Model theory and applications to algebra and analysis

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Programme theme

Model theory is a branch of mathematical logic dealing with abstract structures (models), historically with connections to other areas of mathematics. In the past decade, model theory has reached a new maturity, allowing for a strengthening of these connections and striking applications to diophantine geometry, analytic geometry and Lie theory, as well as strong interactions with group theory, representation theory of finite-dimensional algebras, and the study of the p-adics. The main objective of the semester will be to consolidate these advances by providing the required interdisciplinary collaborations.

Model theory is traditionally divided into two parts pure and applied. Pure model theory studies abstract properties of first order theories, and derives structure theorems for their models. Applied model theory on the other hand studies concrete algebraic structures from a model-theoretic point of view, and uses results from pure model theory to get a better understanding of the structures in question, of the lattice of definable sets, and of various functorialities and uniformities of definition. By its very nature, applied model theory has strong connections to other branches of mathematics, and its results often have non-model-theoretic implications. A substantial knowledge of algebra, and nowadays of algebraic and analytic geometry, is required.

The programme will concentrate on the following areas

Pure model theory.: We expect further developments in the use of stability theory techniques in unstable contexts (simple theories, algebraically closed valued fields) and in non-elementary classes.
Model theory of fields with operators, and connections with arithmetic geometry.: The model theory of differentially closed fields and of other fields with operators has been at the centre of model-theoretic proofs of results in arithmetic geometry. The Zil'ber programme of pseudo-analytic functions is also expected to have some interesting consequences.
O-minimality and related topics.: O-minimality is a property of ordered structures, yielding results akin to traditional real analytic results, such as the classical Finiteness Theorems for subanalytic sets (cell decompositions, Whitney stratifications, etc.). Mathematically central, new examples of o-minimal structures have emerged, and the logical theory has had applications to Lie theory, to asymptotics, and to neural networks.
Henselian fields.: Model theory of Henselian fields, and in particular of p-adic fields and Arc spaces. Connections with algebraic and analytic geometry. Study of cohomology theories and motives, aiming at uniformity results. Study of compact complex manifolds, and uses of stability.
Model theory of groups.: We plan to have a workshop on groups of Finite Morley rank, a topic connected to the Classification of finite simple groups via its techniques and its aims. The recent (and very exciting) developments in the model theory of non-abelian free groups should also be studied, depending on its degree of maturity.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
A note on global $\it p$$^t$$^h$ powers of rigid analytic functions Authors: Zachary Robinson Download PDF	2004	MAA
Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields Authors: Salma Kuhlmann Download PDF	2004	MAA
On fields and colors Authors: Amador Martin-Pizarro, Martin Ziegler, Andreas Baudisch Download PDF	2004	MAA
Supersimplicity and quadratic extensions Authors: Frank Wagner, Amador Martin-Pizarro Download PDF	2004	MAA
The theorem of the complement for sub-Pfaffian sets Authors: Patrick Speissegger, Jean-Marie Lion Download PDF	2004	MAA
Counting and Dimensions Authors: Frank Wagner, Ehud Hrushovski Download PDF	2004	MAA
Complex analytic geometry and analytic-geometric categories Authors: Sergei Starchenko, Yaacov Peterzil Download PDF	2004	MAA
Computing o-minimal topological invariants using differential topology Authors: Yaacov Peterzil, Sergei Starchenko Download PDF	2004	MAA
Compactness of cycle spaces and definability in o-minimal expansions of ${\mathbb R}_{an}$ Authors: Rahim Moosa, Patrick Speissegger, Sergei Starchenko Download PDF	2004	MAA
Model theory of universal covering spaces of complex algebraic varieties Authors: Mikhail Gavrilovich Download PDF	2004	MAA
Exponential and Weierstrass Equations Authors: Jonathan Kirby Download PDF	2004	MAA
Non-commutative geometry and new stable structures Authors: B Zilber Download PDF	2004	MAA
Stable pseudofinite groups Authors: Katrin Tent, Dugald Macpherson Download PDF	2004	MAA
Definable sets in algebraically closed valued fields: elimination of imaginaries Authors: Ehud Hrushovski, Dugald Macpherson, D Hasklell Download PDF	2004	MAA
Reconstruction of homogeneous relational structures Authors: Dugald Macpherson, Silvia Barbina Download PDF	2004	MAA
Marshall's and Milnor's conjectures for preordered von Neumann regular rings Authors: Maximo Dickmann, F Miraglia Download PDF	2004	MAA
Definable relations in the real field with a distinguished subgroup of the unit circle Authors: Oleg Belegradek, B Zilber Download PDF	2004	MAA
Definable groups and compact $\it p$-adic Lie groups Authors: A Pillay, Alf Onshuus Download PDF	2004	MAA
A remark on Zilber's pseudoexponentiation Authors: David Marker Download PDF	2004	MAA
Around unipotence in groups of finite Morley rank Authors: Olivier Frecon Download PDF	2004	MAA
Elimination of ramification I: the generalized stability theorem Authors: Salma Kuhlmann Download PDF	2004	MAA
Additive polynomials and their role in the model theory of valued fields Authors: Salma Kuhlmann Download PDF	2004	MAA
Strongly minimal groups in the theory of compact complex spaces Authors: Rahim Moosa, Matthias Aschenbrenner, Thomas Scanlon Download PDF	2004	MAA
A generic identification theorem for $L^$-groups of finite Morley rank Authors:* Ayse Berkman, Jeffrey Burdges, Alexandre Borovik, ET Al Download PDF	2004	MAA
Involutions in groups of finite Morley rank and degenerate type Authors: Gregory Cherlin, Alexandre Borovik, Jeffrey Burdges Download PDF	2004	MAA
Generix never gives up Authors: Eric Jaligot Download PDF	2004	MAA
The rational points of a definable set Authors: J Pila, Alex Wilkie Download PDF	2004	MAA
Sheaf cohomology in o-minimal structures Authors: Gareth Jones, Nicholas Peatfield, Mario Edmundo Download PDF	2004	MAA

