Further information
Preliminary schedule
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Tuesday 7th July |
Wednesday 8th July |
Thursday 9th July |
Friday 10th July |
9:30-10:00 |
Registration |
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10:00-11:00 |
Angus Macintyre |
Lou van den Dries |
Margaret Thomas |
Adam Harris |
11:00-11:30 |
coffee |
coffee |
coffee |
coffee |
11:30-12:30 |
Patrick Speissegger |
Zoe Chatzidakis |
Philipp Hieronymi |
Vincenzo Mantova |
12:30-13:30 |
lunch |
lunch |
lunch |
Sergei Starchenko |
13:30-14:30 |
lunch |
lunch |
lunch |
finish |
14:30-15:30 |
Kobi Peterzil |
Philipp Habegger |
Boris Zilber |
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15:30-16:00 |
coffee |
coffee |
coffee |
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16:00-17:00 |
Dave Marker |
Giussepina Terzo |
Alex Wilkie |
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17:00- |
reception and posters |
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Location
Talks on Tuesday, Wednesday, Thursday: G51 in the Chemistry building.
Talks on Friday: G.107 in the Alan Turing Building
Reception: outside the Frank Adams rooms in the Alan Turing Building.
Talks
Lou van den Dries
Hardy fields, transseries, surreals, and model theory
Abstract: This is an account of recent joint work with Aschenbrenner and
Van der Hoeven, open problems raised by it, and of work in
progress by Gehret. A year ago we identified the theory of the
differential field T of transseries as the model companion of the
theory of H-fields. One open question to be discussed is whether every
maximal Hardy field is elementarily equivalent to T.
Recently, Berarducci and Mantova constructed a ``simplest
possible'' derivation on Conway's field No of surreal numbers
that makes it an H-field with omega'=1. About a month ago we showed
that No with this derivation is elementarily equivalent to T.
Gehret has been studying the differential field T_log of purely
logarithmic transseries. Various signs point to it also having a viable
model theory.
Philipp Habegger
Unlikely Intersections and o-Minimal Structures
Abstract: The Conjecture on Unlikely Intersections generalizes various classical statements in diophantine geometry. Among them is Faltings's Theorem which states that a smooth curve of genus at least 2 contains only finitely many rational points. These new conjectures originated in work of Bombieri-Masser-Zannier, Pink, and Zilber.
Pila and Zannier used the Pila-Wilkie counting theorem which relies on o-minimal structures to give a new proof of Raynaud's Theorem, another classical diophantine statement.
I will give background information on the Conjecture on Unlikely Intersections. In joint work with Jonathan Pila we proved this conjecture for curves in abelian varieties when everything is defined over a number field. We use a variation of the aforementioned counting theorem.
Philipp Hieronymi
Diophantine approximation, scalar multiplication and decidability
Abstract: It has long been known that the first order theory of the expansion (R,<,+,Z) of the ordered additive group of real numbers by just a predicate for the set of integers is decidable. Arguably due to Skolem, the result can be deduced easily from Buechi's theorem on the decidability of monadic second order theory of one successor, and was later rediscovered independently by Weispfenning and Miller. However, a consequence of Goedel's famous first incompleteness theorem states that when expanding this structure by a symbol for multiplication, the theory of the resulting structure (R,<,+,*,Z) becomes undecidable. This observation gives rise to the following natural and surprisingly still open question: How many traces of multiplication can be added to (R,<,+,Z) without making the first order theory undecidable? We will give a complete answer to this question when "traces of multiplication" is taken to mean scalar multiplication by certain irrational numbers. To make this statement precise: for b in R, let f_b: R -> R be the function that takes x to bx. I will show that the theory of (R,<,+,Z,f_b) is decidable if and only if b is quadratic. The proof rests on the observation that many of the Diophantine properties (in the sense of Diophantine approximation) of b can be coded in these structures. It was in 2007 that Alex pointed out to me (I was still his PhD student at the time) that the approximation functions I was considering, looked to him like Diophantine approximation, and he suggested that I should take a look at a book on this topic. So consider this talk as a long overdue progress update a PhD student is giving his PhD advisor.
Kobi Peterzil
Colloquium talk, see here
Vincenzo Mantova
Surreal numbers, derivations and transseries
Abstract: Conway's "No" is a class of numbers, originally thought as games, equipped with a natural ordered field structure and an exponential function which make it into a monster model of the theory of (R,exp).
In a joint work with Berarducci, we determine the transseries structure of No, and we prove the existence of a natural differential field structure on No similar to the one of Hardy fields.
David Marker
Colloquium talk, see here
Patrick Speissegger
Constructing quasianalytic Ilyashenko algebras
Abstract: If one allows for compositional iterates of log to appear in asymptotic
expansions, Ilyashenko's construction of quasianalytic classes used in
his proof of the Dulac problem can be extended to obtain a larger
quasianalytic algebra. This algebra is still closed under composition,
and it suggests an even more general construction based on monomials
definable in the expansion of the real field by restricted analytic
functions and the exponential function. This is joint work with Tobias Kaiser.
