Abstracts
Daniel Bertrand
Counterexamples with semi-abelian varities. (Slides)Extending conjectures on abelian schemes to semi-abelian schemes can prove hazardous. We will describe several counterexamples which illustrate this principle. They concern :
- the functional analogue of the Lindemann-Weierstrass theorem, where semi-constant semi-abelian schemes must be handled with care (joint work with A. Pillay);
- the analogue of Lehmer’s question on heights, to which Ribet points on semi-abelian surfaces over CM elliptic curves provide a negative answer;
- a recent work on similarly defined Ribet sections, giving a counterexample to the relative version of the Manin-Mumford conjecture on unlikely intersections.
Jean-Benoît Bost
Algebraization of $D$-group schemes and Diophantine geometry. (Slides)I will discuss some constructions involving abelian schemes over curves, their universal vector extensions, and their extensions in the categories of algebraic, analytic, and formal $D$-group schemes. These constructions are directly related with classical conjectures in transcendence theory and Diophantine geometry, and have diverse points of contact with model theory.
Gareth Boxall
Weak one-basedness. (Slides)Berenstein and Vassiliev have defined what it means for an independence relation to be weakly one-based and investigated this notion especially in the setting of geometric theories. I shall discuss their work and some additions made to it in joint work with David Bradley-Williams, Charlotte Kestner, Alexandra Omar Aziz and Davide Penazzi. I shall focus particularly on the finite thorn-rank setting.
Emmanuel Breuillard
Some applications of diophantine geometry and model theory to group theory. (Slides)I will discuss a set of recent results in group theory, whose proof makes key use of a variety of ideas from diophantine geometry and model theory. I will focus on some recent advances regarding geometric and spectral properties of the Cayley graphs of finite simple groups $G$ of Lie type of fixed given rank around the folklore conjecture that they are all uniform expanders. Diophantine geometry enters the game through the proof of a Bogomolov type result for the representation variety of the free group in $GL_n$ (height gap theorem) needed to produce pairs of elements in $G$ with few relations of small length (uniform Tits alternative). Model theory plays a role in uniformity issues and via the related recent work of Hrushovski on approximate groups, a quantitative version of which (joint with Green and Tao) is also needed in the proof.
Lou van den Dries
On the model theory of the differential field of transseries. (Slides)This is joint work with Matthias Aschenbrenner and Joris van der Hoeven. We consider the valued differential field of transseries as a worthy object of model theoretic analysis. Does it have a viable model theory? I will discuss some progress we have made in the last three years towards a positive answer, in particular in coming to grips with immediate extensions of the relevant valued differential fields, and what seems to be the appropriate notion of differential-henselian in this setting.
Deirdre Haskell
Unexpected imaginaries in theories of valued fields with analytic structure. (Slides)Quantifier elimination results for theories of valued fields with analytic structure lead one to believe that these theories might also eliminate imaginaries with the same sorts that suffice to eliminate imaginaries for the algebraic structure. I will exhibit an example of an imaginary which is not coded in the geometric sorts. I will discuss the proof that it is not coded in detail for algebraically closed valued fields, and, more briefly, the changes needed in the argument for real closed and p-adically closed fields. This is joint work of myself with Ehud Hrushovski and Dugald Macpherson.
Moshe Kamensky
Categorical aspects of internality. (Slides)Internality is a model theoretic condition that guarantees (and, indeed, characterises) the existence of definable Galois theory for definable sets in a first order theory. I will explain that this condition can be restated in terms of the categorical properties of the category of definable sets. One result of this is that the notion (and the resulting Galois theory) makes sense in a wider categorical context.
Ehud Hrushovski
Simplicity and pseudofiniteness (Slides)Pseudo-finite theories present surprising structural similarities with supersimple theories. At the base lies a dimension theory, and within each dimension an ideal of definable sets with a form of $3$-amalgamation. There are connections with finite combinatorics and measurable dynamics.
This talk will be an invitation to this area, in part through a number of open problems.
