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Organized by Matthias Aschenbrenner
Minicourse Abstracts
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Introduction to Model Theory
Lecturer: Deirdre Haskell
The goal of these four lectures is to explain to the listener
with no prior knowledge of mathematical logic enough of the key
concepts of model theory to understand the definition and the
power of the notion of o-minimality.
Lectures 1 and 2: language, structure, definable sets, elementary
equivalence, model completeness and quantifier elimination, geometric
interpretation, Hilbert's 17th problem, techniques for proving
quantifier elimination
Lecture 3 and 4: Examples of proving model completeness/ quantifier
elimination: real closed fields; the real field with restricted
analytic functions; the real field with a predicate for integer
powers of 2.
INTRODUCTION TO REAL ANALYTIC GEOMETRY
Lecturer: Krzystof Kurdyka
The goal of this course is to give a quick backround in real
analytic geometry, more precisely to semi-analytic geometry which
was created by S.Lojasiewicz in his celebrated IHES lecture notes
[5] from 1965 (now available on web). Actually the lecture will
be based on a more recent paper [4] of S. Lojasiewicz, M-A. Zurro
and myself.
Lecture 1. Holomorphic and analytic functions in several variables.
Power series, holomorphic functions, Riemann extension theorem,
separate analyticity. References: [3], [6], [8].
Lecture 2. Preparation Theorem and Weierstrass polynomials. Continuity
of roots, discriminant and generalized discriminant. Decomposition
into irreducible factors. Puiseux’s theorem. References:
[3], [6], [7], [8].
Lecture 3. Semi-analytic sets and distinguished stratifications.
Real analytic sets, semi-analytic sets, Thom’s lemma, dimension.
References: [1], [2], [4].
Lecture 4. Regular separation and Lojasiewicz’s inequality.
Metric properties of seminalytic sets, curve selection lemma,
Lojasiewicz’s inequality. References: [5], [1], [4].
References
[1] E. Bierstone and P. D. Milman, Semianalytic and Subanalytic
sets, Publ. I.H.E.S.,
67 (1988), 5-42.
[2] M.Coste, Ensembles semi-alg´ebriques. Real algebraic
geometry and quadratic forms
(Rennes, 1981), pp. 109–138, Lecture Notes in Math., 959,
Springer, Berlin-New York,
1982
[3] R. Gunning, H. Rossi, Hugo Analytic functions of several complex
variables. Prentice-
Hall, Inc., Englewood Cliffs, N.J. 1965 xiv+317 pp.
[4] K. Kurdyka, S. Lojasiewicz, M. Zurro Stratifications distingu´ees
comme un util en
g´eom´etrie semi-analytique, Manuscripta Math. 86,
81–102, (1995).
[5] S. Lojasiewicz, Ensembles semi-analytiques,, preprint, I.H.E.S.
(1965) available on
the web page ”http://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf”.
[6] S. Lojasiewicz, Introduction to complex analytic geometry,
Birkh¨auser, Basel, 1991.
[7] R. Narasimhan, Introduction to the theory of analytic spaces.
Lecture Notes in Math-
ematics, No. 25 Springer-Verlag, Berlin-New York 1966 iii+143
pp
[8] Ruiz, Jesus M. The basic theory of power series. Advanced
Lectures in Mathematics.
Friedr. Vieweg & Sohn, Braunschweig, 1993. x+134 pp.
UNIVERSITE DE SAVOIE, Laboratoire de Mathematiques (LAMA), UMR
5127 CNRS, 73-376 Le Bourget-du-Lac cedex FRANCE
Introduction to o-minimality.
Lecturer: Sergei Starchenko
In these 4 lectures I intend to cover the following subjects.
I. Basics of o-minimality
(1) Monotonicity Theorem
(2) Cell decomposition
(3) Consequences of Cell decomposition.
II. More advanced properties
(1) Triangulation and Euler Characteristic
(2) Piecewise smoothness
III. Hardy Fields, valuations and elementary extensions
(1.) Hardy fields, valuations and elementary extensions
(2.) Growth Dichotomy
(3.) Hausdorff Limits
(4.) Real closed valued fields
(5.) T-convexity
(6.) Bieri-Groves Theorem
Analytic Vector Fields: Singularities
and Limit Cycles
Lecturer: Sergei Yakovenko
The minicourse is planned to provide the audience with the basic
notions of real and complex analytic vector fields. We begin by
describing different notions of normal forms for elementary singularities
of vector fields and desingularization (blow-up) for non-elementary
planar singularities. The minicourse will culminate with the proof
of the finiteness theorem for limit cycles of polynomial vector
fields having only nondegenerate singularities.
The material is covered in the recent textbook: Ilyashenko, Yakovenko,
Lectures on Analytic Differential Equations (AMS 2007), Chapter
I and Chapter IV (sect. 24).
The draft version of this textbook is available online at http://www.wisdom.weizmann.ac.il/~yakov/thebook1.pdf
Several first separate sections of the final version can be found
on the blog http://yakovenko.wordpress.com/ (more specifically,
at http://yakovenko.wordpress.com/category/lecture/ )
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