Sergei Starchenko
On Peterzil-Steinhorn subgroups in o-minimal structures.
Abstract: Peterzil and Steinhorn proved that in o-minimal structures, any unbounded definable curve inside a definable group G gives rise to a one-dimensional torsion-free definable subgroup of G, associated to the curve at ``infinity'’ . In this talk I will present a generalization of the result to definable types of arbitrary dimension: Each definable type p inside G gives rise to a definable, torsion-free subgroup $H_p$ , whose dimension can be read-off the type p.
As we will see, the group $H_p$ is the stabilizer of a partial type associated to p, under the action of G on a certain quotient of the usual types space S(G). It turns out that this quotient of the type space is closely related to the classical construction of Samuel’s compactification of topological groups and more generally of spaces with uniformity.
This is a joint work with Y.Peterzil.
Giussepina Terzo
See here.
Boris Zilber
From Hrushovski's counter-examples to symplectic geometry
Abstract: I will start by recalling Hrushovski's construction, in particular the version by Poizat (coloured fields). This speaker aimed at interpreting these constructions in terms of classical transcendental functions and the geometry on the field of complex numbers such as the field with pseudo-exponentiation, but see also JSL, v.69(4), 2004, and the joint work with J.-D. Caycedo Casallas: arxiv1401.3326. The interesting and somewhat puzzling feature of the last two examples (of ``coloured fields'' type) is that the relevant geometry is not complex analytic.
In the talk I will extend the class of the above examples and
present a family of much richer geometric structures on complex manifolds which have near model-complete superstable theories.
I will explain that these structures can be seen as objects of symplectic geometry and discuss how they fit into the definition of ``generalized complex structures'' introduced in 2002 by N.Hitchin and which play at present a key role in mathematics of string theory.
Posters
Vahagn Aslanyan
Definability of derivations in the reducts of differentially closed fields
Abstract: I consider reducts F_E=(F;+,\cdot,E) of differentially closed fields with a binary relation E for a differential algebraic curve (given by a differential equation of two variables). The main problem is to classify the differential curves for which the derivation D of F is definable in F_E. (Note that the analogous question for pure fields, that is, recovering the field structure from algebraic curves and reducts of ACF in general, is very important in algebraic geometry and in the theory of Zariski geometries.) In this poster I present my results towards this problem. I have found several classes of curves for which the derivation is definable. Using them I also give some necessary and sufficient conditions for definability of D. This also helps to understand the model theoretic properties of the reducts F_E of DCF_0. Note that this is still work in progress and I expect some nice (especially geometric) characterisations of reducts where the derivation is definable.
Levon Haykazyan
Quasiminimality, Pregeometries and Regular Types
Abstract: In a strongly minimal theory, there is a unique nonalgebraic type which induces a homogeneous pregeometry on every model. Different aspects of strongly minimal theories can be generalised to concepts of quasiminimality, homogeneous pregemeotries and strongly regular types. These concepts have applications from classification of nonelementary classes to geometric stability theory. We will explore connections between them, answer some questions and ask yet more questions.
Grzegorz Jagiella
Definable topological dynamics and o-minimality
Abstract: I will present my results on definable topological dynamics of groups definable in o-minimal structures. Given $R$, an o-minimal expansion or a real closed field, I consider the class of $R$-definable groups that admit a definable compact-torsion-free decomposition. This is a large class of groups that properly includes all semisimple definable Lie groups, and every definable group is a central extension of a group with a definable compact-torsion-free decomposition. For a definable group $G$ that admits such decomposition $G=KH$ with $K$ definably compact, $H$ torsion-free, I describe minimal subflows of the universal definable $G(R)$-flow $S_{G,ext}(R)$. In case where $G$ is a definable Lie group, I show that the Ellis group of its universal definable flow is isomorphic to $N_G(H)/H$. I also extend the results to universal covers of definable Lie groups, interpreted in a certain two-sorted structure.
Lubna Shaheen
A geometric model for representations of Z'
Abstract. The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum Spec(Z) defined by Grothendieck serves this purpose. However, it is still not clear what geometry this object carries. E.g. Y I. Manin discusses what the dimension of Spec(Z) could be, speculating that it may be 1, 3 or infinite. A.Connes and C.Consani published recently an important paper which introduces a much more complicated structure called the arithmetic site on the basis of Spec(Z). Our approach is based on the generational of constructions applied by B.Zilber for similar purposes in non-commutative (and commutative) algebraic geometry. The current version is quite basic. We describe a category of certain representations of integral extensions Z[a] of Z and establish its tight connection with the space of elementary theories of pseudo-finite fields. From model-theoretic point of view the category of representations is a multi-sorted structure which we prove to be super-stable with pre-geometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields. Note that the model-theoretic analysis of the structure establishes that the Morley rank of Spec(Z) is infinite while the u-rank is 1, thus identifying formally two of the three Manin’s dimensions.