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Seminars (MAA)

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Event When Speaker Title Presentation Material
MAA 20th January 2005
10:30 to 12:30
An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application

Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.

E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.

M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.

J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)

MAA 20th January 2005
16:00 to 17:30
A Prestel On henselian valued fields
MAA 25th January 2005
10:00 to 12:00
T resplendent models and the Lascar group

Daniel Lascar introduced the group having now its name as a quotient of the group Aut(M) of all automorphisms of the structure M by the normal subgroup Autf(M) of all strong automorphims of M. This construction is independent of the choice of M as far as M is a big saturated model of the complete first-order theory T and can be considered as a model-theoretic invariant of T. It is assumed although it has not been checked in detail that the same construction works for special models M whose cardinality has a big cofinality. We will carry out the construction of the Lascar group in a more general class of models, the class of |T|^{+}-resplendent models. It turns out that the proofs are more easy in this more general setting. We will present the Lascar group as a pure group and we won't discuss its topology, but the topological part adapts easily also to this context.

MAA 25th January 2005
16:00 to 17:30
Groups definable in separably closed fields and algebraic groups: definable simplicity and "pseudo-simple" groups

Many questions which we can answer in the case of superstable groups remain open for stable not superstable groups: existence of an infinite definable abelian subgorup, link between definable simplicity and simplicity for example. Separably closed fields are a good place to test all these questions. In the case of definably simple groups, one is led rapidly to interesting questions about the classification of pseudo-reductive algebraic groups over separably closed fields (in the terminology of Tits).

MAA 27th January 2005
11:00 to 12:30
Expressive power of first-order logic for embedded finite models

I will discuss some questions posed by J.T.Baldwin and M.Benedikt [Trans. AMS 352 (2000)] concerning a relation between expressive power of first-order logic over finite models embedded in a model M and stability-theoretic properties of M.

MAA 27th January 2005
14:30 to 15:30
An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (ll)

Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.

E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.

M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.

J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)

MAA 27th January 2005
16:00 to 17:00
An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (ll)

Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.

E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.

M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.

J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)

MAA 1st February 2005
11:00 to 12:00
Stable pseudofinite groups

I will discuss recent work with Katrin Tent, in which it is shown that any stable pseudofinite group is soluble-by-finite. The `Mekler Construction' gives stable pseudofinite groups which are not abelian-by-finite. We do not know whether every stable pseudofinite group is nilpotent-by-finite.

MAA 1st February 2005
16:30 to 18:30
On the topological degree of functions definable in o-minimal structures (l)

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions.

Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group.

MAA 3rd February 2005
10:30 to 12:30
S Starchenko On the topological degree of functions definable in o-minimal structures (ll)
MAA 3rd February 2005
14:30 to 15:30
An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (III)

Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.

E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.

M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.

J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)

MAA 3rd February 2005
16:00 to 17:00
An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application (III)

Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.

E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.

M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.

J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)

MAA 8th February 2005
11:00 to 12:00
Weak generic types

I will discuss weak generic types in model-theoretic context and also the counterpart of this notion in a more general context of arbitrary groups and compact spaces. I will discuss the relationship with amenable groups, and also state several problems.

MAA 8th February 2005
14:30 to 15:30
Model theory of elliptic functions

The lectures will consider the structure of definitions in the theory of individual Weierstrass elliptic functions, paying as much attention as possible to uniformities as the function, or its associated lattice, varies. The setting will be that of an o-minimal expansion of the real field, and one will interpret therein the elliptic function on a semi-algebraic fundamental parallelogram. This will give o-minimality results, and with more work model- completeness results (related to work of Bianconi). The main novelty will be decidability results for some special elliptic functions, modulo a conjecture of Andre in transcendence theory. The proof will be analogous to that of Wilkie and the author for decidability of the real exponential. Peterzil and Starchenko began the model theory of families of elliptic functions, and showed in particular that this is interpretable in an o-minimal theory, so is certainly not undecidable in any Godelian way. I will outline what one currently knows about the problems of model-completeness and decidability for families of elliptic functions.

Bibliography

1.Bianconi, Ricardo. Some Results in the Model Theory of Analytic Functions, Thesis, Oxford ,1990. 2.Bertolin, Cristiana. Periodes de 1-motifs et transcendance (French) [Periods of 1-motives and transcendence], J. Number Theory 97 (2002), no. 2, 204--221.

3. Gabrielov, Andrei. Complements of subanalytic sets and existential formulas for analytic functions. Invent. Math. 125 (1996), no. 1, 1--12. 4. Gabrielov, Andrei; Vorobjov, Nicolai. Complexity of computations with Pfaffian and Noetherian functions. Normal forms, bifurcations and finiteness problems in differential equations, 211--250, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Pul., Dordrecht, 2004.

5. Macintyre, Angus; Wilkie, A. J. On the decidability of the real exponential field. Kreiseliana, 441--467, A K Peters, Wellesley, MA, 1996. 6. Peterzil, Ya'acov; Starchenko, Sergei. Uniform definability of the Weierstrass p Functions and Generalized Tori of Dimension One, to appear in Selecta Math.

7. Wilkie, A. J. On the theory of the real exponential field. Illinois J. Math. 33 (1989), no. 3, 384--408. 8. Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no. 4, 1051--1094.

MAA 8th February 2005
16:00 to 17:00
Model theory of elliptic functions

The lectures will consider the structure of definitions in the theory of individual Weierstrass elliptic functions, paying as much attention as possible to uniformities as the function, or its associated lattice, varies. The setting will be that of an o-minimal expansion of the real field, and one will interpret therein the elliptic function on a semi-algebraic fundamental parallelogram. This will give o-minimality results, and with more work model- completeness results (related to work of Bianconi). The main novelty will be decidability results for some special elliptic functions, modulo a conjecture of Andre in transcendence theory. The proof will be analogous to that of Wilkie and the author for decidability of the real exponential. Peterzil and Starchenko began the model theory of families of elliptic functions, and showed in particular that this is interpretable in an o-minimal theory, so is certainly not undecidable in any Godelian way. I will outline what one currently knows about the problems of model-completeness and decidability for families of elliptic functions.

Bibliography

1.Bianconi, Ricardo. Some Results in the Model Theory of Analytic Functions, Thesis, Oxford ,1990. 2.Bertolin, Cristiana. Periodes de 1-motifs et transcendance (French) [Periods of 1-motives and transcendence], J. Number Theory 97 (2002), no. 2, 204--221.

3. Gabrielov, Andrei. Complements of subanalytic sets and existential formulas for analytic functions. Invent. Math. 125 (1996), no. 1, 1--12. 4. Gabrielov, Andrei; Vorobjov, Nicolai. Complexity of computations with Pfaffian and Noetherian functions. Normal forms, bifurcations and finiteness problems in differential equations, 211--250, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Pul., Dordrecht, 2004.

5. Macintyre, Angus; Wilkie, A. J. On the decidability of the real exponential field. Kreiseliana, 441--467, A K Peters, Wellesley, MA, 1996. 6. Peterzil, Ya'acov; Starchenko, Sergei. Uniform definability of the Weierstrass p Functions and Generalized Tori of Dimension One, to appear in Selecta Math.

7. Wilkie, A. J. On the theory of the real exponential field. Illinois J. Math. 33 (1989), no. 3, 384--408. 8. Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no. 4, 1051--1094.