Workshops

MAAW01

Seminars

Thursday 20th January 2005
10:30 to 12:30	Alex Wilkie University of Oxford An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application	Room 1
16:00 to 17:30	Alexander Prestel Universität Konstanz On henselian valued fields	Room 1

Tuesday 25th January 2005
10:00 to 12:00	Enrique Casanovas T resplendent models and the Lascar group	Room 1
16:00 to 17:30	Elisabeth Bouscaren Groups definable in separably closed fields and algebraic groups: definable simplicity and "pseudo-simple" groups	Room 1

Thursday 27th January 2005
11:00 to 12:30	Oleg Belegradek Expressive power of first-order logic for embedded finite models	Room 1
14:30 to 15:30	Alex Wilkie University of Oxford An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (ll)	Room 1
16:00 to 17:00	Alex Wilkie University of Oxford An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (ll)	Room 1

Tuesday 1st February 2005
11:00 to 12:00	Dugald Macpherson University of Leeds Stable pseudofinite groups	Room 1
16:30 to 18:30	Yaacov Peterzil University of Haifa On the topological degree of functions definable in o-minimal structures (l)	Room 1

Thursday 3rd February 2005
10:30 to 12:30	Sergei Starchenko University of Notre Dame On the topological degree of functions definable in o-minimal structures (ll)	Room 1
14:30 to 15:30	Alex Wilkie University of Oxford An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (III)	Room 1
16:00 to 17:00	Alex Wilkie University of Oxford An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (III)	Room 1

Tuesday 8th February 2005
11:00 to 12:00	Ludomir Newelski Wroclaw University of Technology Weak generic types	Room 1
14:30 to 15:30	Angus Macintyre Model theory of elliptic functions	Room 2
16:00 to 17:00	Angus Macintyre Model theory of elliptic functions	Room 2

Wednesday 9th February 2005
11:30 to 13:00	Sergei Starchenko University of Notre Dame Tomography in o-minimal structures based on constructible functions	Room 1

