Arnold multiplicity and birational automorphisms Cheltsov I. (Russia) University of Edinburgh cheltsov@yahoo.com Arnold multiplicity is a local invariant of a holomorphic function defined by the square-integrability of fractional powers of the function. In one complex variable, it agrees with the ordinary multiplicity. But for holomorphic functions of several complex variables, Arnold multiplicity is a more subtle invariant which has connections with many problems in various areas of mathematics. Arnold multiplicity can be defined also for holomorphic sections of line bundles on complex manifolds. We will discuss relations between this local invariant of holomorphic sections of line bundles and global birational geometry of Fano varieties.

Linking and causality in globally hyperbolic spacetimesChernov V. V. (USA) Dartmouth College Vladimir.Chernov@dartmouth.edu We construct the invariant alk that is the generalization of the linking number to the case of nonzero homologous submanifolds and apply it to the study of causality in globally hyperbolic spacetimes (X, g). The space N of null geodesics in (X, g) is identified with the spherical cotangent bundle ST M of a Cauchy surface M. All the null geodesics passing through x X form a sky Sx N = ST M of x.

Low observed that if the link (Sx, Sy) is nontrivial, then x, y X are causally related. We show that in many cases (Sx, Sy) = 0 if and only if x, y X are causally related. We show that x, y in a nonrefocussing (X, g) are causally unrelated iff (Sx, Sy) can be deformed to a pair of Sm-1-fibers of ST M M by an isotopy through skies. Low proved that if (S, g) is refocussing, then M is a closed manifolds. We prove that the universal cover of M is also a closed manifold.

based on a joint work with Yuli Rudyak Trigonometric sums in the number theory and analysis Chubarikov V. N. (Russia) Lomonosov Moscow State University This talk is devoted to trigonometric sums in the number theory and analysis, in particular, to the P. L. Chebyshev moment method which is known in the theory of probabilities.

From one side, this method permit to describe some properties of a function on its moments. Often the computation of moments is a simple problem of analysis. From an other side, if the function is large, it is large on the set of the positive measure. These arguments often give the solution of a problem.

I. M. Vinogradov defined the place of analysis in the number theory as follows. He wrote: “Analysis makes it possible to extend considerably the range of problems of the number theory and provides for a more rapid development of this science. I also want to point out one more useful feature of the analytic methods in the number theory. While solving new difficult problems, analysis itsef develops and gets more perfect. Dirichlet’s series and the theory of (s) function can serve as examples as well as some properties of Bessel’s functions of a complex variable (for instance, the theorems of Lindelf, Phragmen, Mellin), discontinuous sums and integrals etc. Thus, the application of the analytic method to the number theory enriches the science with new valuable achievements and, at the same time, develops and perfects the analysis itself”.

We will discuss:

1. The I. M. Vinogradov Mean Value Theorem 2. The Moment Problem for Multiple Trigonometric Sums 3. A upper bound for Weyl sums 4. Multiple Trigonometric Sums on Primes 5. The distribution of values of short trigonometric sums 6. Estimates of trigonometric integrals and complete rational trigonometric sums 7. Some Unsolved Problems.

Vassiliev invriants that do not distinguish mutant knots Chmutov S. V. (USA) Ohio State University chmutov@math.ohio-state.edu The purpose of this presentation is to describe all Vassiliev invariants that do not distinguish mutant knots in terms of their weight systems.

Namely, a (canonical) Vassiliev invariant does not distinguish mutant knots if and only if its weight system depends on the intersection graph of a chord diagram only.

Joint work with Sergei Lando.

Local invariants in real geometry and regularity conditions Comte Georges (France) University of Nice-Sophia Antipolis comte@math.unice.fr For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic (actually definable) set.

Limit cycle bifurcation in thermohaline convection box-model Davydov A. A. (Russia, Austria) Vladimir State University International Institute for Applied Systems Analysis (IIASA) davydov@iiasa.ac.at Melnikov N. B. (Austria, Russia) International Institute for Applied Systems Analysis (IIASA) Central Economics and Mathematics Institute of the Russian Academy of Sciences Lomonosov Moscow State University melnikov@iiasa.ac.at The thermohaline convection box-model proposed by Welander [1] can be written as = 1 - x - q(z)x = (1 - y) - q(z)y (1) where z = -x + ry, 0 < < 1, and 1 < r. Self-sustained oscillations numerically found in this model [1] were used to analyze and explain interdecadal ocean oscillations in general circulation models described by PDEs [2]. Here we prove existence of the limit cycle in the system (1) for a wide class of nonnegative nondecreasing transfer functions q which represent turbulent fluxes. First, we note that for any continuous transfer function the system (1) has at least one steady state, and all its steady states belong to the square = {(x, y) R2 : 0 x 1, 0 y 1}, which is an invariant set of the system (1).