MAA 9th February 2005
11:30 to 13:00
Tomography in o-minimal structures based on constructible functions
MAA 10th February 2005
10:00 to 11:00
A Pillay Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (I)

In the first talk Pillay will discuss various notions of "connected component" for definable groups in arbitrary structures, formulate p-adic versions of the conjectures relating definably compact groups in o-minimal structures to compact Lie groups, and discuss other "good" theories of genericity which generalize stable group theory. In the second talk Onshuus will consider the case of definably compact groups definable in ordered vector spaces over ordered division rings . In the third talk, Onshuus will consider the case of definably compact groups definable in a p-adically closed field K and defined over Q_p. In the fourth talk Pillay will consider the case of elliptic curves defined over a p-adically closed field.

MAA 10th February 2005
11:30 to 12:30
Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (II)
MAA 10th February 2005
14:30 to 15:30
Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (III)
MAA 10th February 2005
16:00 to 17:00
A Pillay Connected components and generics for groups defined in o-minimal structures and p-adically closed fields (IV)
MAA 16th February 2005
10:30 to 12:30
Model theory of elliptic functions (II)
MAA 16th February 2005
14:30 to 15:30
Groups in the theory of compact complex spaces
MAA 16th February 2005
16:00 to 17:00
Groups in the theory of compact complex spaces
MAA 17th February 2005
11:30 to 12:30
P Kowalski E.c. Hasse fields - geometric axioms

Ziegler proved that the theory of fields with e commuting Hasse derivations has a model companion. This model companion eliminates quantifiers and is an expansion of the theory of separably closed fields of inseparable degree e. I wanted to find an axiomatization of this theory which gives criteria whether a system of Hasse-differential equations and inequalities is solvable (e.g. X'=X and X\neq 0 is not solvable). Such axioms are given in terms of higher prolongation spaces (related to arc spaces). I also found axioms of the theories of e.c. fields with truncated Hasse derivations. This theories are higher order analogues of Wood's DCF-p theories.

Related Links

MAA 17th February 2005
14:30 to 15:30
Some exercises in model theory of mathematical physics
MAA 17th February 2005
16:00 to 17:00
Some exercises in model theory of mathematical physics
MAA 23rd February 2005
10:30 to 12:30
Model theory of elliptic functions

The lectures will consider the structure of definitions in the theory of individual Weierstrass elliptic functions, paying as much attention as possible to uniformities as the function, or its associated lattice, varies. The setting will be that of an o-minimal expansion of the real field, and one will interpret therein the elliptic function on a semi-algebraic fundamental parallelogram. This will give o-minimality results, and with more work model- completeness results (related to work of Bianconi). The main novelty will be decidability results for some special elliptic functions, modulo a conjecture of Andre in transcendence theory. The proof will be analogous to that of Wilkie and the author for decidability of the real exponential. Peterzil and Starchenko began the model theory of families of elliptic functions, and showed in particular that this is interpretable in an o-minimal theory, so is certainly not undecidable in any Godelian way. I will outline what one currently knows about the problems of model-completeness and decidability for families of elliptic functions.

Bibliography

1.Bianconi, Ricardo. Some Results in the Model Theory of Analytic Functions, Thesis, Oxford ,1990. 2.Bertolin, Cristiana. Periodes de 1-motifs et transcendance (French) [Periods of 1-motives and transcendence], J. Number Theory 97 (2002), no. 2, 204--221.

3. Gabrielov, Andrei. Complements of subanalytic sets and existential formulas for analytic functions. Invent. Math. 125 (1996), no. 1, 1--12. 4. Gabrielov, Andrei; Vorobjov, Nicolai. Complexity of computations with Pfaffian and Noetherian functions. Normal forms, bifurcations and finiteness problems in differential equations, 211--250, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Pul., Dordrecht, 2004.

5. Macintyre, Angus; Wilkie, A. J. On the decidability of the real exponential field. Kreiseliana, 441--467, A K Peters, Wellesley, MA, 1996. 6. Peterzil, Ya'acov; Starchenko, Sergei. Uniform definability of the Weierstrass p Functions and Generalized Tori of Dimension One, to appear in Selecta Math.

7. Wilkie, A. J. On the theory of the real exponential field. Illinois J. Math. 33 (1989), no. 3, 384--408. 8. Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no. 4, 1051--1094.

MAA 23rd February 2005
14:30 to 15:30
Vanishing lemmas and transcendence

Most transcendence proofs rely on a collection of statements of a purely geometric nature: the ``zero estimates" (or ``vanishing lemmas"), which ensure injectivity (or surjectivity) of an evaluation map on sections of a vector bundle at zero-dimensional subschemes of the base. We shall present (and time permitting, prove) them in a unified way, covering two of the main areas of transcendence: periods of one-motives (Baker-Wustholz theory), and values of solutions of differential equations (Siegel-Shidlovsky theorem).

A new proof of the latter theorem will also be discussed, as well as a $q$-difference analogue of the corresponding ``vanishing lemma".

In each case, the vanishing lemmas exhibit a geometric obstruction which is the key not only to the trancendence results themselves, but also to their conjectural generalizations (Schanuel, Grothendieck, Andr\'e).

Reference:

D. Bertrand: Le th\'eor\`eme de Siegel-Shidlovsky revisit\'e, Pr\'epublication de l'Institut de Math\'ematiques de Jussieu, No 390, Mai 2004.

MAA 23rd February 2005
16:00 to 17:00
New proofs of the Siegel-Shidlovsky theorem
MAA 24th February 2005
11:00 to 12:00
A remark on pseudo-exponentiation

There are a number of interesting open problems about definability in the field of complex numbers with exponentiation. Zilber has proposed a novel approach. He constructed a nonelementary class of exponential algebraically closed fields and showed that in this class definable subsets of the field are countable or co-countable. He also showed the class is categorical in all uncountable cardinalities. The natural question is whether the complex numbers are the unique model in this class of size continuum.

In this talk I will show that, assuming Schanuel's Conjecture, the simplest case of Zilber's strong exponential closure axiom is true in the complex numbers.

MAA 24th February 2005
16:00 to 17:30
Quadratic forms in models of weak arithmetic
MAA 2nd March 2005
11:30 to 12:30
Black box groups and groups of finite Morley rank

I will discuss some strange analogies between some probabilistic methods for recognition of "black box" finite groups and the theory of groups of finite Morley rank. As an application and illustration, I will give a short elementary proof of the following fact.

Let G be a group of finite Morley rank. If a Sylow 2-subgroup S of G is cyclic then S intersects trivially with the connected component of G.

MAA 2nd March 2005
16:00 to 17:30
L Belair Approximation in difference valuation rings

I will discuss a result analogous to a theorem of Greenberg on an approximation property in henselian discrete valuation rings, but for polynomial equations involving also a fixed automorphism. In the case of Witt vectors and their Frobenius, a nullstellensatz for these equations can be deduced.

MAA 3rd March 2005
11:00 to 12:30
Sylow theory for 'p=0' in groups of finite Morley rank
MAA 3rd March 2005
16:00 to 17:30
G Cherlin Conjugacy of good tori

We discuss a conjugacy theorem for maximal "good" tori in groups of finite Morley rank. This result, which generalizes the conjugacy of maximal tori in algebraic groups, plays a role in the classification of simple groups of finite Morley rank and even type. The proof is based on a "generic covering" lemma which has a flavor similar to, but looser than, a conjugacy theorem.

Related Links

MAA 9th March 2005
11:30 to 12:30
Real and motivic integration and oscillating functions

At a basic introductory level, some new results on real and motivic integrals are presented and put in context. It concerns results on Fouriers transforms of analytic functions and on oscillating integrals in general. Instead of giving a rigorous introduction to motivic integrals, we illustrate them by explaining the general philosophy of motivic oscillating integrals. No preliminary knowledge is required.

