the residue field and value group of $M$ are independent from those of $L$ over $C$.
The same theorem holds when "algebraically closed" is replaced by "real closed". I will sketch a proof and discuss some possible generalizations.

**February 15th: FSWFT V - Coordinatizable structures are omega-categorical**, Michael Wan.

**February 17th: VC_n-dimension and a jump in the speed of a hereditary property**, Caroline Terry.

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends n to the number of distinct elements in n with underlying set 1,..., n. There are many wonderful results from combinatorics concerning what functions can occur as speeds of hereditary graph properties. More specifically, these results show there are discrete "jumps" in the possible speeds of hereditary graph properties. In this talk we use VC_n-dimension, a generalization of VC-dimension, to extend one of these results to the setting of arbitrary finite relational languages. In particular, we show that bounded VC_n-dimension characterizes the gap between the fastest and penultimate speeds.

**February 15th: FSWFT IV**, Pierre Simon

**February 10th: Further results on NSOP_1 theories**, Nick Ramsey.

**February 8th: FSWFT III**, Gregory Cherlin.

**February 3rd: FSWFT II**, Pierre Simon

**February 1st: Amalgamating globally valued fields**, Itaï Ben Yaacov.

Globally valued fields (GVFs) are logical structures (in continuous
logic) which attempt to capture global fields: fields equipped with a
family of valuations whose sum always vanishes (the "sum formula"). In
a joint project with E. Hrushovski, we seek to obtain a good model-
theoretic understanding of the class of these objects.

At the technical level, the main effort is to give a model companion,
i.e., to understand extensions of GVFs. There are two main approaches
here: a "global" approach, seeking to describe GVFs and GVF extensions
with global data (-> intersection theory), and a "local" approach,
working at each valuation individually, together with a single local-
global principle (or axiom) called "fullness".

I shall define a full GVF, and explain how one would use fullness in
order to show that a GVF is "rich". This strategy was applied recently
to show that a full GVF is an amalgamation base for GVF extensions (and
even slightly better than this), and it seems reasonable to hope that a
full GVF is existentially closed.

The argument itself is quite involved - within time constraints I may
attempt to give some of its highlights.

**January 27th: Groups with definable generics**, Silvain Rideau

A result of Pillay's state that a group definable in a differentially closed field can be embedded in an algebraic group. Similar theorems have also been proved in various structures of enriched fields: separably closed fields, fields with a generic automorphism, real closed fields... Moreover, the proofs of all these results use similar tools developed to study groups in stable, and then simple, theories.
The goal of my talk will be to explain those results and some of the tools involved in proving them. Then, I will explain how, in certain cases, we can get rid of the stability assumptions in those proofs to use them in valued fields.

**January 25th: FSWFT I**, Carol Wood.

**November 30th: A finitary analogue of the downward Lowenheim-Skolem property**, Abhisekh Sankaran.

We present a new logic based combinatorial property of finite structures, that can be regarded as a finitary analogue of the classical downward Lowenheim-Skolem property from model theory. This property, that we call the *Equivalent Bounded Substructure Property*, abbreviated EBSP, intuitively states that a large structure contains a small "logically similar" substructure. It turns out that this simply stated property is enjoyed by a variety of classes of interest in computer science: examples include classes defined using posets, such as words, trees (unordered, ordered or ranked) and nested words, and classes of graphs, such as graph classes of bounded tree-depth, those of bounded shrub-depth, m-partite cographs, and more generally, graph classes that are well-quasi-ordered under the (isomorphic) embedding relation. Further, EBSP remains preserved under various well-studied operations, such as complementation, transpose, the line-graph operation, disjoint union, join, series and parallel connects, and various products including the cartesian and tensor products. This enables constructing a wide spectrum of interesting classes that satisfy EBSP. We present applications of our investigations into EBSP, to complexity theory and graph minor theory: for the former, we show that many NP-complete problems (such as 3-colorability) become polynomial time solvable over several EBSP classes, and for the latter, we show that every finite graph has a small logically similar minor.

**November 16th: Measuring definable sets in nonarchimedean o-minimal fields and an application to the reals**, Jana Marikova.

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. We then discuss an application to Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field. This is joint work with M. Shiota and with E. Walsberg.

**November 9th: GVF VI**, Pierre Simon

**November 2nd: GVF V**, Pierre Simon

**October 26th: Definable groups in Models of Presburger Arithmetic and G^{00}**, Mariana Vicaria.

Our main motivation is to understand the quotient group $G/G^{00}$, where $G$ is a definable group in Presburger Arithmetic and G^{00} is the smallest type definable subgroup of $G$. Hrushovski, Peterzil and Pillay showed that, in any o-minimal expansion of the real field, for any definably compact connected group $G$, the quotient group $G/G^{00}$ (equipped with the logic topology) is a Lie group, whose dimension is equal to the dimension of $G$ as a definable set in an o-minimal theory.

