Past Logic Seminar

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.

Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham's famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès. The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.

Toward a characterization of modularity using shatter functions, we show that an o-minimal expansion of the  real ordered additive group $(\mathbb{R}; 0, +,<)$ does not define restricted multiplication if and only if the shatter function of every definable family is asymptotic to a polynomial. Our result implies that vc-density can only take integer values in $(\mathbb{R}; 0, +,<)$ confirming a special case of a conjecture by Chernikov. (Joint with Abdul Basit.)

We will define a notion of a semi-retraction between two first-order structures introduced by Scow. We show how a semi-retraction encodes Ramsey problems of finitely-generated substructes of one structure into the other under the most general conditions. We will compare semi-retractions to a category-theoretic notion of pre-adjunction recently popularized by Masulovic. We will accompany the results with examples and questions. This is a joint work with Lynn Scow.

The equivalence of NSOP${}_1$ and NSOP${}_3$, two model-theoretic complexity properties, remains open, and both the classes NSOP${}_1$ and NSOP${}_3$ are more complex than even the simple unstable theories. And yet, it turns out that classical geometric stability theory, in particular the group configuration theorem of Hrushovski (1992), is capable of controlling classification theory on either side of the NSOP${}_1$-SOP${}_3$ dichotomy, via the expansion of stable theories by generic predicates and equivalence relations. This allows us to construct new examples of strictly NSOP${}_1$ theories. We introduce generic expansions corresponding, though universal axioms, to definable relations in the underlying theory, and discuss the existence of model companions for some of these expansions. In the case where the defining relation in the underlying theory $T$ is a ternary relation $R(x, y, z)$ coming from a surface in 3-space, we give a surprising application of the group configuration theorem to classifying the corresponding generic expansion $T^R$. Namely, when $T$ is weakly minimal and eliminates the quantifier $\exists^{\infty}$, $T^R$ is strictly NSOP${}_4$ and TP${}_2$ exactly when $R$ comes from the graph of a type-definable group operation; otherwise, depending on whether the expansion is by a generic predicate or a generic equivalence relation, it is simple or NSOP${}_1$.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

We will construct several algebras of functions definable in R_{an,\exp} which are stable under parametric integration. 

Given one such algebra A, we will study the parametric Mellin and Fourier transforms of the functions in A. These are complex-valued oscillatory functions, thus we leave the realm of o-minimality. However, we will show that some of the geometric tameness of the functions in A passes on to their integral transforms.

I will explain how the model-theoretic algebraic closure in Zilber’s pseudo-exponential field can be described in terms of the self-sufficient closure. I will sketch a proof and show how the Mordell-Lang conjecture for algebraic tori comes into play. If time permits, I’ll also talk about the characterisation of strongly minimal sets and their geometries. This is joint work (still in progress) with Jonathan Kirby.

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

(1) "linear Zarankiewicz's bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in M

(2) M does not define an infinite field.

We prove that the following are equivalent:

(1') linear Zarankiewicz bounds hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in M

(2') M does not define a full field (that is, one whose domain is the whole universe of M).

This is joint work (in progress) with Aris Papadopoulos.

I will discuss the following statement, a definable version of the (p,q) theorem of Jiří Matoušek from combinatorics, conjectured by Chernikov and Simon:

Suppose that T is NIP and that phi(x,b) does not fork over a model M. Then there is some formula psi(y) in tp(b/M) such that the partial type {phi(x,b’) : psi(b’)} is consistent.

Given a cover U of a family of smooth complex algebraic varieties, we associate with it a class C of structures locally definable in an o-minimal expansion of the reals, containing the cover U.  We prove that the class is ℵ0-homogeneous over submodels and stable. It follows that C is categorical in cardinality ℵ1. In the one-dimensional case we prove that a slight modification of C is an abstract elementary class categorical in all uncountable cardinals.
 

We call affine logic the fragment of continuous logic in which the connectives are limited to linear combinations and the constants (but quantification is allowed, in the usual continuous form). This fragment has been introduced and studied by S.M. Bagheri, the first to observe that this is the appropriate framework to consider convex combinations of metric structures and, more generally, ultrameans, i.e., ultraproducts in which the ultrafilter is replaced by a finitely additive probability measure. Bagheri has shown that many fundamental results of continuous logic hold in affine logic in an appropriate form, including Łoś's theorem, the compactness theorem, and the Keisler--Shelah isomorphism theorem.

In affine logic, type spaces are compact convex sets. In this talk I will report on an ongoing work with I. Ben Yaacov and T. Tsankov, in which we initiate the study of extremal models in affine logic, i.e., those that only realize extreme types.