MAA 9th March 2005
16:00 to 17:30
A Hasson Amalgamation with predimension and collapse
MAA 10th March 2005
11:30 to 12:30
Carter subgroups of groups of finite Morley rank

Most transcendence proofs rely on a collection of statements of a purely geometric nature: the ``zero estimates" (or ``vanishing lemmas"), which ensure injectivity (or surjectivity) of an evaluation map on sections of a vector bundle at zero-dimensional subschemes of the base. We shall present (and time permitting, prove) them in a unified way, covering two of the main areas of transcendence: periods of one-motives (Baker-Wustholz theory), and values of solutions of differential equations (Siegel-Shidlovsky theorem).

A new proof of the latter theorem will also be discussed, as well as a $q$-difference analogue of the corresponding ``vanishing lemma".

In each case, the vanishing lemmas exhibit a geometric obstruction which is the key not only to the trancendence results themselves, but also to their conjectural generalizations (Schanuel, Grothendieck, Andr\'e).

Reference:

D. Bertrand: Le th\'eor\`eme de Siegel-Shidlovsky revisit\'e, Pr\'epublication de l'Institut de Math\'ematiques de Jussieu, No 390, Mai 2004.

MAA 10th March 2005
16:00 to 17:15
On commutative von Neumann regular rings with an automorphism
MAAW04 16th March 2005
10:30 to 11:20
T Altinel The programme for classification of groups of finite morley rank
MAAW04 16th March 2005
12:00 to 13:00
G Cherlin Ultraproducts and complex reflection groups
MAAW04 16th March 2005
14:00 to 14:50
E Jaligot Good tori and bad fields
MAAW04 16th March 2005
15:00 to 15:50
The algebraicity conjecture: some thoughts against
MAAW04 16th March 2005
16:30 to 17:30
Finite group theory: The next generation

The underlying methodology of the classification theory of groups of finite Morley rank is the systematic use of ideas from the Classification of Finite Simple Groups (CFSG), both from the original (first generation) papers and from the later revisionism, especially from the Third Generation Proof.

The aim of these notes is to outline some problems arising from an attempt to reverse the transfer of ideas and formulate a fragment of CFSG closest to the classification of groups of finite Morley rank and even type.

MAA 17th March 2005
10:00 to 12:00
Identifications of Lie type groups
MAA 22nd March 2005
11:30 to 12:30
T Altinel Simple groups without involutions

Bad groups are simple groups of finite Morley rank whose proper, definable, connected subgroups are nilpotent. Their existence is a well-known open problem in the analysis of simple groups of finite Morley rank and would refute the Cherlin-Zilber algebraicity conjecture.

One of the striking properties of bad groups is that they do not have involutions. Another one is that their proper, maximal, definable, connected subgroups intersect trivially. These properties are shared by other classes of simple groups of finite Morley rank that are not bad in the above sense. In a sense there is more than one notion of badness. In the talk I will mention some of these bad classes and make some remarks about the reduction of some open problems to a bad configuration.

MAA 22nd March 2005
15:30 to 17:00
Spectra of triangulated categories

Model theory for ``finitary'' (locally finitely presented) abelian categories is well-developed. In particular to such a category one may associate its Ziegler spectrum, sheaf of locally definable scalars, et cetera. I describe how model theory and associated structures have been extended to compactly generated (``finitary'') triangulated categories and how this has been used by various people.

MAA 23rd March 2005
11:00 to 12:30
F Loeser Enlarging motivic constructible functions to exponentials

I shall present recent work with Raf Cluckers about adding exponential functions to constructible motivic functions. We show this enlarged class of functions is stable under integration and develop a Fourier transformation in this setting. We shall end the talk by stating a version of the Ax-Kochen-Ersov Theorem for these functions.

MAA 23rd March 2005
15:30 to 17:00
J Koenigsmann Galois axioms for fields

I will sketch the proof of a new result in birational anabelian geometry: Almost any perfect field K is determined up to isomorphism by the absolute Galois group of th rational function field K(t) over K. This gives rise to an axiomatization of the elementary theory of K by axiomatizing the complete system of finite quotients of this absolute Galois group. As application, I will indicate consequences for the decidability of the perfect hull of F_p((t)).

MAAW01 29th March 2005
10:00 to 11:00
Introduction
MAAW01 29th March 2005
11:30 to 12:30
A Pillay Stability, differential fields, and related structures

We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations.

MAAW01 29th March 2005
14:00 to 15:00
Stability, differential fields, and related structures

We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations.

MAAW01 29th March 2005
15:30 to 16:30
Model theory of algebraically closed valued fields

This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination.

1. D. Haskell, E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint.

2. D. Haskell E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part II: stable domination and independence.'

MAAW01 29th March 2005
16:30 to 17:30
Model theory of algebraically closed valued fields

This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination.

1. D. Haskell, E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint.

2. D. Haskell E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part II: stable domination and independence.'

MAAW01 30th March 2005
09:00 to 10:00
A Pillay Stability, differential fields, and related structures

We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations.

MAAW01 30th March 2005
10:00 to 11:00
Stability, differential fields, and related structures

We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations.

MAAW01 30th March 2005
11:30 to 12:30
Model theory of algebraically closed valued fields

This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination.

1. D. Haskell, E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint.

2. D. Haskell E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part II: stable domination and independence.'

MAAW01 30th March 2005
14:00 to 15:00
Model theory of algebraically closed valued fields

This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination.

1. D. Haskell, E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint.

2. D. Haskell E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part II: stable domination and independence.'

MAAW01 30th March 2005
15:30 to 16:30
JB Bost Some problems arising from the Diophantine study of algebraic foliations

This survey talk will be devoted to various problems arising in the study of Diophantine properties of algebraic foliations.

Hopefully, I will explain (1) how algebraic foliations naturally enters into arithmetic geometry, (2) some known results established notably by means of Diophantine approximation techniques (concerning in particular the Grothendieck-Katz conjecture and its generalizations), and (3) discuss some Diophantine conjectures/problems, and some problems in (differential-)algebraic geometry arising from the use of Diophantine approximation techniques. This last part should present various issues where I expect model theory to be relevant.

MAAW01 31st March 2005
09:00 to 10:00
A Pillay Stability, differential fields, and related structures

We will discuss stability theory and the model theory of differential fields, difference fields, separably closed fields, and compact complex manifolds. We will survey the applications to and connections with diophantine geometry, complex geometry, and the arithmetic of differential equations.

MAAW01 31st March 2005
10:00 to 11:00
Model theory of algebraically closed valued fields

This tutorial of 5 lectures will be an exposition of a proof of elimination of imaginaries for the theory of algebraically closed valued fields (ACVF), when certain extra sorts from M^eq (coset spaces) are added. The proof will be based on that of [1], though further recent ideas of Hrushovski may be incorporated. The tutorial will begin with a general account of the basic model theory of ACVF and the notion of elimination of imaginaries, and will end with further developments from [2]: in particular, the notion of stable domination.

1. D. Haskell, E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries', preprint.

2. D. Haskell E. Hrushovski, H.D. Macpherson, `Definable sets in algebraically closed valued fields. Part II: stable domination and independence.'

MAAW01 31st March 2005
11:30 to 12:30
A local Andre-Oort conjecture via ACFA
MAAW01 31st March 2005
14:00 to 15:00
I Fesenko Two dimensional arithmetic geometry and nonstandard mathematics

This is review of a variety of potential and actual applications of nonstandard mathematics to arithmetic geometry; for the text see www.maths.nott.ac.uk/personal/ibf/rem.pdf

Related Links

MAAW01 31st March 2005
15:30 to 16:30
On a theorem of Simpson
MAAW01 1st April 2005
09:00 to 10:00
Zariski-type structures
MAAW01 1st April 2005
10:00 to 11:00
Zariski-type structures
MAAW01 1st April 2005
11:30 to 12:30
CW Henson & A Berenstein Model theory for metric structures

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc.