Byunghan Kim
Amalgamation functors and homology groups in model theory. (Slides)This is a joint work with John Goodrick and Alexei Kolesnikov. We present definitions of homology groups $H_n$, $n\geq 0$, associated to a family of amalgamation functors. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group $H_2$ for strong types in stable theories and show that any profinite abelian group can occur as the group $H_2$ in the model-theoretic context.
Jonathan Kirby
The first-order theory of pseudoexponentiation. (Slides)Zilber's axiomatization of pseudoexponentiation is in the logic $L_{\omega_1,\omega}(Q)$, which is necessary for a categoricity result. The first-order theory is more difficult to understand because of the presence of arithmetic. However, assuming the Conjecture of Intersections of Tori with subvarieties (CIT), we are able to separate out the effects of arithmetic and give an axiomatization of the complete first-order theory. This is joint work with Boris Zilber.
Angus Macintyre
Model Theory of Adeles (with J.Derakhshan). (Slides)We consider the model theory of the various rings of adeles over number fields K, aiming for as much uniformity as possible in analysis of definitions, as well as analyzing the topology and measure of definable sets. We pay attention to proper extensions of the ring language, and to issues of stable embedding.
Alice Medvedev
QACFA. (Slides)The model theory of fields with one automorphism, or, equivalently, with a $\mathbb{Z}$-action, has been worked out in great detail by Chatzidakis, Hrushovski, and Peterzil. Some of their results (companionability, quantifier elimination, simplicity) extend immediately to the theory of fields with a $\mathbb{Q}$-action. Supersimplicity fails on the fixed fields, but not everywhere.
Ludomir Newelski
Strongly generic sets. (Slides)We consider a group $G$ definable in a model $M$ of a theory $T$. An algebra of subsets of $G$ is called a $G$-algebra if it is closed under left translation by elements of $G$. We call a subset $U$ of $G$ strongly generic if every non-empty element of the $G$-algebra generated by $U$ is (left) generic.
Theorem: Assume $T$ is stable and $U\subseteq G$ is definable. Then $U$ is strongly generic iff $U$ is a union of cosets of a definable subgroup of finite index in $G$. In particular, definable strongly generic subsets of $G$ form a $G$-algebra of sets.
In general let $Def_{ext,G}(M)$ be the $G$-algebra of externally definable subsets of $G$ and $SGen_{ext,G}(M)$ be the family of strongly generic externally definable subsets of $G$. $SGen_{ext,G}(M)$ need not be an algebra of sets, however it contains subsets that are $G$-algebras of sets. Maximal such subsets are called image algebras. Theorem:
- Image algebras are all $G$-isomorphic (that is, isomorphic via a function respecting the action of $G$) and $SGen_{ext,G}(M)$ is a union of them.
- If $\cal A$ is an image algebra, then there is a $G$-epimorphism $Def_{ext,G}(M)\to{\cal A}$.
Item (2) in this theorem may be regarded as a general counterpart of amenability. It points also that there are quite many strongly generic externally definable sets.
Let $S_{ext,G}(M)$ be the Stone space of $Def_{ext,G}(M)$. $S_{ext,G}(M)$ is a $G$-flow, it carries a semigroup structure isomorphic to its Ellis semigroup. Strongly generic sets are related to topological dynamics. For instance, if $\cal A$ is an image algebra, then the Stone space $S({\cal A})$ is isomorphic to any minimal flow in $S_{ext,G}(M)$.
If there are boundedly many strongly generic externally definable subsets of $G$, then the ideal subgroups of $S_{ext,G}(M)$ are absolute (i.e. do not depend on the choice of $M$).
Topological dynamics may be used to explain phenomena related to generic subsets of $G$ outside the stable context, in particular in the o-minimal setting.
Margarita Otero
Cartan subgroups of groups definable in o-minimal structures. (Slides)A subgroup $Q$ of a group $G$ is a Cartan subgroup of $G$ if it is maximal nilpotent and each normal subgroup of finite index of $Q$ has finite index in its normalizer in $G$. We will show various properties of Cartan subgroups of groups definable in o-minimal structures. In particular that there are finitely many conjugacy classes of Cartan subgroups. (Joint work with Elías Baro and Eric Jaligot.)
