## MSRI - Model Theory, Arithmetic Geometry and Number Theory

Seminar programme

Organisers : Z. Chatzidakis, T. Scanlon.

Unless otherwise indicated, all talks take place in Simons auditorium at MSRI, on Tuesdays at **1:30pm and 3:30pm**. **(Please note the change of schedule)**

### Calendar

- 19 May 2014

1:30 - 3pm: Philip Scowcroft (Wesleyan)*Existentially closed prime-model extensions of Abelian lattice-ordered groups*

3:30 - 5pm: Artem Chernikov (Paris 7)*Some results and conjectures about NTP2*

## Abstracts

**Next seminar**

19 May 2014.

1:30 - 3pm: Philip Scowcroft (Wesleyan) *Existentially closed prime-model extensions of Abelian lattice-ordered groups*.

An Abelian lattice-ordered group--``l-group,'' in what follows--is said to be existentially closed (ec) just in case finite systems of l-group equations and inequations over G are solvable in G if solvable in some l-group extending G. An ec prime-model extension of the l-group G is an ec l-group that extends G and embeds over G into every ec l-group extending G. So an l-group G is to an ec prime-model extension of G as a field F is to an algebraic closure of F. This talk will consider the extent to which l-groups have ec prime-model extensions.

3:30 - 5pm: Artem Chernikov (Paris 7)* Some results and conjectures about NTP2*

NTP2 is a large class of first-order theories introduced by Shelah and generalizing both simple and NIP theories. It provides a natural setting for the theory of forking beyond simplicity, and also contains new important algebraic examples: ultraproducts of p-adics, bounded pseudo real closed fields and certain valued difference fields. I will discuss some open questions about the pure theory of NTP2 theories concentrating on burden, Lascar strong types, simple types, resilience and uniform local character of forking over finite sets (ULCFS).

#### Past

**5 May 2014**

1:30 - 3pm: Mei-Chu Chang (UC Riverside), *Multiplicative order mod p : problems and some results*

The questions considered belong to the general theme of 'unlikely intersections' but in this talk we focus on various instances in the mod p setting, where p is a large prime.

We will discuss in particular mod p versions of Lang's conjecture on torsion points on varieties and related questions. Generally speaking, in absence of a direct finite field approach, our basic method consists in lifting the problem to a number field and then rely on finiteness results in this setting. The conclusions for individual p are rather weak but much stronger for 'almost all p'.

*Pseudofinite dimension and generalisations of asymptotic classes of structures*

I will describe joint work with Dario Garcia and Charles Steinhorn on the Hrushovski-Wagner notion of pseudofinite dimension, for, e.g., definable sets in an ultraproduct of finite structures. The focus will be on rather general conditions which ensure that an ultraproduct is (super)simple, or stable, and on the relation between dimension-drop and forking, and on examples.

I will also discuss preliminary work with William Anscombe, Charles Steinhorn, and Daniel Wood on rather general notions of `asymptotic class' of finite structures'. These concern contexts where there is a strong uniformity on the cardinalities of definable sets in a class of finite structures, and take into account multi-sorted behaviour, allow infinite rank, and do not imply simplicity of the ultraproduct theory.

**29 April 2014**

1:30 - 3pm: Barry Mazur (Harvard U.) *Questions about isogenies, automorphisms, and bounds*

Let X be a modular curve - for example, the λ-line - and let α:X → X be an automorphism, say, that doesn't preserve the cusps. How does α affect the general Hecke orbit structure of X? In certain contexts - thanks to the work of a number of people in Model Theory - finiteness (of the intersection of one Hecke orbit with the image under α of another Hecke orbit) is known. I will describe a few of those known results, and formulate some questions about bounds connected to them.

3:30 - 5pm: Kobi Peterzil (Haifa) *Some remarks on definable fundamental sets in o-minimal structures (joint work-in-progress with S. Starchenko)*

Assume that Gamma is a discrete infinite group acting on a complex manifold M. The goal is to realize the quotient space Gamma\M as a definable object in an o-minimal structure. The following is sufficient: the existence of a definable (in some o-minimal structure) subset F of M such that (1) Gamma.F=M and (2) the set of {g in Gamma : g.F ∩ F is non empty} is finite.

Under additional topological assumptions the quotient of F by Gamma, call it M_F, can be endowed with a definable manifold structure, which is naturally biholomorphic to Gamma\M . It turns out that different definable fundamental sets can give rise to definable manifolds which are not definably bi-holomorphic. This can be easily seen by considering the (non-definable) sets Gamma.F in elementary extensions.

