A KAM phenomenon for singular holomorphic vector fields Stolovitch L. (France) Universit Paul Sabatier, CNRS-Institut de Mathmatiques de Toulouse stolo@picard.ups-tlse.fr Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a “good perturbation” of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.

In particular, we show that a volume preserving germ of holomorphic vector field which is a small pertubation of a non-degenerate volume preserving polynomial vector field has a “lot” of invariant manifolds biholomorphic to the intersection of “x1 · · · xn = constant” with a fixed polydisc.

Of course, we also recover the classical KAM theorem for hamiltonians in a neighbourhood of a fixed point.

References [1] Arnold V.I. Proof of a theorem by A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the hamiltonian. Russ. Math. Surv.–1963. N18. p.9–36.

[2] Arnold V.I. Small denominators and the problem of stability of motion in the classical and celestian mechanics. Russ. Math. Surv.– 1963. N 18. p.85-–191.

[3] Bost J.-B. Tores invariants des syst‘emes dynamiques hamiltoniens (d’apr‘es Kolomogorov, Arnol’d, Moser, Rssmann, Zehnder, Herman, Pschel,... ), in Sminaire Bourbaki. Astrisque–1986. N 133– 134. p.113–157.

[4] Stolovitch L. A KAM phenomenon for singular holomorphic vector fields. Publ. Math. I.H.E.S.–2005. N 102. p.99–165.

Classifying spaces in singularity theory and elimination of singularities Szcs A. (Hungary) Etvs Lornd University szucs@cs.elte.hu The classifying spaces of cobordisms of singular maps with an allowed set of singularities form certain fibrations. (See [2].) Using these fibrations we answer explicitely a question of Arnold about the elimination of a singularity - at least up to cobordism.

In [1] V. I. Arnold and his coauthors posed the question: Suppose for a smooth map the homology class of a given singularity stratum of the map vanishes. Does this imply that that singularity type can be eliminated by a homotopy The question in this form has a negative answer.

We shall consider an analogous question, namely: When the given singularity type can be eliminated by a restricted cobordism of the map. The cobordism will be called restricted if it has no singularities not equivalent to the singularities of the original map.

We will give a necessary and sufficcient condition for the elimination of a top singularity of a smooth map by a restricted cobordism. This answer turns out to be very similar to the one suggested by the above question of Arnold.

Before giving the precise formulation let us recall that the Thom polynomial of a singularity type gives the cohomology class dual to a given singularity stratum. Such a stratum has a special normal structure and higher Thom polynomials express the normal characteristic classes of this stratum.

Theorem:

n+k n+k Let Mn and P be smooth manifolds, k > 0, and let f : Mn P be a generic smooth map, and let be its top (i.e. most complicated) singularity type. Then n+k 1) the restrited cobordism classes of n-manifolds into P form an Abelian group.

2) A non-zero multiple of the element of this group represented by the map f contains an -free smooth map if and only if the Gysin map f! annihilates the Thom polynomial of the stratum of f together with all the higher Thom polynomials of this stratum.

References [1] V. I. Arnold, V.V. Goryunov, O.V. Lyashko, V. A. Vasil’ev: Singularity theory I, 1998. Springer [2] A. Szcs: Cobordism of singular maps. // Arxiv math. GT/Binary quadratic forms with semigroup propertyTimorin V. A. (Russia) IUM, Moscow timorin@math.sunysb.edu A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x, y)) = f(x)f(y) for all vectors x and y from the integer 2dimensional lattice, then the form f has semigroup property. We give an explicit integer parameterization of all pairs (f, s) with the property stated above. We do not know any other examples of forms with semigroup property.

joint work with F. Aicardi Gibbs entropy and dynamics Treschev D. V. (Russia) Steklov Mathematical Institute, RAS treschev@mi.ras.ru Let (M, µ) be a measure space and let be another measure on M:

d = dµ. We define the Gibbs entropy s() = log dµ and some its modifications (the coarse-grained entropy).

We plan to discuss the question: how physical are these objects in the sense:

– do they grow if a µ-preserving dynamics appear – what properties of the dynamics are responsible for this growth – in what extent this “growth” can be independent of arbitrariness of the construction Here one should keep in mind that if the dynamics is reversible, some strong restrictions are imposed on any entropy-like dynamical quantity.