From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities. This includes new joint work with Sylvy Anscombe and Philip Dittmann.

A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q $(where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and $a$ vary: for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu q^d$ for some $(\mu,d) \in E$.

This led in work of Elwes, Steinhorn and myself to the notion of 'asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of  'measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem, or certain pseudofinite difference fields).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite 'generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.

The Skolem Problem asks to decide whether a linearly recurrent sequence (LRS) over the rationals has a zero term.  It is sometimes considered as the halting problem for linear loops.   In this talk we will give an overview of two current approaches to establishing decidability of this problem.  First, we observe that the Skolem Problem for LRS with simple characteristic roots is decidable subject to the $p$-adic Schanuel conjecture and the exponential-local-global principle.  Next, we define a set $S$ of positive integers such that (i) $S$ has positive lower density and (ii) The Skolem Problem is decidable relative to $S$, i.e., one can effectively determine the set of all zeros of a given LRS that lie in $S$.

The talk is based on joint work with Y. Bilu, F. Luca, J. Ouaknine, D. Pursar, and J. Nieuwveld.  

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

In this talk, we discuss existentially closed measure preserving actions of countable groups.  A classical result of Berenstein and Henson shows that the model companion for this class exists for the group of integers and their analysis readily extends to cover all amenable groups.  Outside of the class of amenable groups, relatively little was known until recently, when Berenstein, Henson, and Ibarlucía proved the existence of the model companion for the case of finitely generated free groups.  Their proof relies on techniques from stability theory and is particular to the case of free groups.  In this talk, we will discuss the existence of model companions for measure preserving actions for the much larger class of universally free groups (also known as fully residually free groups), that is, groups which model the universal theory of the free group.  We also give concrete axioms for the subclass of elementarily free groups, that is, those groups with the same first-order theory as the free group.  Our techniques are ergodic-theoretic and rely on the notion of a definable cocycle.  This talk represents ongoing work with Brandon Seward and Robin Tucker-Drob.

MSO can be used not only to accept/reject words, but also to transform words into other words, e.g. the doubling function w $\mapsto$ ww. The traditional model for this is called MSO transductions; the idea is that each position of the output word is interpreted in some position of the input word, and MSO is used to define the order on output positions and their labels. I will explain that an extension, where output positions are interpreted using $k$-tuples of input positions, is (a) is also well behaved; and (b) this is surprising.

We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal theory of any $L$-model of $T$.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.

Previous joint work with Caroline Terry had identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups, pioneered by Ben Green around 2003. 
Higher-order arithmetic regularity decompositions, based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, are an essential part of today's arithmetic combinatorics toolkit.
In this talk, I will describe recent joint work with Caroline Terry in which we define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of finite elementary abelian groups. If time permits, I will briefly outline some analogous results we obtain in the context of hypergraph regularity decompositions.

Several natural measures of complexity can be attached to an
existentially definable ("diophantine") subset of a field. One of these
is the minimal number of existential quantifiers required to define it,
while others are of a more geometric nature. I shall define these
measures and discuss interesting interactions and behaviours, some of
which depend on properties of the field (e.g. imperfection and
ampleness). We shall see for instance that the set of n-tuples of field
elements consisting of n squares is usually definable with a single
quantifier, but not always. I will also discuss connections with
Hilbert's 10th Problem and a number of open questions.
This is joint work with Nicolas Daans and Arno Fehm.

In this talk, we describe some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics.

The motivations for this work come from Computer Science, but there may be something of interest for model theorists and other logicians.

The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings.

One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way:

- the Ehrenfeucht-Fraisse game

- the quantifier rank fragments of first-order logic

- the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction to the existential-positive part, and (iii) the extension with counting quantifiers

- the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez).

Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth.

Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters.

This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers.

Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages.

Examples include semantic characterisation and preservation theorems, Lovasz-type results on  isomorphisms, and classification of constraint satisfaction problems.

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.   

(This is Part II of a two-part talk.)

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

This talk will be about the stable boundary seen from different recent points of view.

Sela proved in 2006 that the (non abelian) free groups are stable. This implies the existence of a well-behaved forking independence relation, and raises the natural question of giving an algebraic description in the free group of this model-theoretic notion. In a joint work with Rizos Sklinos we give such a description (in a standard fg model F, over any set A of parameters) in terms of the JSJ decomposition of F over A, a geometric group theoretic tool giving a group presentation of F in terms of a graph of groups which encodes much information about its automorphism group relative to A. The main result states that two tuples of elements of F are forking independent over A if and only if they live in essentially disjoint parts of such a JSJ decomposition.