One now considers the structure obtained by endowing M_F with all definable analytic subsets of its Cartesian powers. In the basic case of (C,+) and the group of integers (“the exponential case”), the various fundamental sets give rise to three types of strongly minimal structures: trivial, linear and non-locally modular. We use a theorem of Bishop and o-minimality to establish a GAGA principle in this setting, even when the covering map is not definable on F.

**22 April 2014**, 9:30am - 5:45pm, Simons auditorium. Differential/difference day.

**Friday 18 April 2014, 4:10 - 5:30pm, 891 Evans Hall** (Number theory seminar)

Daniel** **Bertrand (Paris 6), *Unlikely intersections in semi-abelian schemes.*

Given a "non-special" section of a semi-abelian scheme over a curve, the relative Manin-Mumford conjecture (RMM) asserts that its image W meets only finitely many torsion curves. I will explain how a relative version of the points constructed (in a Kummer theoretical setting) by the organizer of this seminar provide ``very special" examples of infinite intersection. However, the corresponding curves W recover a "normally special" status, when viewed in the setting of Pink's conjecture on mixed Shimura varieties. Furthermore, these sections form the only counterexample to the standard version of RMM for semi-abelian surfaces. These are joint results with B. Edixhoven, and with D. Masser, A. Pillay and U. Zannier.

**15 Apr****il 2014.**

1:30 - 3pm: Alf Onshuus (U. Andes), *Torsion free groups definable in o-minimal structures.*

We will analyze torsion free groups definable in o-minimal structures, using the concept of "supersolvability". Among other things we will prove (or sketch the proof) that:

Every torsion free group definable in an o-minimal structure is supersolvable.

- Every connected simply connected supersolvable real Lie group is Lie isomorphic to a group definable in the real exponential field.

3:30 - 5pm: Chris Miller (Ohio State) *Tameness and coincidence of dimensions in expansions of the real field*

Philipp Hieronymi and I have recently established that if E is a boolean combination of open subsets of real euclidean n-space, and the expansion of the real field by E does not define the set of all integers, then the Lebesgue covering dimension of E is equal to its Assouad dimension (hence also to its Hausdorff and packing dimensions, and also to its upper Minkowski dimension if E is bounded). The proof is too technical to attempt in a seminar talk, but the result is surprisingly easy to prove for the case n=1. Indeed, I will prove the (possibly) stronger result that all reasonable (in a way that I will make precise) Lipschitz invariant metric dimensions then coincide on E.

**8 April 2014**

1:30 - 3pm: Maryanthe Malliaris (U. Chicago), *On the fundamental complexity of simple theories*

Abstract: The talk will be about a recent theorem of Malliaris and Shelah (part of ongoing joint work on Keisler's order) which characterizes the simple theories in terms of saturation of ultrapowers.

3:30 - 5pm: Cameron Hill (Wesleyan), *The Finite Submodel Property and Definability in Classes of Finite Structures. *

I will discuss several notions that are intended to pull classes of finite structures into the framework of "classical'' model theory. These notions include robust chains (due to Macpherson-Steinhorn) and their cousins robust classes. To give these ideas a context, I will discuss them in relation to (variants of) the finite submodel property, zero-one laws, and (possibly) Ramsey theory.

**Tuesday April 1**

11am - 12:30pm: George Comte (Chambery), *Deformation of tame sets and additive invariants.*

We present classical or more recent additive invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and motivic type invariants are encompassed under this point of view. We then explain how all these invariants are related to each other as well as we propose a general conjectural principle explaining why such invariants have to be related. This last principle may appear as the incarnation, in tame definable geometries, of deep finiteness results in convex geometry, according to which additive invariants in convex geometry are very few.

1:30 - 3pm: Adam Topaz (UC Berkeley), *Towards a Z/ell analogue of Bogomolov's birational anabelian geometry.*

In the early 90's, F. Bogomolov introduced a program whose ultimate goal is to reconstruct function fields of dimension > 1 over algebraically closed fields from their pro-ell 2-step nilpotent Galois groups. Although it is far from being resolved in full generality, this program has since been carried through for function fields over the algebraic closure of a finite field by Bogomolov-Tschinkel and by Pop.

After an introduction to Bogomolov's program, I will describe the possibility of a Z/ell-analogue of Bogomolov's program, the inherent problems/difficulties that come with this analogue, and some partial results in this direction.