Several tools that were used to proved this theorem can be recovered for Presburger Arithmetic, such as the cell decomposition theorem and a notion of dimension for definable sets (proved by R. Cluckers). These tools allow us to characterize the definable groups in Presburger Arithmetic. We show that any definable group in this theory has an abelian subgroup of finite index.

In Presburger Arithmetic, the analoguous notion to being definably compact is being bounded. Every such groups has the fsg property (finitely satisfiable generics). This allows us to characterize $G^{00}$ as the stabilizer of a generic type.

This work was developed during my master thesis at Universidad de los Andes, under the supervision of Prof. Alf Onshuus.

**October 19th: GVF IV**, Pierre Simon.

We will explain the classical examples: the algebraic closure of the rationals and the algebraic closure of fields of rational functions.

**October 12th: Towards a model theory of almost complex manifolds**, Michael Wan.

I will outline progress towards generalizing the model theory of compact complex manifolds to the setting of almost complex manifolds.
Zilber showed that any compact complex manifold equipped with its complex analytic topology has a nice model-theoretic structure theory, satisfying the axioms of a so-called "Zariski structure". His proof relies heavily on classical results in complex analytic geometry. I will present almost complex analogues of those results, including an "identity theorem" for almost complex maps, building on Peterzil and Starchenko's o-minimal theory of complex analysis. I will sketch some proofs, which make use of elementary pseudoholomorphic curve theory to bridge the gap between the complex and almost complex worlds.

**October 5th: GVF III**, Silvain Rideau.

We will describe the language in which globally valued fields are considered, give an axiomatization and then describe some of the open problems.

**September 28th: GVF II**, Silvain Rideau.

I will present the continous logic setting in which globally valued fields are considered. Then I will define globally valued fields.

**September 21st: The logical complexity of finitely generated commutative rings**, Tom Scanlon.

From Gödel's Incompleteness Theorems and Lagrange's Four Squares Theorem we know that the theory of the of ring Z of integers is very complicated. From old theorems of J. Robinson, R. Robinson and R. Rumely it follows that every infinite finitely generated commutative ring interprets the integers, and, hence have theories which are at least as complicated as those of Z. Conversely, from the Gödel's proof, one sees that every such ring is interpreted in the integers and therefore has a theory which is no more complicated than that of the integers. However, this observation about mutual interpretability does permit us to fully describe the class of definable sets in such rings. For example, if R is a finitely generated commutative ring interpreted in the integers, is it the case that every arithmetic subset of R is definable? For example, if we assume in addition that R is an integral domain, the answer is affirmative. However, the Feferman-Vaught theorem implies that the answer is negative for the ring Z * Z. More surprisingly, constructions of derivations on nonstandard models of arithmetic show that the answer is negative for the dual numbers Z[e]/(e^2). From the structural point of view, the real question is for which finitely generated commutative rings R is R parametrically bi-interpretable with Z? We show that the answer is positive if and only if there is some positive integer d which annihilates the nilradical of R and the punctured spectrum Spec*(R) := Spec(R)\Max(R) is nonempty and connected. (This is a report on joint work with Matthias Aschenbrenner, Anatole Khélif, and Eudes Naziazeno Galvão.)

**September 14th: GVF I**, Tom Scanlon.

We will start our investigation of globally valued fields with van den Dries's work on the Rumely principle.

**September 7th:Imagnaries in pseudo-p-adically closed fields**, Silvain Rideau.

During her PhD, Montenegro showed that pseudo-real-closed fields and pseudo-p-adically closed fields share many model theoretic properties. The main asymmetry is that she does not prove elimination of imaginaries for pseudo-p-adically closed fields. In this talk, I will more or less restore that symmetry by showing that provided we add geometric sorts for each of the valuations, we do get weak elimination of imaginaries. The proof is essentially a lifting of the classic proof for pseudo-algebraically closed fields to the unstable setting.

This work is joint with Samaria Montenegro.

**August 31st: Kim-Independence**, Nick Ramsey.

The success of the theory of simple theories is built upon a number of theorems showing that forking has many desirable properties in a simple theory. Most of these properties, moreover, *characterize* simplicity (e.g., symmetry, transitivity, Kim's lemma), which means that a naïve generalization of these theorems to a broader class of theories is impossible. We will describe some recent work, now joint with Itay Kaplan, which shows that a simplicity-style structure theory exists for NSOP1 theories after replacing forking and dividing with their generic analogues, Kim-forking and Kim-dividing. This yields many new characterizations of NSOP1 theories and generalizes the known independence theory in several key examples.

**August 24th: Groups definable in PRC fields**, Pierre Simon.

We show that solvable groups definable in PRC fields are virtually isogeneous with a (multi) semi-algebraic group. To prove this, we adapt the work of Hrushovski and Pillay on local and pseudofinite fields. Additional ingredients include certain variations on Hrushovski’s stabilizer theorem and a study of definably amenable NTP2 groups. (joint work with Samaria Montenegro and Alf Onshuus.)