The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role.

The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.

MAAW01 1st April 2005
14:00 to 15:00
CW Henson & A Berenstein Model theory for metric structures

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc.

The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role.

The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.

MAAW01 1st April 2005
15:30 to 16:30
Closed trajectories of plane systems of ODE's, moments, iterated integrals, and compositions - with the stress on some "formal" aspects

Recently some new connections have been found between the structure of the closed trajectories of plane systems of ODE's and certain questions in classical analysis and algebra. In particular, this concerns the vanishing problem of certain moment-like expressions and of iterated integrals on one side, and the structure of the composition factorization of analytic functions on the other. These connections provide a useful information on the Center conditions (for all the trajectories around a critical point to be closed) and on the distribution of the isolated closed trajectories (limit cycles). Some of the above effects are rather "structural" or "formal" in their nature, and, as the preliminary considerations show, they possibly can be considered in the framework of the Model Theory.

MAAW01 4th April 2005
09:00 to 10:00
Zariski-type structures
MAAW01 4th April 2005
10:00 to 11:00
Zariski-type structures
MAAW01 4th April 2005
11:30 to 12:30
CW Henson & A Berenstein Model theory for metric structures

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc.

The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role.

The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.

MAAW01 4th April 2005
14:00 to 15:00
CW Henson & A Berenstein Model theory for metric structures

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc.

The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role.

The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.

MAAW01 4th April 2005
15:30 to 16:30
Y Raynaud Ultrapowers and ultraroots in Banach spaces theory
MAAW01 5th April 2005
09:00 to 10:00
Zariski-type structures
MAAW01 5th April 2005
10:00 to 11:00
CW Henson & A Berenstein Model theory for metric structures

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally, there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A,B) = m(A*B), where * is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*-algebras, etc.

The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role.

The development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.

MAAW01 5th April 2005
11:30 to 12:30
F Loeser Operations on constructible functions (I)

Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation.

MAAW01 5th April 2005
14:00 to 15:00
F Loeser Operations on constructible functions (II)

Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation.

MAAW01 5th April 2005
15:30 to 16:30
The theorem of the complement for nested sub-Pfaffian sets

We show that the Pfaffian closure of an o-minimal structure with analytic cell decomposition is model complete. This is achieved by proving a theorem of the complement for nested sub-Pfaffian sets over the o-minimal structure in question.

MAAW01 5th April 2005
16:30 to 17:30
Nonstandard 1-dimensional tori are locally modular

In earlier work we showed how a uniform family of biholomorphisms of 1-dimensional complex tori and algebraic cubics is definable in $R_{an,exp}$, covering in this way all smooth cubics, but not all tori.

As a corollary, one obtains in elementary extensions of $R_{an,exp}$ some ``nonstandard'' 1-dimensional tori. I will discuss the induced analytic structure on these tori and show that these nonstandard tori are strongly minimal and locally modular.

MAAW01 6th April 2005
09:00 to 10:00
Isotriviality criteria for families of non-algebraic compact K\" ahler manifolds, and model-theoretic nonmultidimensionality of the class C.

A question raised by A.Pillay is whether the class $\calC$ of compact complex manifolds $F$ bimeromorphic to some compact K\" ahler manifold $F'$ (depending on $F$) is nonmultidimensional in the model theoretic sense.

Specialised to the case of {\it simple} manifolds $F$ (those which are not covered by proper compact analytic subsets, and of complex dimension at least $2$), this means that if $f:X\to S$ is a surjective holomorphic map with $X$ in $\calC$, and general smooth fibre $X_s$ simple, then $f$ is {\it isotrivial}, which means that any two such fibres are isomorphic.

We show that this is indeed the case for (most of) the known simple manifolds: the non-projective hyperk\" ahler manifolds, and the general complex tori.

The talk is intended for non-specialists in complex geometry.

MAAW01 6th April 2005
10:00 to 11:00
Analogues of Hilbert's tenth problem

This presentation is part of the tutorial on Hilbert's Tenth Problem, which will be presented jointly with Thanases Pheidas. The aim is to give a comprehensive overview of results concerning decidability questions related to solving diophantine equations. For more information I refer to the abstract submitted by Thanases Pheidas.

MAAW01 6th April 2005
11:30 to 12:30
Analogues of Hilbert's tenth problem

This presentation is part of the tutorial on Hilbert's Tenth Problem, which will be presented jointly with Thanases Pheidas. The aim is to give a comprehensive overview of results concerning decidability questions related to solving diophantine equations. For more information I refer to the abstract submitted by Thanases Pheidas.

MAAW01 6th April 2005
14:00 to 15:00
Hilbert's tenth problem for function fields

We will discuss how elliptic curves of rank one can be used to prove undecidability of Hilbert's Tenth Problem for function fields of characteristic zero, such as R(t) and C(t_1,...,t_n) (n at least 2).

The approach for function fields of positive characteristic is very different, and we will sketch some of the methods used there.

MAAW01 6th April 2005
15:30 to 16:30
F Loeser Operations on constructible functions (III)

Constructible functions appear under various guises in many different setting. Starting from real and o-minimal geometry, we shall journey to the p-adic and motivic setting. We plan to focus mainly on the construction of direct image (pushforward) but we intend also taking some time to discuss other operations, like the Fourier transformation.

MAAW01 6th April 2005
16:30 to 17:30
Real integration of oscillating functions

We present new results on real integration of oscillating functions, for example, Fourier transforms of subanalytic functions; it is joint work with Aschenbrenner and Rolin. We begin to control very good the transcendental functions we have to add, (namely, oscillating versions of basic Abelian integrals), in order to describe these parameterized integrals. The method is in principle algorithmic, with a similar algorithm as to compute motivic oscillating integrals. Yet, conjectures linking real and motivic integrals remain unsolvable.

MAAW01 7th April 2005
09:00 to 09:30
Analytic Zariski geometries and the Hrushovski Collapse

In recent years much progress has been made along the 'Zilber Programme' intended to explain Hrushovski-type constructions in terms of analytic geometry. These results, however, have only dealt with Hrushovski-type structures of infinite rank. we will show that considering appropriate expansions of the infinite rank ab initio structrue, Hrushovski's strongly minimal sets can be obtained as infinitesimal neighborhoods (in terms of specializations) of carefully enough chosen points. We will also show that these results can be extended to other Hrushovski-type structres, most notably to ones supporting a structure of an algebraically closed field.

If time allows we will discuss briefly: (a) The obstacles in generalizing these results to more complicated structures (e.g. the fusion of two strongly minimal Zariski geometries). (b) The nature of a possible analytic prototype in which our interpretation of the collapse could be described in analytic terms.

MAAW01 7th April 2005
09:30 to 10:00
S Barbina Reconstruction of homogeneous relational structures

Reconstruction results give conditions under which the automorphism group of a structure determines the structure up to bi-interpretability or bi-definability. Here we examine a large class of omega-categorical combinatorial structures, which was isolated by Herwig and contains K_n-free graps, k-hypergraphs and Henson digraphs. Using a Baire category approach we show how to obtain reconstruction for this class, proving that a reconstruction condition, developed by M. Rubin, holds. The method rests on the existence of a generic pair of automorphisms.