Thursday 10th February 2005
08:00 to 10:00	Elisabeth Bouscaren MAA - Meeting	Room 2
10:00 to 11:00	Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (I)	Room 1
11:30 to 12:30	Alf Onshuus University of Illinois Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (II)	Room 1
14:30 to 15:30	Alf Onshuus University of Illinois Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (III)	Room 1
16:00 to 17:00	Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (IV)	Room 1

Wednesday 16th February 2005
10:30 to 12:30	Angus Macintyre Model theory of elliptic functions (II)	Room 1
14:30 to 15:30	Thomas Scanlon University of California, Berkeley Groups in the theory of compact complex spaces	Room 1
16:00 to 17:00	Thomas Scanlon University of California, Berkeley Groups in the theory of compact complex spaces	Room 1

Thursday 17th February 2005
11:30 to 12:30	E.c. Hasse fields - geometric axioms	Room 1
14:30 to 15:30	Some exercises in model theory of mathematical physics	Room 1
16:00 to 17:00	Some exercises in model theory of mathematical physics	Room 1

Wednesday 23rd February 2005
10:30 to 12:30	Angus Macintyre Model theory of elliptic functions	Room 1
14:30 to 15:30	Daniel Bertrand Vanishing lemmas and transcendence	Room 1
16:00 to 17:00	Daniel Bertrand New proofs of the Siegel-Shidlovsky theorem	Room 1

Thursday 24th February 2005
11:00 to 12:00	David Marker A remark on pseudo-exponentiation	Room 1
16:00 to 17:30	Paola D'Aquino Quadratic forms in models of weak arithmetic	Room 1

Wednesday 2nd March 2005
11:30 to 12:30	Alexandre Borovik University of Manchester Black box groups and groups of finite Morley rank	Room 1
16:00 to 17:30	Luc Bélair UQAM - Université du Québec à Montréal Approximation in difference valuation rings	Room 1

Thursday 3rd March 2005
11:00 to 12:30	Jeffrey Burdges Rutgers, The State University of New Jersey Sylow theory for 'p=0' in groups of finite Morley rank	Room 1
16:00 to 17:30	Gregory Cherlin Conjugacy of good tori	Room 1

Wednesday 9th March 2005
11:30 to 12:30	Raf Cluckers CNRS - Ecole Normale Superieure Paris Real and motivic integration and oscillating functions	Room 1
16:00 to 17:30	Assaf Hasson University of Oxford Amalgamation with predimension and collapse	Room 1

Thursday 10th March 2005
11:30 to 12:30	Eric Jaligot Université Paris 7 - Denis-Diderot Carter subgroups of groups of finite Morley rank	Room 2
16:00 to 17:15	Francoise Point On commutative von Neumann regular rings with an automorphism	Room 2

Wednesday 16th March 2005
10:30 to 11:20	Tuna Altinel The programme for classification of groups of finite morley rank	Room 1
11:30 to 11:50	Coffee	No Room Required
12:00 to 13:00	Gregory Cherlin Ultraproducts and complex reflection groups	Room 1
13:00 to 14:00	Lunch at the Newton Institute	Discussion Room
14:00 to 14:50	Eric Jaligot Université Paris 7 - Denis-Diderot Good tori and bad fields	Room 1
15:00 to 15:50	The algebraicity conjecture: some thoughts against	Room 1
16:00 to 16:30	Tea	No Room Required
16:30 to 17:30	Alexandre Borovik University of Manchester Finite group theory: The next generation	Room 1
17:30 to 18:30	Wine and Beer Reception	No Room Required

Thursday 17th March 2005
10:00 to 12:00	Franz Timmesfeld Mathematisches Institut, Giessen Identifications of Lie type groups	Room 2

Tuesday 22nd March 2005
11:30 to 12:30	Tuna Altinel Simple groups without involutions	Room 1
15:30 to 17:00	Spectra of triangulated categories	Room 2

Wednesday 23rd March 2005
11:00 to 12:30	Francois Loeser Enlarging motivic constructible functions to exponentials	Room 2
15:30 to 17:00	Galois axioms for fields	Room 1