I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), to uniformity of honest definitions and to the construction of compressible models in the context of (local) NIP. All notions will be defined during the talk.
Joint with Martin Bays and Pierre Simon.

(Joint with Y. Halevi and A. Hasson.) We consider two kinds of expansions of a valued field $K$:

(1) A $T$-convex expansion of real closed field, for $T$ a polynomially bounded o-minimal expansion of $K$.

(2) A $P$-minimal field $K$ in which definable functions are PW differentiable.

We prove that any interpretable infinite field $F$ in $K$ is definably isomorphic to a finite extension of either $K$ or, in case (1), its residue field $k$. The method we use bypasses general elimination of imaginaries and is based on analysis of one dimensional quotients of the form $I=K/E$ inside $F$ and their connection to one of 4 possible sorts: $K$, $k$ (in case (1)), the value group, or the quotient of $K$ by its valuation ring. The last two cases turn out to be impossible and in the first two cases we use local differentiability to embed $F$ into the matrix ring over $K$ (or $k$).

The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a constraint for the finite sets witnessing ''many'' intersections with V, namely a condition called in general position, which plays a key role in forcing the group to be abelian.  In this talk, I will present a result which shows that in the case of the graph of complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:

1. The graph of G has ''many'' intersections with finite sets in weak general position;

2. G is nilpotent;

3. The ultrapower of G has a pseudofinite coarse approixmate subgroup in weak general position.

Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups.

This is joint work with Martin Bays and Jan Dobrowolski.

We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory.  Given structures $A$ and $B$, a semi-retraction is a pair of  quantifier-free type respecting maps $f: A \rightarrow B$ and $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is quantifier-free type preserving, i.e. an embedding.  In the case that $A$ and $B$ are locally finite ordered structures, if $A$ is a semi-retraction of $B$ and the age of $B$ has the Ramsey property, then the age of $A$ has the Ramsey property.

There are many open conjectures about the algebraic behaviour of transcendental functions in arithmetic geometry, one of which is the Existential Closedness problem. In this talk I will review recent developments made on this question: the cases where we have unconditional existence of solutions, the conditional existence of generic solutions (depending on the conjecture of periods and Zilber-Pink), and even a few cases of unconditional existence of generic solutions. Many of the results I will mention are joint work with (different subsets of) Vahagn Aslanyan, Jonathan Kibry, Sebastián Herrero, and Roy Zhao. 

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.

Zilber's Restricted Trichotomy Conjecture predicts that every sufficiently rich strongly minimal structure which can be interpreted from an algebraically closed field K, must itself interpret K. Progress toward this conjecture began in 1993 with the work of Rabinovich, and recently Hasson and Sustretov gave a full proof for structures with universe of dimension 1. In this talk I will discuss a partial result in characteristic zero for universes of dimension greater than 1: namely, the conjecture holds in this case under certain geometric restrictions on definable sets. Time permitting, I will discuss how this result implies the full conjecture for expansions of abelian varieties.

In 2010, Hrushovski--Loeser studied the homotopy type of the Berkovich analytification of a quasi-projective variety over a valued field. In this talk, we explore the extent to which some of their results might hold in a relative setting. More precisely, given a morphism of quasi-projective varieties over a valued field, we ask if we might construct deformation retractions of the analytifications of the source and target which are compatible with the analytification of the morphism and whose images are finite simplicial complexes. 

The Zilber-Pink conjecture predicts how large the intersection of a d-dimensional subvariety of an abelian variety/algebraic torus/Shimura variety/... with the union of special subvarieties of codimension > d can be (where the definition of "special" depends on the setting). In joint work with Fabrizio Barroero, we have reduced this conjecture for complex abelian varieties to the same conjecture for abelian varieties defined over the algebraic numbers. In work in progress, we introduce the notion of a distinguished category, which contains both connected commutative algebraic groups and connected mixed Shimura varieties. In any distinguished category, special subvarieties can be defined and a Zilber-Pink statement can be formulated. We show that any distinguished category satisfies the defect condition, introduced as a useful technical tool by Habegger and Pila. Under an additional assumption, which makes the category "very distinguished", we show furthermore that the Zilber-Pink statement in general follows from the case where the subvariety is defined over the algebraic closure of the field of definition of the distinguished variety. The proof closely follows our proof in the case of abelian varieties and leads also to unconditional results in the moduli space of principally polarized abelian surfaces as well as in fibered powers of the Legendre family of elliptic curves.