Chloé Perin
Homogeneity of the free group. (Slides)We show with R. Sklinos that the free groups are all homogeneous, that is, that any two tuples of elements which have the same first-order properties can be sent one onto the other by an automorphism of the group. On the other hand, we showed that this is not true of most fundamental groups of surfaces. Some of the techniques used in the proof are geometric in nature, and rely on tools developed by Sela in his solution to the Tarski problem.
Kobi Peterzil
Definable quotients of locally definable groups. (Slides)(Joint work with P. Eleftheriou) A locally definable group $G$ is given as a countable union of definable sets, with the group operation definable on each definable subset of $G\times G$. The quotient of $G$ by a locally definable subgroup $G_0$ is called definable if there is locally definable, surjective homomorphism, with kernel $G_0$, from $G$ onto a definable group.
In the o-minimal setting there are various natural constructions which give rise to such definable quotients. For example, in an ordered vector space (o.v.s) $V$ every definable definably compact group is obtained as the quotient of an open, locally definable subset of $V^n$ by a discrete subgroup (a theorem of Eleftheriou and Starchenko).
It turns out that this result can be generalized to arbitrary expansions of ordered groups:
Theorem: Every definably compact group $H$ is a definable quotient of a locally definable group $G$ by a discrete subgroup, where the group $G$ itself is an extension of a locally definable group in an o-minimal expansion of a real closed field, by an open locally definable subgroup of $V^n$, for some o.v.s $V$.
The theorem can be used to analyze definable groups in o-minimal expansions of ordered groups using results in expansions of real closed fields and in ordered vector spaces (e.g Compact Domination for such groups follows).
Along the way, some interesting open questions arise about locally definable groups in o-minimal structures: For example, assume $G$ is a locally definable abelian group, which is generated by a definably connected subset. Show that $G$ is divisible. These and other questions turn out to be related to the existence of $G^{00}$, the minimal type definable subgroup of bounded index, which is related to the existence of definable generic subsets of $G$.
Jonathan Pila
Some unlikely intersections beyond Andre-Oort. (Slides)I will report on further applications of o-minimality to diophantine geometry, in particular on some recent work joint with Philipp Habegger. We obtain some special (and partial) cases of the Zilber-Pink conjecture relating to unlikely intersections in products of the $j$-line.
Anand Pillay
Connected components, universal covers, and the Lascar group. (Slides)I discuss recent examples of definable groups $G$ where $G^{00} \neq G^{000}$. Also connections to the Lascar group, and questions around the Borel complexity of the corresponding equivalence relations. Joint work with A. Conversano and K. Krupinski.
Damian Rossler
About the Mordell-Lang conjecture in positive characteristic. (Slides)The Mordell-Lang conjecture in positive characteristic is the statement that in an abelian variety $A$ over an algebraically closed field of char. $p>0$, the prime-to-$p$ divisible envelope of a finitely generated subgroup cannot meet a subvariety $X$ in a dense set, unless $X$ is the translate of an abelian subvariety or if $A$ has a subquotient, which is defined over a finite field.
The Manin-Mumford conjecture in positive characteristic is defined by the same sentence, with "prime-to-$p$ divisible envelope of a finitely generated subgroup" replaced by "group of torsion point of $A$". The Mordell-Lang conjecture was first proved by Hrushovski in 1996, using deep results of Model-theory, in particular his work with Zilber on Zariski geometries. Following a suggestion of A. Pillay, we shall give an algebraic proof of the fact that the Manin-Mumford conjecture actually implies the Mordell-Lang conjecture.