On the richness of the Hamiltonian chaos Turaev Dmitry (Israel) Ben Gurion University turaev@math.bgu.ac.il We show an ultimate richness of chaotic dynamics of symplectic diffeomorphisms that have an elliptic point. These maps form an open subset Dn in the space of all C-smooth symplectic diffeomorphisms of R2n (a widely believed conjecture is that the C-closure of Dn coincides with the set of all symplectic diffomorphisms that are not uniformly partiallyhyperbolic). Our main theorem is that a C-generic map from Dn is universal in the sense of the following definition. Let f be a symplectic Cdiffeomorphism R2n R2n. Let be a C-diffeomorphism R2n R2n such that it preserves the standard symplectic form modulo a constant factor (i.e., is a symplectic diffeomorphism times a constant). Let B2n be the unit ball in R2n.

We call the map fk, = -1 fk |B2n a renormalized iteration of f.

This is a symplectic diffeomorphism B2n R2n.

We call the map f universal, if the set of all its renomalized iterations is C-dense in the set of all symplectic C-diffeomorphisms B2n R2n.

We stress that the renormalized iterations are just iterations up to a coordinate transformation. Therefore, the iterations of any universal map approximate arbitrary well all symplectic dynamics possible in R2n.

By proving the genericity of the universal maps, we thus show an ultimate richness of the chaotic behavior near elliptic points of a typical 2n-dimensional symplectic diffeomorphism: the dynamics of any single such diffeomorphism is as complicated as the dynamics of all symplectic diffeomorphisms of R2n altogether.

On double Hurwitz numbers in genus Vainshtein A. (Israel) University of Haifa alek@math.haifa.ac.il We study double Hurwitz numbers in genus zero counting the number 1 of covers CP CP with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers.

Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz Varchenko A. (USA) University of North Carolina at Chapel Hill anv@email.unc.edu I shall discuss the proof of Shapiro’s conjecture by methods of math physics. The Shapiro’s conjecture says the following. If the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This statement, in particular, implies the following result. If 1 r all ramification points of a parametrized rational curve f : CP CP lie on a circle in the Riemann sphere CP, then f maps this circle into a r r suitable real subspace RP CP. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum.

Equations and solutions for moving of a rotating body in chemical processes. Spiral trajectories and photophoresis Vedenyapin V. V. (Russia) Keldysh Institute of Applied Mathematics, Moscow State Region University vicveden@kiam.ru The book [1] contains an assertion about spiral trajectories of a rigid body and consideration of movement with changing of inertia ellipsoid.

Those topics have applications in moving of sputniks [7] and in chemical kinetics of big reacting particles.

The case of a big particle, moving in chemically reacting gas, was considered. It was called chemojet motion and was found experimentally by o a group of scientists from Chemical faculty of Moscow State University [8].

A model of a chemical reaction on the surface of a body was created and a system of equations of motion was written and investigated.[2,3] We got an ideal cylindrical spiral trajectory in asymptotic when time tends to infinity. This justified qualitatively the result of experiment of European Cosmic Agency and in Moscow State University. A step and a diameter of those cones were calculated in exact form.

This model was used for explanation of photoforesis [4]. The term photophoresis was proposed by Felix Ehrenhaft [5]. In his experiments dust, silver and copper particles in gases irradiated by light “strongly exhibited a tremendous lightnegative movement, although they ought to be most heated on the side toward the light, and would expect a movement away from the light” ([6]).

Movement away from light was called lightpositive or positive photophoresis and towards it lightnegative or negative photophoresis.

“During the course of the experiment, the motion of the particle traced out a “spiral” path. However upon magnification of a given section of a given spiral, one saw a “spiral” path within the path of the larger spiral...

In viewing these microphotographs, one had the distinct impression that something phenomenal was happening, but no definitive explanation for the observation was presented” [6].

Now it is clear that Felix Ehrenhaft spiral paths and so both positive and negative photophoresia have their explanation in the framework of chemojet motion: the former as a consequence of reactive forces and the latter of counter reactive ones. On the other hand Ehrenhaft spiral paths strongly support all mathematical theory of motion of any big particle in reacting gas, constructed in papers [2-4].

References [1] Арнольд В.И. Математические методы классической механики, М.: Эдиториал УРСС, 2000.

[2] Vedenyapin V.V., Batysheva Y.G., Melihov I.V. and Gorbatchevski A.Ya. On a motion of solids in chemically active gaz // Doclady Physics, V. 48, No 10, P. 556-558 (2003). Translated from Doklady Akademii Nauk, Vol. 392, No 6, pp. 738-780 (2003).

[3] J.G.Batisheva. On the Derivation of Dynamic Equations for a Rigid Body in a Gas Reacting Nonuniformly with Its Surface // Doklady Physics, Vol. 48, No. 10, 2003, pp. 587-589. Translated from Doklady Akademii Nauk, vol. 392, No. 5, 2003, pp. 631-633.

[6] Photophoresis phenomenon. Archive message from “Physics Forum”. Posted by Alan Marshall on November 11 (2001).

[7] Белецкий В.В., Яншин А.М. Влияние аэродинамических сил на вращательное движение спутников. Киев: Наукова думка, 1984.

[8] Melihov I.V., Simonov E.F., Vedernikov A.A., Berdonosov S.S., Chemojet Motion of Rigid Bodies, Rus. Chem. Jour., v. 41, No 3, pp. 5-16, 1997.

What does Lebesque’s measure in the infinite dimensional space mean Vershik A. M. (Russia) St. Petersburg Branch of Steklov Mathematical Institute We will introduce so called infinite-dimensional Lebesgue measure which is the limit of measures on some homogeneous manifolds, – this is parallel to the classical Maxwell–Poincare lemma, which presents the Gaussian distribution as a weak limit of the measures on spheres Sn.

This Lebesgue measure opens new possibilities in the combinatorics, representation theory, random processes, and in the theory of Fock space.

Cohomology of the braid groups and special involutions Veselov A. P. (UK, Russia) Loughborough Landau Institute A.P.Veselov@lboro.ac.uk We give an explicit description of the action of a Coxeter group G on the total cohomology of the complement to the corresponding complexified reflection hyperplanes. The answer is given in terms of a geometric class of the involutions in G called special. The relation with the classical results by Arnold and Brieskorn on the cohomology of the braid groups will be discussed.

The talk is based on a joint work with G. Felder.

The 16th Hilbert problem, a story of mystery, mistakes and solution Viro O. Ya. (Russia) Petersburg Department of Steklov Institute of Mathematics oleg.viro@gmail.com Hilbert’s problem of the topology of algebraic curves and surfaces (the 16th problem from the famous list presented at the second International Congress of Mathematicians in 1900) was difficult to formulate. The way it was formulated made it difficult to anticipate that it has been solved.

I believe it has, and this happened more than thirty years ago, although the World Mathematical Community missed to acknowledge this.

Positivity of Schur function expansions of Thom polynomials Weber Andrzej (Poland) Uniwersytet Warszawski aweber@mimuw.edu.pl We consider a holomorphic map f of complex manifolds. For a prescribed singularity type let f denote the locus of the points in which the map f is of that type. The cohomology class of f depends on the Chern classes of the manifolds. It is expressed by the Thom polynomial. We will show that the coefficients of the Thom polynomials in the Schur basis are nonnegative. The proof is obtained by a combination of the approach to Thom polynomials via classifying spaces with the Fulton-Lazarsfeld theory of cone classes for ample vector bundles. Further generalizations based on the representation theory will be sketched.

References [1] P. Pragacz, A. Weber Positivity of Schur function expansions of Thom polynomials, arXiv:math/0605308, to appear in Fund. Math.

Differential invariants of 2-order ODEs Yumaguzhin V. A. (Russia) Program Systems Institute of RAS, Pereslavl’-Zalesskiy yuma@diffiety.botik.ru Every ODE E of the form y = a(x, y)y 3 + b(x, y)y 2 + c(x, y)y + d(x, y) (1) can be identified with the section SE : (x, y) a(x, y), b(x, y), c(x, y), d(x, y) of the product bundle : R2 R4 R2.

Thus the set of all equations (1) is identified with the set of all sections of.

It is well known that an arbitrary point transformation f transforms an arbitrary equation (1) to the equation of the same form. Coefficients of the transformed equation are expressed in terms of the coefficients of the initial one and 2-jet of f. That is f defines the natural transformation of sections of. This defines the natural lifting of f to the diffeomorphism f(0) of the total space of. In turn, f(0) is naturally lifted to diffeomorphism f(k) of the bundle Jk of k-jets of sections of, k = 1, 2,... Thus the pseudogroup of all point transformations of the base of acts by its lifted transformations on every Jk.

The lifting of point transformations generates the natural lifting of any vector field X from the base of to the vector field X(k) on Jk.

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