Wednesday 13th April 2005
14:00 to 15:00	Mario Edmundo Sheaf cohomology in arbitrary o-minimal structures	Room 1
15:30 to 16:30	Mario Edmundo Sheaf cohomology in arbitrary o-minimal structures	Room 1

Thursday 14th April 2005
11:30 to 12:30	Galois cohomology for surgical fields	Room 2
15:30 to 17:00	A twisted Chebotarev Theorem	Room 1

Wednesday 20th April 2005
11:30 to 12:30	Frank Wagner Counting in bad fields	Room 2
15:30 to 17:00	Mikhail Gavrilovich University of Oxford On the model theory of universal covers of algebraic varieties	Room 1

Thursday 21st April 2005
13:30 to 15:00	Tristram de Piro University of Edinburgh A non-standard Bezout theorem	Room 2
15:30 to 17:00	Askold Khovanskii University of Toronto Solvability and unsolvability of equations in finite terms	Room 2

Wednesday 27th April 2005
10:30 to 12:30	Ehud Hrushovski Hebrew University of Jerusalem Model theoretic topics in valued fields III	Room 1
15:30 to 17:00	Askold Khovanskii University of Toronto On a one-dimensional topological variant of Galois theory	Room 1

Thursday 28th April 2005
09:30 to 10:30	Yuri Ershov Novosibirsk State University Model theory of multi-valued fields	Room 1
11:00 to 12:30	Maximo Dickmann Université Paris 7 - Denis-Diderot The Marshall and Milnor conjectures for formally real von Neumann regular rings	Room 1
17:00 to 18:00	Yuri Ershov Novosibirsk State University Model theory of multi-valued fields	Room 1

Tuesday 3rd May 2005
11:00 to 12:30	Christian Michaux Université de Mons Bisimulations for o-minimal hybrid systems	Room 1
16:30 to 18:00	Andrei Gabrielov Indiana University-Purdue University Indianapolis Betti numbers of sets defined by quantifier-free formulas	Room 2

Wednesday 4th May 2005
10:00 to 11:00	Margarita Otero Universidad Autonoma de Madrid Definably compact groups	Room 1
11:30 to 12:30	Margarita Otero Universidad Autonoma de Madrid Definably compact groups	Room 1
15:30 to 17:00	Andrei Gabrielov Indiana University-Purdue University Indianapolis Betti numbers of sets defined by formulas with quantifiers	Room 1

Wednesday 11th May 2005
11:30 to 12:30	Franz-Viktor Kuhlmann University of Saskatchewan Truncation integer parts of valued fields	Room 2
14:00 to 15:00	Franz-Viktor Kuhlmann University of Saskatchewan Truncation integer parts of valued fields	Room 2

Thursday 12th May 2005
11:30 to 12:45	Philip Scowcroft Wesleyan University Non-negative solvability of linear equations in ordered groups	Room 2
15:30 to 17:00	Christian Michaux Université de Mons Dimension theory for BSS - r.e. sets	Room 2

Wednesday 18th May 2005
11:30 to 12:30	Byunghan Kim Relative computability for curves	Room 2

Thursday 19th May 2005
11:30 to 12:30	Timothy Mellor Universität Regensburg Elimination of imaginaries for real closed valued fields	Room 2
14:00 to 15:00	Topological properties of sets definable in weakly o-minimal structures	Room 2
15:30 to 16:30	Groups and fields definable in weakly o-minimal structures	Room 2

Wednesday 25th May 2005
10:30 to 12:30	Jean-Philippe Rolin Université de Bourgogne Non-oscillating solutions of differential equations and o-minimality	Room 2
16:30 to 17:30	Andrei Gabrielov Indiana University-Purdue University Indianapolis Mystery of point charges	Room 2

Thursday 26th May 2005
09:00 to 10:00	Andrei Gabrielov Indiana University-Purdue University Indianapolis Rational functions and real Schubert calculus	Room 1
10:30 to 11:30	Michael Singer North Carolina State University Galois theory of parameterised linear differential equations and linear differential algebraic groups	Room 2