MAAW01 7th April 2005
10:00 to 10:30
Arc-properties of functions definable in o-minimal structures

There are two topics at the base of this talk: arc-analytic functions (mostly studied in the subanalytic setting) and the relationships between o-minimal structures and Hardy fields. They lead us to study the following problem: for a function definable in some o-minimal structure (over the field of reals), what kind of property may be detected in restriction to some "small" space of definable arcs. We shall prove for instance that for most classical polynomially bounded o-minimal structures, continuity is equivalent to continuity on restriction to polynomial arcs. We will finish with problems of arc-definability for which we will give several open questions.

MAAW01 7th April 2005
10:30 to 11:00
The theory of a Hilbert space with a generic automorphism

I will present an axiomatization for the continuous theory of a generic unitary representation of the group Z (integers), i.e. the theory of a Hilbert space with a generic automorphism. This is also the theory of the regular unitary representation of Z. I will describe the properties of this theory (e.g. superstable, non-multidimensional, non-omega stable) and present a full characterization of types using the spectral decomposition theorem. This is a joint work with Itay Ben-Yaacov and Moshe Zadka.

MAAW01 7th April 2005
11:30 to 12:00
T Mellor An Euler characteristic for real closed valued fields

I shall consider a real closed valued field together with certain extensions. There are four classes of definable set: considering both structures, both as fields and valued fileds. The relationships between these classes allow one to prove the existance of an Euler characteristic for the (valued field) definable sets of the origional structure.

Related Links

MAAW01 7th April 2005
15:30 to 16:30
Application of arc spaces to the model theory of fields with commuting derivations
MAAW01 7th April 2005
16:30 to 17:30
Model theoretic topics in valued fields I
MAAW01 8th April 2005
09:00 to 10:00
On the quasi-minimality of certain expansions of the complex field

I shall consider various natural pre-geometries on expansions of the complex field inspired by the work of Peterzil and Starchenko on the development of complex analysis within an o-minimal structure. Unfortunately, I still cannot realise my original aim of using such methods to show that the complex exponential field is quasi-minimal (ie every definable subset of the complex numbers is either countable or co-countable) but I can at least show that we do have quasi-minimality if we only allow the operations of raising to real powers. (I should point out, however, that if one assumes postive answers to certain conjectures from diophantine geometry and transcendence theory then Zilber has already shown this,and more. See 'Raising to powers in algebraically closed fields', J Math Logic vol 3(2), 2003, 217-238.)

Another aspect of the talk is that it gives some sort of answer to a question of Hrushovski (private communication a couple of years ago) which asks whether elimation of quantifiers for algebraically closed fields may be naturally deduced from elimination of quantifiers for real closed ordered fields. I show that even though the definable closure operator on an o-minimal structure (expanding a real closed field) does not satisfy the modular law, it may nevertheless be linearised and thereby induce a pregeometry on an expansion of its algebraic closure. The Cauchy-Riemann equations play a role here so that, for example, in the pure field case this pregeometry IS algebraic closure (and not,say, "algebraic closure of the set of real and imaginary parts").

MAAW01 8th April 2005
10:00 to 10:30
Grothendieck-Katz conjecture for elliptic curves

We will consider a special case of Grothendieck-Katz conjecture given by the logarithmic derivative.

MAAW01 8th April 2005
10:30 to 11:00
Schanuel conditions for Weierstrass differential equations

I will discuss a version of Schanuel's conjecture for Weierstrass equations in differential fields. This gives a necessary and sufficient condition for a system of Weierstrass differential equations to have a solution.

The necessity part builds on work by James Ax, who proved the equivalent statement for the exponential equation, and by Brownawell and Kubota who proved an analogue for complex power series. The sufficiency part builds on work of Cecily Crampin.

I hope also to show connections to the theory of the complex Weierstrass p-functions and to structures constructed via Hrushovski's amalgamation technique.

MAAW01 8th April 2005
11:30 to 12:00
Mordell-Lang theorem for Drinfeld modules and minimal groups in the theory of separably closed fields

We study the rings of quasi-endomorphisms of certain minimal groups in the theory of separably closed fields. These groups are associated to Drinfeld modules of finite characteristic. Based on our results, we are able to prove a Mordell-Lang statement for Drinfeld modules of finite characteristic.

MAAW01 8th April 2005
12:00 to 12:30
Classes of (Lie) Differential Fields without Model Companions

A Lie differential field is a field F given with some Lie algebra L acting on F as derivations. If we fix L we get the class of L-differential fields, which has amalgamation when the characteristic is 0. If in addition L is finite dimensional over F then the above class has a model companion (and hence a model completion).

However if L isn't finitely presented (at least locally), i.e. if there is a finitely generated sub Lie algebra of L without a finite presentation, then the class of L-differential fields does NOT have a model companion. I will describe how this result is proved by producing a non-eliminable quantifier, using a system of linear PDEs. The question of a tighter connection between companionability and finite presentability remains open.

MAAW01 8th April 2005
14:00 to 15:00
Model theoretic topics in valued fields II
MAAW01 8th April 2005
15:30 to 16:30
Model theory of elliptic functions: model completeness, uniformity, decidability
MAA 13th April 2005
14:00 to 15:00
M Edmundo Sheaf cohomology in arbitrary o-minimal structures

In joint work with Nicholas Peatfield, we show the existence of sheaf cohomology in arbitrary o-minimal structures satisfying the o-minimal Eilenberg-Steenrod axioms.

MAA 13th April 2005
15:30 to 16:30
M Edmundo Sheaf cohomology in arbitrary o-minimal structures

In joint work with Nicholas Peatfield, we show the existence of sheaf cohomology in arbitrary o-minimal structures satisfying the o-minimal Eilenberg-Steenrod axioms.

MAA 14th April 2005
11:30 to 12:30
Galois cohomology for surgical fields

In 1995, Pillay and Poizat introduced the notion of a surgical structure (translated from the french chirurgicale), a structure such that to each definable set there was an element of a poset attached to it (denoted h by its dimension) in such a way that if there was a partition of a set X into finite pieces and each of these pieces could be sent via some finite-to-one map to another set Y, then dim(X)is bounded above by dim(Y). Moreover, an accumulation character was required on this assignment, i.e, given a definable equivalence relation on a definable set, there were only finitely many classes of dimension the dimension of the ambient set.

Under these weak assumptions, they proved that a field interpreted in such a structure is perfect and has small absolute Galois group.

I will show how these techniques can be extended to consider certain Galois cohomological groups relative to the field, and discuss their geometrical meaning.

The talk is intended to be self contained and for a general audience in Model Theory.

MAA 14th April 2005
15:30 to 17:00
A twisted Chebotarev Theorem
MAA 20th April 2005
11:30 to 12:30
Counting in bad fields
MAA 20th April 2005
15:30 to 17:00
On the model theory of universal covers of algebraic varieties
MAA 21st April 2005
13:30 to 15:00
T de Piro A non-standard Bezout theorem
MAA 21st April 2005
15:30 to 17:00
A Khovanskii Solvability and unsolvability of equations in finite terms

My talk will be dedicated to the question of unsolvability of equations in finite terms. This question has a rich history.

First proofs of unsolvability of algebraic equations by radicals were found by Abel and Galois. Thinking on the problem of explicit indefinite integration of an algebraic differential form, Abel founded the theory of algebraic curves. Liouville continued Abel's work and proved the non-elementarity of indefinite integrals of many algebraic and elementary differential forms. Liouville was also the first to prove the unsolvability of many linear differential equations by quadratures.

The relationship between the solvability by radicals and the properties of a certain finite group goes back to Galois. The notion of finite group introduced by Galois was motivated exactly by this question. Sophus Lie introduced the notion of continuous transformation group while trying to solve differential equations explicitly and to reduce them to a simper form. With each linear differential equation, Picard associated its Galois group, which is a Lie group (and, moreover, a linear algebraic group). Picard and Vessiot showed that this particular group is responsible for the solvability of equations by quadratures. Kolchin developed the theory of algebraic groups and elaborated the Picard--Vessiot theory.

