## Summer School on Real Geometry

Grand Hotel Bellavista, Levico (Trento - Italy), June 12-16, 2000

This is the second announcement of the Summer School on Real Geometry that will be held in Levico (Trento-Italy) from June 12 to June 16 2000, mainly sponsored by CIRM and GNSAGA.
The School intends to present introductory courses at advanced gradued and postgraduate level in the area of Real Geometry and its applications. It is addressed to students and researchers interested in the field. It will be organized in three ten hours courses with the following teachers and topics:
J.Kollàr (Princeton University): The interplay between real and complex algebraic geometry.

S.Orevkov (Universitè de Toulouse): Topology of real algebraic and real pseudo-holomorphic curves

Y.Yomdin (Weizman Institute of Science, Rehovot): Uniform bounds in real algebraic geometry.

(More detailed programs can be found at the end of this announcement)
Since the first announcement, Prof. J.Kollàr has informed us that, due to some health problems, it is unlikely that he will be able to travel in June, so that his presence at the School is not assured. Nevertheless we hope sincerely him to recover promptly and to be in Levico with us. We leave the program unchanged, at least for the moment.
The school will take place at Grand Hotel Bellavista. Price per person and day on a full-board basis will be approximately of 50 Euros.
Lodging and board of Ph.D.students and recently graduated researchers without a position will be funded by the organization; unfortunately travel expenses are not included and should be paid by their home institutions.
Funding for other participants cannot be assumed and will depend on the funds obtained from the different institutions we are applying for.
We stress the fact that we are organizing a School, not a Conference; for this reason there will be no talks besides the lectures. Instead, some tutorials could be given depending on the schedule. One of the aims is also the interaction among young researchers. Therefore communication of their research or Ph.D topics will be promoted either in the form of posters or informal afterhours sessions.
Any comunication may be sent to Mr. Augusto Micheletti at the following address
michelet@science.unitn.it

Hoping to seeing you soon

The Organizing Committee:

Fabrizio Broglia (broglia@dm.unipi.it)

Michel Coste (coste@maths.univ-rennes1.fr)

## Program

### J.KOLLÀR: "The interplay between real and complex algebraic geometry"

The aim of these lectures is to concentrate on some topics of real and complex algebraic geometry which are especially interesting from the point of view of the other field.
Some examples of these are:
Lefschetz principles, minimal models, Lojasiewicz inequalities and the interaction between the real and complex topology of varieties.

### Y.YOMDIN: "Uniform bounds in real algebraic geometry"

Contents
1. Introduction
Some basic examples
Examples of geometric bounds
Examples of $C^k$ bounds
Examples of Bernstein-type inequalities
2. Geometric and Entropy bounds and analytic applications.
Vitushkin variations and Entropy bounds.
Qualitative Sard theorem
High dimensions and concentration effect
3. C^k and analytic complexity bounds
C^k and analytic "complexity units"
Quantitative "resolution of singularities"
Applications in dynamics
4. Bernstein-type inequalities
Bernstein inequality for algebraic functions
Generalizations
Applications in Center-Focus problem
Summary
The main purpose of this course is to stress a certain type of properties of real semialgebraic sets, which we call "uniform bounds", to present some examples of such properties and to give thier applications in analysis and differential equations. The most important feature of the "uniform bounds" is that they depend only on the discrete data (i.e. on the degrees and the combinatorial structure of the equations and inequalities, defining the set, but not on the specific values of the coefficients).
We consider several basic examples, as stated in the Content above. Most of them will be presented without proofs, while at present we plan to concentrate and to give a more detailed presentation of the third and of the fourth topics above: C^k and analytic complexity bounds for real semialgebraic sets and Bernstein-type inequalities.

### S.OREVKOV: "Topology of real algebraic and real pseudo-holomorphic curves"

Real pseudo-holomorphic curve is a natural generalization of the notion of real algebraic curve. This is a Conj-anti-invariant 2-surface in CP^2 which is a J-holomorphic curve with respect to some tame almost Conj-anti-invariant complex structure J. The classification of real pseudo-holomorphic curves seems to be simpler than the classification of real algebraic curves. An arrangement of ovals on RP^2 is realizable pseudo-holomorphically if and only if a certain class of braids contains a quasipositive braid (a braid b is called quasipositive if $b=\prod a_i\sigma_1 a_i^{-1})$. We shall discuss some old and new methods to decide if a braid is quasipositive or not. We shall discuss also examples of algebraically unrealizable pseudo-holomorphic curves. Each such example requires very fine methods of prohibition, because the most of methods used in the topology of real algebraic curves work also for real pseudo-holomorphic curves (e.g. Bezout's theorem for axiliary lines and conics, complex orientations, arithmetics of the intersection form on branched coverings etc).