Forthcoming events in this series

Thu, 25 Feb 2016

Extremal fields and tame fields

Franz-Viktor Kuhlmann
(Katovice University)

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on
his notion of ``extremal valued fields''. He proved that algebraically
complete discretely valued fields are extremal. However, the proof
contained a mistake, and it turned out in 2009 through an observation by
Sergej Starchenko that Ershov's original definition leads to all
extremal fields being algebraically closed. In joint work with Salih
Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate
definition and then characterized extremal valued fields in several
important cases.

We call a valued field (K,v) extremal if for all natural numbers n and
all polynomials f in K[X_1,...,X_n], the set of values {vf(a_1,...,a_n)
| a_1,...,a_n in the valuation ring} has a maximum (which is allowed to
be infinity, attained if f has a zero in the valuation ring). This is
such a natural property of valued fields that it is in fact surprising
that it has apparently not been studied much earlier. It is also an
important property because Ershov's original statement is true under the
revised definition, which implies that in particular all Laurent Series
Fields over finite fields are extremal. As it is a deep open problem
whether these fields have a decidable elementary theory and as we are
therefore looking for complete recursive axiomatizations, it is
important to know the elementary properties of them well. That these
fields are extremal could be an important ingredient in the
determination of their structure theory, which in turn is an essential
tool in the proof of model theoretic properties.

The notion of "tame valued field" and their model theoretic properties
play a crucial role in the characterization of extremal fields. A valued
field K with separable-algebraic closure K^sep is tame if it is
henselian and the ramification field of the extension K^sep|K coincides
with the algebraic closure. Open problems in the classification of
extremal fields have recently led to new insights about elementary
equivalence of tame fields in the unequal characteristic case. This led
to a follow-up paper. Major suggestions from the referee were worked out
jointly with Sylvy Anscombe and led to stunning insights about the role
of extremal fields as ``atoms'' from which all aleph_1-saturated valued
fields are pieced together.

Thu, 18 Feb 2016

Joint Number Theory/Logic Seminar: On a modular Fermat equation

Jonathan Pila
((Oxford University))
`I will describe some diophantine problems and results motivated
by the analogy between powers of the modular curve and powers of the
multiplicative group in the context of the Zilber-Pink conjecture.
Thu, 04 Feb 2016

Joint Number Theory/Logic Seminar: Strongly semistable sheaves and the Mordell-Lang conjecture over function fields

Damian Rössler
((Oxford University))

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

Thu, 28 Jan 2016

Characterizing diophantine henselian valuation rings and ideals

Sylvy Anscombe
(University of Central Lancashire)

I will report on joint work with Arno Fehm in which we apply
our previous `existential transfer' results to the problem of
determining which fields admit diophantine nontrivial henselian
valuation rings and ideals. Using our characterization we are able to
re-derive all the results in the literature. Also, I will explain a
connection with Pop's large fields.

Thu, 03 Dec 2015

Near-henselian fields - valuation theory in the language of rings

Franziska Jahnke

Abstract: (Joint work with Sylvy Anscombe) We consider four properties 
of a field K related to the existence of (definable) henselian 
valuations on K and on elementarily equivalent fields and study the 
implications between them. Surprisingly, the full pictures look very 
different in equicharacteristic and mixed characteristic.

Thu, 26 Nov 2015


Benedikt Loewe

 If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame.  We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases.  Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing:  in this case, we are looking at the so-called "generic multiverse".

Thu, 19 Nov 2015

Real, p-adic, and motivic oscillatory integrals

Raf Cluckers

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations. 

Thu, 12 Nov 2015

Restricted trochotomy in dimension 1

Dmitri Sustretov
(Hebrew University of Jerusalem)

Let M be an algebraic curve over an algebraically closed field and let
$(M, ...)$ be a strongly minimal non-locally modular structure with
basic relations definable in the full Zariski language on $M$. In this
talk I will present the proof of the fact that $(M, ...)$ interprets
an algebraically closed field.

Thu, 05 Nov 2015

Decidability of the Zero Problem for Exponential Polynomials

James Worrell
(Computing Laboratory, Oxford)

We consider the decision problem of determining whether an exponential
polynomial has a real zero.  This is motivated by reachability questions
for continuous-time linear dynamical systems, where exponential
polynomials naturally arise as solutions of linear differential equations.

The decidability of the Zero Problem is open in general and our results
concern restricted versions.  We show decidability of a bounded
variant---asking for a zero in a given bounded interval---subject to
Schanuel's conjecture.  In the unbounded case, we obtain partial
decidability results, using Baker's Theorem on linear forms in logarithms
as a key tool.  We show also that decidability of the Zero Problem in full
generality would entail powerful new effectiveness results concerning
Diophantine approximation of algebraic numbers.

This is joint work with Ventsislav Chonev and Joel Ouaknine.

Thu, 29 Oct 2015

Joint Number Theroy/Logic Seminar: A minimalistic p-adic Artin-Schreier

Florian Pop
(University of Pennsylvania)

In contrast to the Artin-Schreier Theorem, its p-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of p-adic fields.  We plan to give a characterization of p-adic p-Henselian valuations in an essentially finite way. This relates to the Z/p metabelian form of the birational p-adic Grothendieck section conjecture.