Arnold discovered that many classical mathematical questions are unsolvable for topological reasons. In particular, he showed that the general algebraic equation of degree at least 5 is unsolvable by radicals exactly for topological reasons. While developing Arnold's approach, in the beginning of 70s, I constructed a peculiar one-dimensional topological variant of the Galois theory. According to this theory, the way how the Riemann surface of an analytic function covers the complex plane can obstruct the representability of this function by explicit formulas. In this way, the strongest known results on non-expressibility of functions by explicit formulas are obtained. Recently, I succeeded to generalize these topological results to the multivariable case.

MAA 27th April 2005
10:30 to 12:30
Model theoretic topics in valued fields III
MAA 27th April 2005
15:30 to 17:00
A Khovanskii On a one-dimensional topological variant of Galois theory
MAA 28th April 2005
09:30 to 10:30
Model theory of multi-valued fields
MAA 28th April 2005
11:00 to 12:30
The Marshall and Milnor conjectures for formally real von Neumann regular rings

[Joint work with F. Miraglia, Univ. of Sao Paulo, Brazil]

I'll outline a proof of Milnor's Witt ring conjecture and of Marshall's signature conjecture for the (reduced) theory of quadratic forms on rings of the type mentioned in the title. The proof is carried out in the context of (reduced) special groups --an axiomatic theory of quadratic forms--, and is an application of k-theoretic techniques in this context (developed in a forthcoming paper in Journal of Pure and Applied Algebra), which I will explain in the talk. The proof also resorts to our solution of Lam's conjecture for preordered fields.

MAA 28th April 2005
17:00 to 18:00
Model theory of multi-valued fields
MAA 3rd May 2005
11:00 to 12:30
C Michaux Bisimulations for o-minimal hybrid systems

An hybrid system roughly consists in finitely many continuous dynamical systems coupled with a finite automaton which governs the discrete transitions between the continuous dynamical systems. Behaviors of hybrid systems model air traffic control, flight behavior... The main decision problem for hybrid systems is the reachability problem: given a set I of initial states, a set F of final states and a set A of states to avoid, does the system securely behave on inputs from I, i.e. does the system always reach a final state, avoiding the set of prohibited states?

A first global approach to this question is to build finite bisimulations of the system. We will show how to prove existence of finite bisimulations when the hybrid system is definable in a o-minimal theory. This is joint work with Thomas Brihaye, Cedric Riviere and Christophe Troestler. Our results generalizes previous results of Lafferriere, Pappas and Sastry.

Recent advances on the effectivenesss of our construction have been obtained by Maragarita Korovina and Nicolai Vorobjov in the pfaffian case.

MAA 3rd May 2005
16:30 to 18:00
Betti numbers of sets defined by quantifier-free formulas

(Joint work with N. Vorobjov)

Upper bounds for the Betti numbers of real algebraic sets were obtained by Oleinik-Petrovskii (1949), Milnor (1964) and Thom (1965). These bounds, based on Morse theory, were single exponential in the number of variables. Basu (1999) extended these results to real semialgebraic sets defined by equations and non-strict inequalities. However, the best previously known upper bounds for general semialgebraic sets were double exponential. Gabrielov and Vorobjov (2005) obtained a single exponential upper bound on the Betti numbers of a general semialgebraic set. Given a semialgebraic set X, another semialgebraic set Y defined by equations and non-strict inequalities is constructed, with the same Betti numbers as X. Basu's theorem applied to Y provides the upper bound for the Betti numbers of X. The method easily generalizes to non-algebraic functions, such as Pfaffian functions.

MAA 4th May 2005
10:00 to 11:00
Definably compact groups
MAA 4th May 2005
11:30 to 12:30
Definably compact groups
MAA 4th May 2005
15:30 to 17:00
Betti numbers of sets defined by formulas with quantifiers

(Joint work with D. Novikov, B. Shapiro)

In his famous Treatise on Electricity and Magnetism (1873) J.C. Maxwell considered the following problem: How many equilibrium points a potential of n fixed point charges in R^3 may have? Maxwell applied Morse theory (developed 50 years later) to approach this problem. He claimed that there should be at most (n-1)^2 equilibrium points. In a footnote to the third edition (1891) of the Maxwell's book, J.J. Thomson stated that he could not find any proof of this. In the talk, recent progress in this direction will be reported, based on fewnomial theory and Voronoi diagrams.

MAA 11th May 2005
11:30 to 12:30
S Kuhlmann Truncation integer parts of valued fields

We study the problem of constructing complements to the valuation ring of a valued field, which are $k$-algebras over the residue field $k$.

These $k$-algebras have many important arithmetic properties, and are closely related to truncation integer parts.

We work out the case of henselian fields and study the canonical integer part $\Neg(F)\oplus\Z$ of any truncation closed subfield $F$ of the field of power series $k((G))$, where $\Neg(F):=F\cap k((G^{<0}))$. In particular, we prove that $k((G^{<0}))\oplus\Z$ has (cofinally many) prime elements for any ordered divisible abelian group $G$.

Addressing a question in a paper of Berarducci, we show that every truncation integer part of a non-archimedean exponential field has a cofinal set of irreducible elements. Finally, we apply our results to two important classes of exponential fields:exponential algebraic power series and exponential-logarithmic power series.

MAA 11th May 2005
14:00 to 15:00
S Kuhlmann Truncation integer parts of valued fields

We study the problem of constructing complements to the valuation ring of a valued field, which are $k$-algebras over the residue field $k$.

These $k$-algebras have many important arithmetic properties, and are closely related to truncation integer parts.

We work out the case of henselian fields and study the canonical integer part $\Neg(F)\oplus\Z$ of any truncation closed subfield $F$ of the field of power series $k((G))$, where $\Neg(F):=F\cap k((G^{<0}))$. In particular, we prove that $k((G^{<0}))\oplus\Z$ has (cofinally many) prime elements for any ordered divisible abelian group $G$.

Addressing a question in a paper of Berarducci, we show that every truncation integer part of a non-archimedean exponential field has a cofinal set of irreducible elements. Finally, we apply our results to two important classes of exponential fields:exponential algebraic power series and exponential-logarithmic power series.

MAA 12th May 2005
11:30 to 12:45
P Scowcroft Non-negative solvability of linear equations in ordered groups

Over an ordered field, the class of solution sets of finite systems of homogeneous weak linear inequalities is closed under projection, and this fact yields a simple proof of Farkas' theorem characterizing the nonnegative solvability of systems of linear equations. If one generalizes the notion of inequality to that of congruence inequality--which combines an inequality with a congruence in a special way--then one may exploit techniques from model theory to prove that over any ordered Abelian group, the class of solution sets of finite systems of congruence inequalities is closed under projection. Thus ordered rings other than fields obey versions of Farkas' theorem, and this talk will describe such results for dense subrings of the reals and for the integers.

MAA 12th May 2005
15:30 to 17:00
C Michaux Dimension theory for BSS - r.e. sets
MAA 18th May 2005
11:30 to 12:30
M Kim Relative computability for curves

We will discuss the relationship between a number of computability/decidability problems for equations in two variables.