Wednesday 1st June 2005
11:00 to 12:30	Rahim Moosa University of Waterloo Model theory of compact complex spaces I: introduction	Room 1
14:00 to 15:00	Jean-Pierre Ressayre Université Paris 7 - Denis-Diderot Transseries and polynomially bounded o-minimality	Room 1

Thursday 2nd June 2005
11:30 to 12:30	Charles Steinhorn Vassar College On close to tame expansions of densely ordered groups	Room 1
14:00 to 15:30	Rahim Moosa University of Waterloo Model theory of compact complex spaces II: essentially saturated space	Room 1

Wednesday 8th June 2005
10:30 to 12:30	Lou van den Dries University of Illinois Algebraically closed and real closed fields with small multiplicative group	Room 1
14:30 to 15:30	Model theory of compact complex spaces III: Campana's work on the nonmultidimensionality conjecture	Room 1

Thursday 9th June 2005
11:00 to 12:00	Franz-Viktor Kuhlmann University of Saskatchewan Additive polynomials	Room 1
14:00 to 15:00	Groups, measures and NIP	Room 1
15:30 to 16:30	Groups, measures and NIP	Room 1

Wednesday 15th June 2005
10:30 to 12:00	Fields with finitely many definable subsets	Room 2
14:00 to 15:00	Alex Wilkie University of Oxford Some model theory of complex analytic functions	Room 2
15:30 to 16:30	Alex Wilkie University of Oxford Some model theory of complex analytic functions	Room 2

Thursday 16th June 2005
10:30 to 12:00	Uniform first order definitions in finitely generated fields	Room 2
14:00 to 15:00	Franz-Viktor Kuhlmann University of Saskatchewan Dense subfields of henselian fields, integer parts, and nasty valuations on rational function fields	Room 2

Wednesday 22nd June 2005
11:30 to 12:30	A Higman subgroup theorem for lattice-ordered groups	Room 1
16:00 to 17:00	Matthias Aschenbrenner University of Illinois A generalisation of the Hilbert Basis Theorem	Room 1

Thursday 23rd June 2005
15:30 to 17:00	Dugald Macpherson University of Leeds Measurable structures, and asymptotics in finite structures	Room 1

Wednesday 29th June 2005
11:00 to 12:30	John Wilson Characterisation and axiomatisation of finite soluble groups	Room 1

Thursday 30th June 2005
14:00 to 15:30	Uniform first order definitions in finitely generated fields II	Room 2
16:00 to 17:00	Franz-Viktor Kuhlmann University of Saskatchewan More on additive polynomials	Room 1

Model theory and applications to algebra and analysis

Organisers

Zoe Chatzidakis
Dugald Macpherson University of Leeds
Anand Pillay
Alex Wilkie University of Oxford

Speakers

Tuna Altinel
Matthias Aschenbrenner University of Illinois
Oleg Belegradek
Daniel Bertrand
Alexandre Borovik University of Manchester
Elisabeth Bouscaren
Jeffrey Burdges Rutgers, The State University of New Jersey
Luc Bélair UQAM - Université du Québec à Montréal
Enrique Casanovas
Gregory Cherlin
Raf Cluckers CNRS - Ecole Normale Superieure Paris
Paola D'Aquino
Tristram de Piro University of Edinburgh
Maximo Dickmann Université Paris 7 - Denis-Diderot
Mario Edmundo
Yuri Ershov Novosibirsk State University
Andrei Gabrielov Indiana University-Purdue University Indianapolis
Mikhail Gavrilovich University of Oxford
Assaf Hasson University of Oxford
Ehud Hrushovski Hebrew University of Jerusalem
Eric Jaligot Université Paris 7 - Denis-Diderot
Askold Khovanskii University of Toronto
Byunghan Kim
Franz-Viktor Kuhlmann University of Saskatchewan
Francois Loeser
Angus Macintyre
Dugald Macpherson University of Leeds
David Marker
Timothy Mellor Universität Regensburg
Christian Michaux Université de Mons
Rahim Moosa University of Waterloo
Ludomir Newelski Wroclaw University of Technology
Alf Onshuus University of Illinois
Margarita Otero Universidad Autonoma de Madrid
Yaacov Peterzil University of Haifa
Francoise Point
Alexander Prestel Universität Konstanz
Jean-Pierre Ressayre Université Paris 7 - Denis-Diderot
Jean-Philippe Rolin Université de Bourgogne
Thomas Scanlon University of California, Berkeley
Philip Scowcroft Wesleyan University
Michael Singer North Carolina State University
Sergei Starchenko University of Notre Dame
Charles Steinhorn Vassar College
Franz Timmesfeld Mathematisches Institut, Giessen
Lou van den Dries University of Illinois
Frank Wagner
Alex Wilkie University of Oxford
John Wilson