Ahuva Shkop
Finding something real in Zilber’s Field. (Slides)In 2004, Zilber constructed a class of exponential fields, known as pseudoexponential fields, and proved that there is exactly one pseudoexponential field in every uncountable cardinality up to isomorphism. He conjectured that the pseudoexponential field of size continuum, $K$, is isomorphic to the classic complex exponential field. Since the complex exponential field contains the real exponential field, one consequence of this conjecture is the existence of a real closed exponential subfield of $K$. In this talk, I will sketch the proof of the existence of uncountably many non-isomorphic countable real closed exponential subfields of $K$ and discuss some of their properties.
Pierre Simon
Distal NIP theories. (Slides)NIP theories are often thought of as being built, in some vague sense, of stable and ordered pieces. One side of the picture, the stable part, is well understood. We are interested in the other extreme. I will present a notion of pure instability called "distality" and give equivalent definitions. The class of distal theories includes for example o-minimal theories and the p-adics. I will also briefly explain how in non-distal theories one can pick out the stable part of types.
Michael Singer
Jordan-Hoelder Theorem for Differential Algebraic Groups and Factorization of Partial Differential Operators. (Slides)It is well known that an ordinary differential operator factors as a product of irreducible operators and that in any such factorization the number of such factors is unique. This uniqueness no longer holds for partial differential operators. Nonetheless, solutions of linear partial differential operators are the most basic examples of differential algebraic groups (DAGs) and a Jordan-Hoelder type theorem for differential groups gives a kind of factorization into irreducibles where in any such factorization the number of "factors" are unique and, after a possible permutation are equivalent in a suitable sense. The Jordan-Hoelder Theorem for DAG’s decomposes a DAG into a descending chain of DAG’s with successive quotients being almost simple and unique up to isogeny. I will discuss this result, giving many examples and compare it to similar results coming from model theory. In addition I will give a classification, in the case of one derivation, of almost simple DAGs based on results of Altinel and Cherlin.
Charles Steinhorn
On perspectives and trends in model theory through one (aging) individual’s looking glass. (Slides)We take a (necessarily) personal historical look at model theory, reflecting on important themes past, present, and, from this speaker’s perspective, future. This talk comes with the automatic disclaimer that any errors of fact, as well as the views expressed, are those of the speaker alone.
Margaret Thomas
Integer-valued functions and rational points on definable sets. (Slides)We consider analytic functions which are definable in the real exponential field and which take integer values at natural number inputs. The analysis of such functions can be connected to a conjecture of Wilkie concerning the density of rational points on sets definable in this structure. We shall review some results in this latter area and demonstrate how a proven case of Wilkie’s conjecture, for one-dimensional sets, contributes to our understanding of the behaviour of integer-valued definable functions.
Frank Wagner
Ample questions and simple answers. (Slides)n-ampleness is a hierarchy for the geometyric complexity of forking independence defined by Pillay (and slightly modified by Evans); the first two levels correspond to the better-known one-basedness and CM-triviality. We shall abord various questions around ampleness in the context of simple theories (notably preservation under analyses), and we shall discuss related notions (a relative version, the canonical base property, a theory of levels).
Alex Wilkie
Analytic continuation for algebro-logarithmic germs. (Slides)Let $O$ denote the ring of germs (at 0) of complex analytic functions. Let $L$ denote the smallest subring of $O$ containing all (germs at 0 of) complex polynomials and which is closed under taking logarithms (i.e. if $f$ is in $O$ and $f(0)$ is nonzero, then any determination of $\log(f)$ is also in $O$), taking real powers and, further, is algebraically closed in $O$. I show that any $f$ in $O$ can be analytically continued along all but finitely rays emanating from $0$. The proof is an application of the o-minimality of $\mathbb{R}_\text{an,exp}$.
Boris Zilber
Curve and its Jacobian in the light of model theory (Slides)I will recall a recent theorem by Bogomolov, Korotaev and Tschinkel about Jacobians of curves over finite fields and explain how this result in a more general form can be derived from the known theorem of E.Rabinovich of 1986. Moreover, the model theoretic proof answers a question posed by the three authors. I am also going to discuss the open problem of obtaining and strengthening Rabinovich's theorem using later notions and ideas of Zariski geometries.