MAA 19th May 2005
11:30 to 12:30
T Mellor Elimination of imaginaries for real closed valued fields
MAA 19th May 2005
14:00 to 15:00
Topological properties of sets definable in weakly o-minimal structures

A first order structure M equipped with a dense linear ordering is called weakly o-minimal iff all definable subsets of M are finite unions of convex sets. In the first part of the talk we will discuss some properties of the topological dimension of sets definable in weakly o-minimal structures. This will constitute a basis for the second part which will be focused on the problem of topologisation of groups, group actions and fields definable in weakly o-minimal structures.

MAA 19th May 2005
15:30 to 16:30
Groups and fields definable in weakly o-minimal structures
MAA 25th May 2005
10:30 to 12:30
J-P Rolin Non-oscillating solutions of differential equations and o-minimality

We study the behaviour of solutions of analytic differential equations from the point of vue of o-minimality. It is well known that the structure generated by the non spiraling leaves of codimension 1 analytic foliations is o-minimal. We investigate the properties of non oscillating trajectories of analytic vector fields. We show that, under some sufficient conditions related to the notion of quasi-analyticity, these trajectories belong to o-minimal structures. We also give some examples of non oscillating trajectories which do not belong to any o-minimal structure, and examples of infinite families of o-minimal structures such that any two of them do not admit an o-minimal common extension.

MAA 25th May 2005
16:30 to 17:30
Mystery of point charges

(Joint work with D. Novikov, B. Shapiro)

In his famous Treatise on Electricity and Magnetism (1873) J.C. Maxwell considered the following problem: How many equilibrium points a potential of n fixed point charges in R^3 may have? Maxwell applied Morse theory (developed 50 years later) to approach this problem. He claimed that there should be at most (n-1)^2 equilibrium points. In a footnote to the third edition (1891) of the Maxwell's book, J.J. Thomson stated that he could not find any proof of this. In the talk, recent progress in this direction will be reported, based on fewnomial theory and Voronoi diagrams.

MAA 26th May 2005
09:00 to 10:00
Rational functions and real Schubert calculus
MAA 26th May 2005
10:30 to 11:30
Galois theory of parameterised linear differential equations and linear differential algebraic groups

I will describe a Galois theory of differential equations of the form

dY/dx = A(x,t_1, ... , t_n) Y

where A(x,t_1, ... , t_n) is an m x m matrix with entries that are functions of the principal variable x and parameters t_1, ... , t_n. The Galois groups in this theory are linear differential algebraic groups, that is, groups of m x m matrices (f_{i,j}(t_1, ..., t_n)) whose entries satisfy a fixed set of differential equations. For example, in this theory the equation

dy/d/x = (t/x) y

has Galois group

G = { (f(t)) | (log f(t))'' = 0} .

I will give an introduction to the theory of linear differential algebraic groups and the above Galois theory and discuss the place of isomonodromic famillies in this theory and the relation to Pillay's differential Galois theory. This is joint work with Phyllis Cassidy.

MAA 1st June 2005
11:00 to 12:30
Model theory of compact complex spaces I: introduction
MAA 1st June 2005
14:00 to 15:00
Transseries and polynomially bounded o-minimality

`Transseries' is the short name coined by Ecalle for certain generalised power series - here the name is appropriate to denote all such series. The use of transeries allows the study of o-minimal expansions of the reals to rest on fully model theoretic methods. We develop this theme in case the o-minimal expansion of the reals is polynomially bounded and we prove results of quantifier(s) elimination, of axiomatisation, of cell decomposition and relative computability - in the `restricted case' where the primitive functions added to the real field are C^\infty with arguments ranging over [0,1].

MAA 2nd June 2005
11:30 to 12:30
On close to tame expansions of densely ordered groups

Most of this talk concerns joint work with C. Miller. Linearly ordered structures having an o-minimal open core enjoy several "tameness" properties. We discuss conditions under which an expansion of a densely ordered group has an o-minimal open core, and consider examples as well as further results.

MAA 2nd June 2005
14:00 to 15:30
Model theory of compact complex spaces II: essentially saturated space
MAA 8th June 2005
10:30 to 12:30
Algebraically closed and real closed fields with small multiplicative group
MAA 8th June 2005
14:30 to 15:30
Model theory of compact complex spaces III: Campana's work on the nonmultidimensionality conjecture
MAA 9th June 2005
11:00 to 12:00
F-V Kuhlmann Additive polynomials

I will give a survey on additive polynomials, their role in valuation theory and algebraic geometry and what we know and do not know about them. I show their connection with the defect of valued field extensions and the meaning of the defect for the model theory of valued fields and for local uniformization in positive (residue) characteristic. I will mention a classification of Artin-Schreier extensions and its applications. If time permits, I will also talk about maximality properties of valued fields and Ershov's notion of extremal fields.

MAA 9th June 2005
14:00 to 15:00
Groups, measures and NIP

(Joint work with Hrushovski and Peterzil). I will give a proof of the conjectures relating [definably compact groups definable in saturated o-minimal expansion of RFC], to [compact Lie groups]. I will expand on some of the ingredients of the proof and related notions which may be of independent interest (Keisler's work on measures, consequences of "not the independence property", model theory of the standard part map,...)

MAA 9th June 2005
15:30 to 16:30
Groups, measures and NIP
MAA 15th June 2005
10:30 to 12:00
Fields with finitely many definable subsets

We prove that a field with finitely many definable subsets is finite. We also conjecture a relative version of this statement: If K is a field extension of k, and the collection of sets obtained by intersecting each k-definable subset of K with K-k is finite, then k and K are either both finite or both algebraically closed. This is joint work with Kiran Kedlaya.

MAA 15th June 2005
14:00 to 15:00
Some model theory of complex analytic functions
MAA 15th June 2005
15:30 to 16:30
Some model theory of complex analytic functions
MAA 16th June 2005
10:30 to 12:00
Uniform first order definitions in finitely generated fields

We construct a first order sentence that is true for all finitely generated fields of characteristic 0 and false for all finitely generated fields of positive characteristic. Also, for each n we construct a first order formula with n free variables that for any

MAA 16th June 2005
14:00 to 15:00
F-V Kuhlmann Dense subfields of henselian fields, integer parts, and nasty valuations on rational function fields

We show that every henselian field of residue characteristic 0 admits a dense subfield. Certain special cases allow subfields with very interesting additional properties. We also discuss the existence and size of integer parts and of complements to valuation rings.

MAA 22nd June 2005
11:30 to 12:30
A Higman subgroup theorem for lattice-ordered groups
MAA 22nd June 2005
16:00 to 17:00
M Aschenbrenner A generalisation of the Hilbert Basis Theorem
MAA 23rd June 2005
15:30 to 17:00
Measurable structures, and asymptotics in finite structures

Work of Chatzidakis, van den Dries and Macintyre [CDM] shows that in finite fields, the sizes of definable sets (defined by a fixed formula with parameters) have a uniform asymptotic behaviour as the field and parameters vary. This enables one to associate a dimension (the natural one) and a measure to any definable set in a pseudofinite field. I and Steinhorn have investigated arbitrary classes of finite structures for which the conclusion of the [CDM] theorem holds, and the corresponding notion of (supersimple) measurable structure. In this talk I will describe examples, but will mainly discuss more recent work of Ivan Tomasic, Richard Elwes, and Mark Ryten.

MAA 29th June 2005
11:00 to 12:30
Characterisation and axiomatisation of finite soluble groups

Ever since the result of Philip Hall characterizing solubility for finite groups in terms of arithmetical statements about orders of subgroups, there has been interest in finding properties that distinguish the finite groups that are soluble from those that are not. I will discuss three such properties in the lecture.