Show All

Model theory and applications to algebra and analysis

Programme theme

Preprints

A note on global $\it p$$^t$$^h$ powers of rigid analytic functions

Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields

On fields and colors

Supersimplicity and quadratic extensions

The theorem of the complement for sub-Pfaffian sets

Counting and Dimensions

Complex analytic geometry and analytic-geometric categories

Computing o-minimal topological invariants using differential topology

Compactness of cycle spaces and definability in o-minimal expansions of ${\mathbb R}_{an}$

Model theory of universal covering spaces of complex algebraic varieties

Exponential and Weierstrass Equations

Non-commutative geometry and new stable structures

Stable pseudofinite groups

Definable sets in algebraically closed valued fields: elimination of imaginaries

Reconstruction of homogeneous relational structures

Marshall's and Milnor's conjectures for preordered von Neumann regular rings

Definable relations in the real field with a distinguished subgroup of the unit circle

Definable groups and compact $\it p$-adic Lie groups

A remark on Zilber's pseudoexponentiation

Around unipotence in groups of finite Morley rank

Elimination of ramification I: the generalized stability theorem

Additive polynomials and their role in the model theory of valued fields

Strongly minimal groups in the theory of compact complex spaces

A generic identification theorem for $L^*$-groups of finite Morley rank

Involutions in groups of finite Morley rank and degenerate type

Generix never gives up

The rational points of a definable set

Sheaf cohomology in o-minimal structures

Workshops

An introduction to recent applications of model theory

Pure model theory

Model theory, algebraic and analytic geometry

Seminars

Organisers

Speakers

Supported By

Model theory and applications to algebra and analysis

Programme theme

Preprints

A note on global $\it p$$^t$$^h$ powers of rigid analytic functions

Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields

On fields and colors

Supersimplicity and quadratic extensions

The theorem of the complement for sub-Pfaffian sets

Counting and Dimensions

Complex analytic geometry and analytic-geometric categories

Computing o-minimal topological invariants using differential topology

Compactness of cycle spaces and definability in o-minimal expansions of ${\mathbb R}_{an}$

Model theory of universal covering spaces of complex algebraic varieties

Exponential and Weierstrass Equations

Non-commutative geometry and new stable structures

Stable pseudofinite groups

Definable sets in algebraically closed valued fields: elimination of imaginaries

Reconstruction of homogeneous relational structures

Marshall's and Milnor's conjectures for preordered von Neumann regular rings

Definable relations in the real field with a distinguished subgroup of the unit circle

Definable groups and compact $\it p$-adic Lie groups

A remark on Zilber's pseudoexponentiation

Around unipotence in groups of finite Morley rank

Elimination of ramification I: the generalized stability theorem

Additive polynomials and their role in the model theory of valued fields

Strongly minimal groups in the theory of compact complex spaces

A generic identification theorem for $L^*$-groups of finite Morley rank

Involutions in groups of finite Morley rank and degenerate type

Generix never gives up

The rational points of a definable set

Sheaf cohomology in o-minimal structures

Workshops

An introduction to recent applications of model theory

Pure model theory

Model theory, algebraic and analytic geometry

Seminars

Organisers

Speakers

Supported By

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