Thu, 22 Oct 2015

Definability in algebraic extensions of p-adic fields

Angus Macintyre
(Queen Mary University London)

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Tue, 23 Jun 2015

17:00 - 18:00

Almost small absolute Galois groups

Arno Fehm

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

Thu, 18 Jun 2015

17:30 - 18:30

On the Consistency Problem for Quine's New Foundations, NF

Peter Aczel

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

In my talk I will introduce TST^+ and its possible models and discuss some of the interesting ideas, that I have understood, that Holmes uses in one of his possible proofs.  If there is time in my talk I will also mention a more recent approach of Jamie Gabbay who is taking a nominal sets approach to the problem.
Thu, 11 Jun 2015

17:30 - 18:30

Examples of quasiminimal classes

Jonathan Kirby

I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and open questions. Then I will explain some joint work with Martin Bays in which we have constructed variants of the pseudo-exponential fields (originally due to Boris Zilber) which are quasimininal and discuss progress towards the problem of showing that complex exponentiation is quasiminimal. I will also discuss some joint work with Adam Harris in which we try to build a pseudo-j-function.

Thu, 04 Jun 2015

17:30 - 18:30

Some effective instances of relative Manin-Mumford

Gareth Jones

In a series of recent papers David Masser and Umberto Zannier proved the relative Manin-Mumford conjecture for abelian surfaces, at least when everything is defined over the algebraic numbers. In a further paper with Daniel Bertrand and Anand Pillay they have explained what happens in the semiabelian situation, under the same restriction as above.

At present it is not clear that these results are effective. I'll discuss joint work with Philipp Habegger and Masser and with Harry Schimdt in which we show that certain very special cases can be made effective. For instance, we can effectively compute a bound on the order of a root of unity t such that the point with abscissa 2 is torsion on the Legendre curve with parameter t.


**Note change of room**



Thu, 21 May 2015

16:00 - 17:00

Anabelian Geometry with étale homotopy types

Jakob Stix
(University of Heidelberg)

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.


**Joint seminar with Number Theory. Note unusual time and place**

Thu, 14 May 2015

17:30 - 18:30

Commutative 2-algebra, operads and analytic functors

Nicola Gambino

Standard commutative algebra is based on the notions of commutative monoid, Abelian group and commutative ring. In recent years, motivations from category theory, algebraic geometry, and mathematical logic led to the development of an area that may be called commutative 2-algebra, in which the notions used in commutative algebra are replaced by their category-theoretic counterparts (e.g. commutative monoids are replaced by  symmetric monoidal categories). The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra, and to outline how this suggests counterparts of basic aspects of algebraic geometry. In particular, I will describe some joint work with Andre’ Joyal on operads and analytic functors in this context.

Thu, 07 May 2015

17:30 - 18:30

Free actions of free groups on countable structures and property (T)

David Evans

In joint work with Todor Tsankov, we show that the automorphism groups of countable, omega-categorical structures have Kazhdan's property (T). The proof uses Tsankov's work on the unitary representations of these groups, together with a construction of a particular free subgroup of the automorphism group.

Thu, 30 Apr 2015

17:30 - 18:30

Strong type theories and their set-theoretic incarnations

Michael Rathjen

There is a tight fit between type theories à la Martin-Löf and constructive set theories such as Constructive Zermelo-Fraenkel set theory, CZF, and its extension as well as classical Kripke-Platek set theory and extensions thereof. The technology for determining their (exact) proof-theoretic strength was developed in the 1990s. The situation is rather different when it comes to type theories (with universes) having the impredicative type of propositions Prop from the Calculus of Constructions that features in some powerful proof assistants. Aczel's sets-as-types interpretation into these type theories gives rise to  rather unusual set-theoretic axioms: negative power set and negative separation. But it is not known how to determine the proof-theoretic strengths of intuitionistic set theories with such axioms via familiar classical set theories (though it is not difficult to see that ZFC plus infinitely many inaccessibles provides an upper bound). The first part of the talk will be a survey of known results from this area. The second part will be concerned with the rather special computational and proof-theoretic behavior of such theories.

Thu, 12 Mar 2015

17:30 - 18:30

Rosenthal compacta and NIP formulas

Pierre Simon
(Université Lyon I)

A compact space is a Rosenthal compactum if it can be embedded into the space of Baire class 1 functions on a Polish space. Those objects have been well studied in functional analysis and set theory. In this talk, I will explain the link between them and the model-theoretic notion of NIP and how they can be used to prove new results in model theory on the topology of the space of types.

Thu, 05 Mar 2015

11:00 - 12:30

QE in ACFA is PR

Ivan Tomasic


It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.


Thu, 26 Feb 2015

17:30 - 18:30

The existential theory of equicharacteristic henselian valued fields

William Anscombe

We present some recent work - joint with Arno Fehm - in which we give an `existential Ax-Kochen-Ershov principle' for equicharacteristic henselian valued fields. More precisely, we show that the existential theory of such a valued field depends only on the existential theory of the residue field. In residue characteristic zero, this result is well-known and follows from the classical Ax-Kochen-Ershov Theorems. In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's theory of tame fields. One corollary is an unconditional proof that the existential theory of F_q((t)) is decidable. We will explain how this relates to the earlier conditional proof of this result, due to Denef and Schoutens.

Thu, 19 Feb 2015

17:30 - 18:30

Hardy type derivations on the surreal numbers

Alessandro Berarducci
(Universita di Pisa)

The field of transseries was introduced by Ecalle to give a solution to Dulac's problem, a weakening of Hilbert's 16th problem. They form an elementary extension of the real exponential field and have received the attention of model theorists. Another such elementary extension is given by Conway's surreal numbers, and various connections with the transseries have been conjectured, among which the possibility of introducing a Hardy type derivation on the surreal numbers. I will present a complete solution to these conjectures obtained in collaboration with Vincenzo Mantova.