MAA 30th June 2005
14:00 to 15:30
Uniform first order definitions in finitely generated fields II
MAA 30th June 2005
16:00 to 17:00
F-V Kuhlmann More on additive polynomials
MAAW03 11th July 2005
10:00 to 11:00
L van den Dries Asymptotic differential algebra and H-fields

The differential field of LE-series has very strong closure properties, and seems a perfect arena for asymptotic differential algebra. The theory of H-fields is an approach to develop a model theory for this differential field. I will discuss the field of LE-series, the subject of H-fields in general, and review the present state of an ongoing project on these matters in collaboration with Matthias Aschenbrenner and Joris van der Hoeven. The talk by Aschenbrenner will focus on recent progress.

MAAW03 11th July 2005
11:30 to 12:30
Definable groups over valued fields
MAAW03 11th July 2005
14:30 to 15:30
J-M Lion The Haefliger theorem for foliations and o-minimal structures
MAAW03 11th July 2005
16:00 to 17:00
J-P Rolin Quasi-analytic solutions of differential equations and o-minimal structures

We present the results of a joint work with F. Sanz and R. Schaefke. Consider a non-oscillating trajectory of real analytic vector field. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in that is not definable in any o-minimal extension of the reals.

MAAW03 11th July 2005
17:00 to 18:00
M Aschenbrenner Solving linear differential equations over H-fields

Many basic questions about algebraic differential equations over H-fields remain open. I will talk about recent work (joint with Lou van den Dries and Joris van der Hoeven) which answer some of these questions in the case of linear differential equations.

MAAW03 12th July 2005
09:00 to 10:00
R Pink A common generalisation of the conjectures of Andre-Oort, Manin-Mumford and Mordell-Lang

The formal similiarity between the Manin-Mumford and Mordell-Lang conjectures on the one hand, and the Andre-Oort conjecture on the other hand, suggests that a common generalization should exist for subvarieties of mixed Shimura varieties. We propose such a conjecture, explain why it implies all the stated conjectures, and explain its relation with existing results.

Related Links

MAAW03 12th July 2005
10:00 to 11:00
D Roessler Two applications of automatic uniformity
MAAW03 12th July 2005
11:30 to 12:30
Strongly minimial groups in the theory of compact complex spaces

A compact complex space is viewed as a first-order structure in the language where all analytic subsets of the cartesian powers are named. Anand Pillay and Thomas Scanlon have characterised all strongly minimal groups definable in such a structure as being either a simple complex torus or the additive/multiplicative group of the complex field. I will discuss joint work with Matthias Aschenbrenner and Thomas Scanlon in which we give a uniform version of this result thereby characterising strongly minimal groups in elementary extensions of compact complex spaces.

MAAW03 12th July 2005
14:30 to 15:30
Aspects of the algebraic structure of groups definable in o-minimal structures

Let M be an o-minimal expansion of a real closed field. A definable group is a group that both the set and the graph of the operation are definable in M. Let G be a closed and bounded definable group. I will show the following:

(1) G is divisible if and only if G is definably connected.

(2) (Joint work with M.Edmundo) If G is abelian then the group structure of the torsion subgroups of G is determined.

Both proofs require the understanding of the o-minimal cohomology algebra of G.

I will also discuss the role played by the o-minimal Euler characteristic in aspects of the algebraic structure of definable groups.

MAAW03 12th July 2005
16:00 to 17:00
On the number and shape of asymptotic cones of semialgebraic groups

Asymptotic cones are a useful concept in order to study large scale geometric invariants of metric spaces. I will explain the construction of asymptotic cones and will indicate why the number of homeomorphism types of asymptotic cones of a Lie group depends on the continuum hypothesis.

MAAW03 12th July 2005
17:00 to 18:00
E Jaligot Groups of finite Morley rank and genericity

The ultimate Algebricity Conjecture concerning groups of finite Morley rank postulates that simple groups of this class are algebraic. The weaker Genericity Conjecture postulates that they contain a generous Carter subgroup. These definable connected nilpotent subgroups of finite index in their normalizers exist in any group of finite Morley rank and they are a good approximation of maximal tori in the algebraic context. Such a subgroup is said to be generous if its conjugates form a generic subset of the ambiant group. I will explain a conjugacy theorem of generous Carter subgroups and show some striking consequences of the presence a generous Carter subgroup in a group of finite Morley rank.

Related Links

MAAW03 13th July 2005
09:00 to 10:00
Uniform first order definitions in finitely generated fields
MAAW03 13th July 2005
10:00 to 11:00
Orbital integrals for linear groups
MAAW03 13th July 2005
11:30 to 12:30
An application of motivic integration to representations of p-adic groups
MAAW03 14th July 2005
09:00 to 10:00
Non-oscillating trajectories of vector fields and Zariski's local uniformization

A transcendent non-oscillating trajectory of an analytic germ of real vector field induces a structure of Hardy field for the meromorphic functions. It has a natural valuation associated to it. The study of this valuation allows to get a reduction of singularities of the vector field following the strict transform of the trajectory. More generally, for a holomorphic complex vector field, the above results can be generalized for a given valuation of the field of the meromorphic functions. We obtain in this way a local uniformization in the sense of Zariski, that should be globalized in dimension three, following the classical results of Zariski. The key of these results is a construction (due to J. Cano and Grigoriev-Singer) based on the Newton Polygon of a differential operator, that assures finiteness results on the valuation allowing the local uniformization.

MAAW03 14th July 2005
10:00 to 11:00
Homotopy types of fibres of pfaffian maps

We prove a tight upper bound on the number of different homotopy types of fibres of semialgebraic, quadratic or semi-Pfaffian maps in terms of formats of these maps. A similar argument leads to a tight upper bound on the sizes of finite bisimulations of Pfaffian dynamical and hybrid systems.

MAAW03 14th July 2005
11:30 to 12:30
On the topological degree of functions definable in o-minimal structures
MAAW03 14th July 2005
14:30 to 15:30
Analytic Denef-Pas cell decomposition

We prove a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable defined on annuli over K.

We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et int\'egration motivic I, Comptes rendus de l'Acad\'emie des Sciences, 339 (2004) 411 - 416].

Related Links

MAAW03 14th July 2005
16:00 to 17:00
L Lipshitz Overconvergent real closed quantifier elimination

Let K be the (real closed) field of Puiseux series in t over the reals, R, endowed with the natural linear order. Then the elements of the formal power series rings R[[x_1,...,x_n]] converge t-adically on [-t,t]^n, and hence define functions [-t,t]^n to K. Let L be the language of ordered fields, enriched with symbols for these functions. We show that K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

Related Links

MAAW03 15th July 2005
09:00 to 10:00
Model theory of difference fields and some remarks on Galois groups

In this talk I will present a short survey of the results known to-date on generic difference fields. I will also discuss Galois groups of difference equations, and why the model-theoretic Galois group coincides with the classical Galois group defined using Picard-Vessiot extensions

MAAW03 15th July 2005
10:00 to 11:00
Small profinite groups
MAAW03 15th July 2005
11:30 to 12:30
Elementary equivalence vs isomorphism over large fields

During the last time there was some progress in tackling the problem of "elementary equivalence vs isomorphism" for function fields in both the arithmetical and the geometric situation. We will show that actually similar results can be obtained in the case of function fields over the so called large fields (under some supplementary "finiteness" hypothesis).

Related Links

MAAW03 15th July 2005
14:30 to 15:30
TW Scanlon Additive groups
MAAW03 15th July 2005
16:00 to 17:00
Betti numbers of definable sets

A spectral sequence associated with a surjective closed mapping allows one to provide upper bounds for the Betti numbers of a wide class of sets defined by formulas with quantifiers in terms of the Betti numbers of auxiliary sets defined by quantifier-free formulas. A review of the recent development in this direction will be presented

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons