[9] Aminova A. V. Projective transformations of pseudo-Riemannian manifolds. Moscow: Yanus-K. 2003.

Enumerating stair-shaped young tableaux Baryshnikov Yuliy (USA) Bell Laboratories ymb@research.bell-labs.com Up-down permutations – permutations where increases and decreases alternate – feature prominently in Arnold’s work, see e.g. [1, 2]. One can view these permutations as standard Young tableaux with stair-like Young diagram shapes.

The problem of enumerations of standard skew Young tableaux of given shape is rather nontrivial (existing explicit formulae are unwieldy and do not lend themselves readily to asymptotic analysis). In this work we count Young tableaux filling a natural family of stair-like Young diagrams whose north-western and south-eastern boundaries are parallel to the bisector of the first quadrant (we use the orientation where the entries of the tableau increase left-to-right and top-to-bottom). We will encode such Young tableaux by the vector of column heights; examples of Young diagrams and their encodings are shown below:

] [1 2 2 2 2 2 1] [2 3 3 3 Figure 1: Stair-like Young diagrams and their height vectors.

To find the number of Young tableaux with stair-like shapes we adopt a variant of the transfer operator method generalizing the approach of [3];

it leads to a problem of diagonalization of the certain integral operator.

Thus, for the shapes with typical columns being of height 2m, the operator acts on the subspace of even totally antisymmetric functions in L2([-1, 1]m) as (Sf)(x1, x2,..., xm) = (1) 1-xm 1-xm-1 1-x= · · · f(y1,..., ym)dymdym-1 · · · dy1, 0 0 The operator S is diagonal in the orthonormal basis kjxl fk1,k2,...,km(x1, x2,..., xm) = 2m/2 det cos (2) l,j=1...m P (kj-1)/j 2m(-1) with the eigenvalues k1,k2,...,km =.

m kj j Similar approach works also for shapes with odd typical column heights; here the eigenfunction are the determinants det (sin(kjxl))k1

These general results lead to dozens of theorems on enumeration of stair-like Young tableaux for small values of column heights. For example, for the shape with the vector [44... 4] (n columns altogether), the number of Young tableaux is E2n E2n-2 E2n+n N[4 ] = (4n)! -, (2n)! (2n - 2)! (2n + 2)! E2nz2n where E2n are the Euler numbers, given by sec(z) =.

(2n)! This and many others of the resulting identities beg for a combinatorial explanation.

This is joint work with Dan Romik, Hebrew University.

References [1] Arnold V. I., Bernoulli–Euler updown numbers associated with function singularities, their combinatorics and arithmetics // Duke Math. J. – 1991. 63, no. 2, pp. 537–555.

[2] Arnold V. I., Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups.// Uspekhi Mat.

Nauk – 1992, v. 47, no. 1(283), 3–45.

[3] Elkies N., On the Sums (4k+1)-n.// Amer. Math. Monthly, k=– 2003. Vol. 110, N.7, pp. 561–573.

Finite rank approximations of chaotic dynamical systems with neutral singularities Blank M. L. (Russia) Russian Academy of Sci., Inst. for Information Transm. Problems blank@iitp.ru In 1960 S. Ulam [1] has formulated a hypothesis about the possibility of an approximation of an action of a chaotic dynamical system by means of a sequence of finite state Markov chains and proposed the sim plest scheme which can be described in modern terms as follows. Let T be a transfer-operator corresponding to the dynamical system (T, X), i.e.

-T µ(A) := µ(T A) for any Borel set A X and a probabilistic measure µ. Let := {i} be a finite measurable partition of X with the diameter. Consider an operator acting on probabilistic measures (generalized m(A i) functions): Q µ(A) := µ(i). Then the Ulam’s approx m(i) i imation can be written as a superposition of the operators Q T and his hypothesis says that for a “good” enough map and a “good” enough partition statistical properties of the original dynamical system can be obtained from the limit properties of the operators Q T when the par tition diameter vanishes. Observe that numerically the complete spectral analysis of the finite stochastic matrix corresponding to Q T is a routine procedure.

It turns out that for a broad class of dynamical systems having some hyperbolicity properties (piecewise expanding maps, Anosov torus diffeomorphisms, random maps) one might show that both the corresponding transfer-operator and its perturbation are quasi-compact, which leads to the direct operator analysis of the spectrum stability with respect to perturbations generated by the operator Q.

Strictly speaking even for a very “good” hyperbolic dynamical system some additional assumptions are needed to prove the hypothesis for the complete spectrum. Surprisingly, a similar statement about the leading eigenfunction turns out to be extremely robust. In fact, the only known counterexample [2] given by the map:

x + if 0 x < 4 5 T x := -2x + 1 if x < x + 1 otherwise. 2 is not only discontinuous but this discontinuity occurs at a periodic turning point (compare to instability results about general random perturbations [3]).

Up to now there were no mathematical results about the nonhyperbolic situation and our aim is to show that despite the conventional technics mentioned above no longer works in this case, yet the stability of the leading eigenfunction can be proven. Consider a family of expanding maps with neutral singularities. A typical example of this type is the so called Manneville-Pomeau map Tx := x + x(mod1) from the unit interval X := [0, 1] into itself with > 1. It is known that the map T possesses the only one SRB measure µ which is absolutely continuous (but has an unbounded density) if 1 < < 2, and is the Dirac measure at the origin 1 if > 2. The following result demonstrates that the Ulam scheme {0} works correctly for this nonhyperbolic map.

Theorem. For any 1 and small enough 0 < 1 the Markov chain generated by the transfer operator Q T is uniquely ergodic and its unique invariant distribution µ satisfies the relations: (a) µ(1) 0 C2- 1, (b) µ(1)/ - > 1, (c) µ - 1 > 2.

{0} Here 1 is the element of containing the origin.

To prove this result we developed a completely new approach based on the analysis of the action of the corresponding transfer operators on “monotonic measures” µ defined by the property that µ(A) µ((A + x) X) for any Borel set A X and x X.

We shall discuss also the generalization of the above result for a much more general class of piecewise convex maps with neutral singularities.

References [1] Ulam S. Problems in modern mathematics, Interscience Publishers, New York, 1960.

[2] Blank M. Perron-Frobenius spectrum for random maps and its approximation. // Moscow Math J., 1:3(2001), 315–344.

[3] Blank M. Stability and localization in chaotic dynamics, MCCME, Moscow, 2001.

Caustics of interior scattering Bogaevsky I. A. (Russia) Moscow State University bogaevsk@mccme.ru The geometric optics of linear short waves is described by a Fresnel hypersurface, which is defined by an eikonal equation and situated in a contact space. A Fresnel hypersurface can have conical singularities, which are locally diffeomorphic to the product of the two-dimensional cone and a real vector space. Its Legendre submanifolds describing the propagation of wave fronts can have singularities as well because the Fresnel hypersurface itself is not smooth. V.I.Arnold has discovered that there are two types of typical conical singularities of a generic Fresnel hypersurface up to contact diffeomorphisms - elliptic and hyperbolic. Besides, he has found a normal form of a typical Legendre submanifold in a neighborhood of a hyperbolic conical point of the Fresnel hypersurface. We describe all typical caustics of this Legendre submanifold in three-dimensional space.

Symbolic dynamics of almost collision orbits of the elliptic 3 body problem Bolotin Sergey (Russia) Steklov Mathematical Institute, Moscow bolotin@mi.ras.ru Suppose Sun of mass 1 and Jupiter of mass µ move with period along ellipses with eccentricity, and an Asteroid of negligible mass moves in the gravitational field of Sun and Jupiter. For µ = 0 Jupiter disappears and we obtain Kepler’s problem. We prove the existence of chaotic almost collision orbits of the Asteroid which, as µ 0, shadow chains of collision orbits of Kepler’s problem. Periodic orbits of this type were first considered by Poincar for the general 3 body problem.

One of the results is as follows. Let G be the angular momentum of the Asteroid and E the energy. For = 0 Jacobi’s constant J = E - G is a first integral. Let = {(g, h) : g2/2 - g - 1 < h < -g} be the set of (g, h) such that Kepler’s orbit with G = g and J = h is an ellipse crossing the unit circle which is Jupiter’s orbit for = µ = 0.

Theorem 1. Let > 0. There exist C, 0, > 0 such that for any (0, 0), any µ (0, ) and any sequence (gi, hi) in, there exists i=an almost collision orbit of the elliptic 3 body problem and a sequence (ti) such that 0 < ti - ti-1 < C-1 and |J(ti) - hi| + |G(ti) - gi| < i=for all i.

Thus the angular momentum and Jacobi’s constant wander “randomly” in. In fact J changes much slower than G. In contrast with regular perturbations of an integrable system for which “diffusion speed” is exponentially slow, the rate of change of J is of order.

The proof is based on a reduction to hyperbolic dynamics of skew product of almost integrable symplectic maps fk of the annuli:

fk(t, h) = (t + k(h) + O(), h + O()), t mod 2, ak < h < bk.

Abelian functions and singularity theory Buchstaber Victor M. (Russia) Steklov Mathematical Institute, RAS Theory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics.

In a cycle of our joint papers with V. Enolskii and D. Leykin we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function.

A sigma-function is defined on a cover of U, where U is the space of a bundle p: U B defined by a family of plane algebraic curves of fixed genus. The base B of the bundle is the space of the family parameters and a fiber Jb over b B is the Jacobi variety of the curve with the parameters b. A second logarithmic derivative of the sigma-function along the fiber is an Abelian function on Jb.

Thus, one can generate a ring F of fiber-wise Abelian functions on U.

The problem to find derivations of the ring F along the base B is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications.

The talk presents a solution of this problem recently found by the speaker and D. Leykin.

A precise modern formulation of the problem involves the language of Differential Geometry. We obtained explicit expressions for the generators of the module of differentiations of a ring of Abelian functions.

The families of curves, which we work with, are special deformations of the singularities yn - xs, where gcd(n, s) = 1. The choice of this type of families allows us to use methods and results of Singularity Theory, especially Arnold’s convolution of invariants and the theorem of Zakalyukin on holomorphic vector fields tangent to the discriminant variety.

References [1] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Kleinian functions, hyperelliptic Jacobians and applications, Reviews in Mathematics and Math. Physics, I. M. Krichever, S. P. Novikov Editors, v. 10, part 2, Gordon and Breach, London, 1997, 3–120.

[2] V. M. Buchstaber, D. V. Leykin, Polynomial Lie algebras, Funct.

Anal. Appl. 36 (2002), no. 4, 267–280.

[3] V. M. Buchstaber, D. V. Leykin, The heat equations in a nonholonomic frame, Funct. Anal. Appl. 38 (2004), no. 2, 88–101.

[4] V. M. Buchstaber, D. V. Leykin, Addition laws on Jacobian varieties of plane algebraic curves, Proc. Steklov Math. Inst. 251 (2005), 49– 120.

[5] V. M. Buchstaber, D. V. Leykin, Differentiation of Abelian functions over its parameters, Russian Math. Surveys, v. 62, Issue 4, 2007.

Poincar series and the monodromy zeta functions Campillo Antonio (Spain) Valladolid University Poincar series associated to multi-index filtrations are studied in joint work with F. Delgado and S. Gussein-Zad. One gets that for some natural filtrations associated to the inner structure of several singularity types, the Poincar series provides direct information on the geometry or the topology of the singularity. For quasi-homogeneous singularities W. Ebeling and S. Gussein-Zad have shown that the Poincar series associated to the weight filtration is related to the monodromy zeta function. This leads to consider also multi-index filtrations for given embedded singularities, and their associated Poincar series, which is done in current work by A. Lemahieu. Thus natural Poincar series for singularities, both non embedded and embedded ones, can be studied. We review those results and show how any of those Poincar series has significant information on the geometry or topology of the singularities, and compare such information in several cases. In particular, relations with monodromy zeta functions are emphasized.

On the index of the subgroup generated by the Heegner divisors Castao-Bernard C. (Italy) ICTP (Trieste) ccastano@ictp.it Let E be an elliptic curve defined over Q and assume E has rank + one. So there is a non-trivial morphism p : X0 (N) E defined over Q + such that i OE E, where X0 (N) is the quotient of X0(N) by the + Fricke involution wN, and N is the conductor of E. Denote by J0 (N) the + Jacobian of X0 (N) and let P be a generator of the subgroup generated + + + by the set of p(yD) E(Q), where yD J0 (N)(Q) runs through all Heegner divisors as in [2]. (Cf. [1].) This talk is about a conjectural + relation between the index [E(Q) : ZP ] and the real locus X0 (N)(R) of + the quotient curve X0 (N) suggested by new numerical evidence.

References [1] Borcherds, R.E., The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J., 97 (1999), no. 2, p. 219–233.

[2] Gross, B.H., Kohnen, W., and Zagier, D.B., Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), no. 1–4, p.

497–562.

Singularities of dynamical systems:

a catastrophic viewpoint Chaperon M. (France) Universit Paris chaperon@math.jussieu.fr The idea of stratifying function spaces is not as familiar in dynamics as in differential topology or in the theory of singularities of smooth maps.

The aim of this talk is to present in that framework various results, some of which are quite recent, and various questions, some of which must be very hard and might be ill-posed.

An aspect of the subject is that the birth of invariant circles in the “Hopf” bifurcation generalizes to generic families in dimension 2n, depending on at least n parameters, as the birth not only of invariant n–tori [1] but also of various other higher dimensional compact invariant manifolds [2,3,4,5], the largest of which are (2n - 1)–spheres. Their study has to do with Lotka-Volterra systems, on which it brings some apparently new information [3].

References [1] Broer H., Huitema G.B., Sevryuk M.B. Quasi-periodic motions in families of dynamical systems. Order amidst chaos.

Lecture Notes in Mathematics 1645 (1997), Springer-Verlag.

[2] Chaperon M., Kammerer-Colin de Verdire M., Lpez de Medrano S. More compact invariant manifolds appearing in the non-linear coupling of oscillators.

C. R. Acad. Sci., Paris, Sr. I, 342 (2006), 301–305.

[3] Chaperon M., Lpez de Medrano S. On generalized Hopf bifurcations and the regularity of carrying simplices.

To appear.

[4] Kammerer-Colin de Verdire M. Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions.

C. R. Acad. Sc. Paris 339 (2004), Groupe 1, 625–629.

[5] Kammerer-Colin de Verdire M. Bifurcations de varits invariantes.

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