MSRI/Evans lectures.

4 - 5pm: Alex Wilkie (U Manchester), *Bounding the density of rational points on transcendental hypersurfaces via model theory.*

I shall begin by describing the solution (by Bombieri and Pila) to the following problem of Sarnak: prove that if f is a real analytic, but not algebraic, function defined on the closed interval [0,1], then the equation f(s)=q has rather few solutions in rational numbers.

Once the terms here have been made precise, one can then formulate a natural conjecture for analytic functions f of several variables. It turns out that the number-theoretic part of the argument in the one variable case generalises easily. However, the difficulty comes in the analysis, and it is here that techniques from model theory, specifically o-minimality, play a role.

**25 March 2014. (Warning: change of schedule)**

1:30 - 3pm: Johannes Nicaise (Leuven), *Weight functions on Berkovich spaces and poles of maximal order of motivic zeta functions.*

I will explain some connections between Berkovich spaces of degenerations of Calabi-Yau varieties and the Minimal Model Program in birational geometry. The central object in this theory is the so-called weight function on the Berkovich space. This function has some interesting properties that suggest that one can use it to contract the Berkovich space onto its canonical skeleton. I will also show how analogous properties of weight functions of hypersurface singularities yield a proof of a 1999 conjecture of Veys on poles of maximal order of motivic zeta functions. This is based on joint work with Mircea Mustata and Chenyang Xu.

3:30 - 5pm: Matthias Aschenbrenner (UCLA) *Newtonian fields*

In joint work with Lou van den Dries and Joris van der Hoeven, we are investigating algebraic and model-theoretic properties of certain valued differential fields in which the valuation and derivation interact in a very strong way. Prime examples of interest are various fields of "transseries," first introduced independently by J. Écalle in his work on Hilbert’s 16th Problem and by the model theorists Dahn and Göring in their work around Tarski’s problem on real exponentiation, as well as Hardy fields (such as fields of germs of functions definable in an o-minimal expansion of the real field). One important question in this subject is what the right notion of "differentially henselian" should be. In this talk I will propose an answer to this question; I will introduce the concept of "newtonian" valued differential field and explain some of its fundamental properties. Lurking in the background is a version of the Newton polygon process for differential polynomials, which I also hope to describe.

**19 March 2014, 3:40 - 5pm, room 740 Evans Hall (Number Theory seminar)**: Umberto Zannier (Pisa), *Unlikely Intersections and Pell's equation in polynomials*

We will begin with a discussion of some recent results obtained with David Masser. These results are simultaneously "relative" cases of the Manin-Mumford conjecture (a theorem of Raynaud) and special cases of the so-called Pink Conjecture.

We will continue with a presentation of applications of our results to the solvability of the Pell equation X^2-DY^2 = 1 in nonconstant polynomials X(t),Y (t) when D = D(t) is also a polynomial.

This analogue of Pell's equation for integers was studied already by Abel. In the polynomial situation, solvability is no longer ensured by simple conditions on D, and in fact may be considered "exceptional." In our treatment, we let D(t) = DL(t) vary over a pencil. In the particular case where DL(t) = t^6 + t + L, our results imply that the Pell equation is solvable with nonzero Y(t) only for finitely many complex values of L.

**18 March 2014**

• 11am - 12:30pm: Thomas Tucker (Rochester U.), *Preperiodic portraits modul primes*

Let F be a rational function of degree > 1 over a number field or function field K and let z be a point that is not preperiodic. Ingram and Silverman conjecture that for all but finitely many positive integers (m,n), there is a prime p such that z has exact preperiodic m and exact period n (we call this pair (m,n) the portrait of z modulo p). We present some counterexamples to this conjecture and show that a generalized form of abc implies -- one that is true for function fields -- implies that these are the only counterexamples. We also present a connection with Douady-Thursto-Hubbard rigidity. This represents joint work with several authors.

• 1:30 - 3pm: Holly Krieger (MIT) *The arithmetic of dynamical sequences*

We study sequences which are forward orbits under iteration of certain dynamical systems, and I will discuss results on the arithmetic of such sequences, which rely on techniques from Diophantine approximation, arithmetic dynamics, and complex dynamics.

**11 March 2014**

• 11am - 12:30pm: Dragos Ghioca (U British Columbia), *The Dynamical Mordell-Lang problem.*

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point in X. We show that the set containing all positive integers n such that the n-th iterate of x under f lands in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density 0. This is joint work with Jason Bell and Tom Tucker.

• 1:30 - 3pm: Uri Andrews (U. Wisconsin), *Relative computability of models of a strongly minimal theory.*

We show that if a strongly minimal theory has one recursive model then every model is recursive in 0^(3). In a special cases we can lower this bound to 0^(2). This answers a long-standing open question in recursive model theory. The analysis uses both model theoretic ideas as well as some recursion theoretic techniques. I will try to explain this interplay without assuming recursion theory knowledge beyond the existence of a *Halting set*, whose role I will briefly review. (Work joint with Julia F. Knight)

**4 March 2014**

• 11am - 12:30pm: Michael Larsen (Indiana U.), *Local vs. finite Galois obstructions to rational points.*

Let X be a non-singular curve over a number field K. Consider two kinds of obstructions to the existence of K-points on X: local obstructions (e.g., X has no points over R or over Q_p), and finite Galois obstructions (e.g., there exists an element s of Gal(\bar K/K) such that no element of X(\bar K) is s-stable). I will ask whether these two obstructions are the same and discuss evidence coming from algebraic number theory, Diophantine geometry, and additive combinatorics.

• 1:30 - 3pm: Anand Pillay (Notre Dame U.), *Nash groups. *

I will discuss the old problem of describing Nash groups up to Nash isomorphism, equivalently groups definable in the real field up to definable isomorphism, and mention some current work (with Starchenko).

**25 February 2014**

• 11am - 12:30pm: Paola D'Aquino (Napoli 2), *Exponential polynomials.*

I will present some results on the zero sets of exponential polynomials over certain exponential fields.

• 1:30 - 3pm: David Masser, *The unlikelihood of integrability in elementary terms*. Abstract

**18 February 2014**

• 2 - 3:30pm: Florent Martin (Lille), *A topological tameness result for Berkovich spaces*

Let k be a complete discrete valuation field and f : X -> Y an analytic morphism between k-affinoid spaces (X and Y might be closed polydisc for instance). We prove that the complementary set of the image of f has finitely many connected components with respect to the Berkovich topology.

We will explain this result, which relies essentially on a quantifier elimination theorem for fields with analytic structures due to Leonard Lipshitz.

**13 February 2014**

• 3:30 - 5pm: Jacob Tsimerman (Harvard), *An Ax-Schanuel theorem for the modular curve and the j-function.*

The classical Ax-Schanuel theorem states that, in a differential field, any algebraic relations involving the exponential function must arise in a 'trivial' manner. It turns out that one can formulate natural analogues of this theorem in the context of uniformization maps arising from Shimura varieties, the simplest case of which is the j-function. Besides their inherent appeal, such analogues have applications to the Zilber-Pink conjecture in number theory; a far reaching generalization of Andre-Oort.

We will explain these analogues and sketch a proof in the case of the j-function. This is joint work with J.Pila.

**28 January 2014.**

• 11am - 12:30pm: Antoine Ducros (UPMC), *Extension of Gauss valuations, stably dominated types and piecewise-linear subsets of Berkovich spaces.*

Let k be a valued field, and let L be a finite extension of k(T_1,...T_n). If G is an abelian ordered group G containing |k^*|, every n-uple (g_1,..g_n) of elements of G gives rise to the so-called Gauss valuation on k(T_1,...T_n), defined by the formula \sum a_I T^I \mapsto \max |a_I| g^I . I will explain how one can use a finiteness result by Hrushovski and Loeser concerning the space of stably dominated types (in the theory ACVF) on an algebraic curve to prove the following: there exists a finite subset E of L such that for every (g_1,...,g_n) as above, all extensions of the corresponding Gauss valuation to L are separated by elements of E.

I will also say a few words about an application of this result (which was my main motivation for proving it) concerning subsets of Berkovich spaces that inherit a natural structure of piecewise-linear space.

• 1:30 pm - 3:00pm: Martin Bays (Mc Master), *Universal covers of commutative finite Morley rank groups*

As part of his programme to tackle the model theory of complex exponentation, Zilber (2002) obtained a categoricity result for the structure of the exponential map in the “Lie algebra” language <>C;+> --> <C^*;+,*>, where the domain has only linear structure.

I will present an abstract version of this result, where C^* is replaced by an almost arbitrary commutative finite Morley rank group - for example by a semiabelian variety in arbitrary characteristic, or a Manin kernel.

I will aim to give some details of the proof.

This is work-in-preparation with Bradd Hart and Anand Pillay.