TBA

This is a joint work with C.Daw in progress. We study the L_{omega_1,omega}-theory of the modular functions j_n: H -> Y(n). In other words, H is seen here as the universal cover in the class of modular curves. The setting is different from one considered before by Adam Harris and Chris Daw: GL(2,Q) is given here as the sort without naming its individual elements. As usual in the study of 'pseudo-analytic cover structures', the statement of categoricity is equivalent to certain arithmetic conditions, the most challenging of which is to determine the Galois action on CM-points. This turns out to be equivalent to determining the Galois action on SL(2,\hat{Z})/(-1), that is a subgroup of

Out SL(2,\hat{Z})/(-1)   induced by the action of  Gal_Q. We find by direct matrix calculations a subgroup Out_* of the outer automorphisms group which contains the image of Gal_Q. Moreover, we prove that Out_* is the image of Drinfeld's group GT (Grothendieck-Teichmuller group) under a natural homomorphism.

It is a reasonable to conjecture that Out_* is equal to the image of Gal_Q, which would imply the categoricity statement. It follows from the above that our conjecture is a consequence of Drinfeld's conjecture which states that GT is isomorphic to Gal_Q.  

Recently in a joint work with J. Dobrowolski and N. Ramsey it is shown that in any NSOP1 theory with existence,
Kim-independence satisfies all the basic axioms over sets (except base monotonicity) that hold in simple theories with forking-independence. This is an extension of the earlier work by I. Kaplan and N. Ramsey that such hold over models in any NSOP1 theory. All simple theories; unbounded PAC fields; vector spaces over ACF with bilinear maps; the model companion of the empty theory in any language are typical NSOP1 examples.

   An important issue now is to know the existence of canonical bases. In stable and simple theories well-behaving notion of canonical bases for types over models exists, which is used in almost all the advanced studies. But there are a couple of crucial obstacles in finding canonical bases in NSOP1 theories. In this talk I will report a partial success/limit of the project. Namely, a type of a certain Morley sequence over a model has the weak canonical base. In my talk I will try to explain all the related notions.

 I will report on my recent work on dimension, definable groups, and definable fields in vector spaces over algebraically closed [real closed] fields equipped with a non-degenerate alternating bilinear form or a non-degenerate [positive-definite] symmetric bilinear form. After a brief overview of the background, I will discuss a notion of dimension and some other ingredients of the proof of the main result, which states that, in the above context, every definable group is (algebraic-by-abelian)-by-algebraic [(semialgebraic-by-abelian)-by-semialgebraic]. It follows from this result that every definable field is definable in the field of scalars, hence either finite or definably isomorphic to it [finite or algebraically closed or real closed].
 

This is joint with Gabe Conant. We give a structure theorem for finite subsets A of arbitrary groups G such that A has "small tripling" and "bounded VC dimension". Roughly, A will be a union of a bounded number of translates of a coset nilprogession of bounded rank and step (up to a small error).

I will present a conjectural picture of what a classification theory for NIP could look like, in the spirit of Shelah's classification theory for stable structures. Though most of it is speculative, there are some encouraging initial results about the lower levels of the classification, in particular concerning structures which, in some strong sense, do not contain trees.

The classification of NIP fields is a major open problem in model theory.  This talk will be an overview of an ongoing attempt to classify NIP fields of finite dp-rank.  Let $K$ be an NIP field that is neither finite nor separably closed.  Conjecturally, $K$ admits exactly one definable, valuation-type field topology (V-topology).  By work of Anscombe, Halevi, Hasson, Jahnke, and others, this conjecture implies a full classification of NIP fields.  We will sketch how this technique was used to classify fields of dp-rank 1, and what goes wrong in higher ranks.  At present, there are two main results generalizing the rank 1 case.  First, if $K$ is an NIP field of positive characteristic (and any rank), then $K$ admits at most one definable V-topology.  Second, if $K$ is an unstable NIP field of finite dp-rank (and any characteristic), then $K$ admits at least one definable V-topology.  These statements combine to yield the classification of positive characteristic fields of finite dp-rank. In characteristic 0, things go awry in a surprising way, and it becomes necessary to study a new class of "finite rank" field topologies, generalizing V-topologies.  The talk will include background information on V-topologies, NIP fields, and dp-rank.

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 an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

 This is a longstanding problem. We will present candidate axioms that we believe form a complete axiom system satisfied by $F_p^{alg}((t))$.  A currently "unfinished" proof will be discussed.   This work is joint with Françoise Delon (Paris) and Arno Fehm (Dresden).