Theorem 1 [3,4] Let the family q = Q(., ), > 0, of continuous transfer functions converge point-wise outside zero as 0+ to the function, > 0, such that (z) = 0, z < 0, and (z) =, z 0.

Then for sufficiently small there exists a unique steady state (x, ) in the system (1). If additionally this function is differentiable at the point, = - + r, and the inequality q() < -1 - - 2q() is true, then x this steady state is a hyperbolic repeller.

Poincare-Bendixson theorem yields the following Corollary [3,4] Let the function q satisfy the assumptions of Theorem 1, so that there is a unique steady state (x, ) which is a hyperbolic repeller.

Then the system (1) has a limit cycle inside the square that encloses the steady state (x, ).

Consider the flip-flop model, i. e. the transfer function has a jump at zero. Solutions to the system (1) with q = are understood in the Filippov sense.

Theorem 2 [3,4] Let 1 < r < 1/ and (r - 1)/(1 - r) < then the system (1) with q = has the unique steady state 1 - r 1 - r (x, y) =,, (2) 1 - r(1 - ) and this steady state is topologically equivalent to a stable focus.

The result preserves if we allow for nonzero left and right derivatives of the transfer function at zero.

Theorem 3 [4] Let 1 < r < 1/, and q is a piece-wise differentiable transfer function continuous outside of zero with a jump at zero such that q(0-) < (r-1)/(1-r) < q(0+). Then the point (2) is the unique steady state of the system (1) and it is topologically equivalent to a stable focus.

Moreover, the 2-get of the respective Poincare map at the point (2) does not depend on one-sided derivatives of the function q at zero.

Consider now a family Q(., ), > 0, of differentiable transfer functions that smoothes a function q from Theorem 3 and satisfies Theorem 1.

Then on coming out of zero a limit cycle bifurcation takes place analogous to the classical soft loss of stability.

References [1] Welander P. A. Simple Heat-Salt Oscillator, Dynamics of Atmosphere and Oceans, 6:4 (1982), 233–242.

[2] Rahmstorf S. Bifurcations of the Atlantic thermohaline circulation in response to changes in the hydrological cycle, Nature, 378 (1995), 145–149.

[3] Davydov A. A., Melnikov N. B. Andronov–Hopf bifurcation in simple double diffusion models, Uspekhi Matematicheskikh Nauk, 61:(2007), 175–176 (in Russian).

[4] Davydov A. A., Melnikov N. B. Soft loss of stability in ocean circulation box model with turbulent fluxes, Proceedings of the Steklov Institute of Mathematics, 2007, 16 pp. (in press).

Self-averaging and critical exponents in random spin systems De Sanctis Luca (Italy) ICTP lde@math.princeton.edu We illustrate a quite general setting in which it is easy to obtain typical results in random spin systems such as mean field and finite connectivity spin glasses. From stochastic stability we can obtain self-averaging with respect to the Gibbs measure and as a consequence a family of constraints on the distribution of multi-overlaps, which are the physical quantities encoding the thermodynamic properties of the models. From the selfaveraging with respect to the quenched-Gibbs measure we obtain further (and stronger) factorization properties. The same convexity arguments at the basis of the stochastic stability provide information on the free energy from which one can find the critical points and exponents of the overlaps.

Symplectic singularities of varieties: the method of algebraic restrictions Domitrz Wojciec (Poland) Institute of Mathematics, Polish Academy of Sciences domitrz@mini.pw.edu.pl This is the joint work with S. Janeczko and M. Zhitomirskii. We study germs of singular varieties in a symplectic space. In [1] V. Arnol’d discovered so called “ghost” symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of differential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of Darboux-Givental’ theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We prove that a quasi-homogeneous variety N is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 singularity, and for regular union singularities.

References [1] V. I. Arnold, First step of local symplectic algebra, Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs’ 60th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 194(44), 1999, 1–8.

On mono-monostatic bodies and turtles Domokos G. (Hungary) Budapest University of Technology and Economics, Department of Mechanics, Materials and Structures domokos@iit.bme.hu Varkonyi P.L. (Hungary) Budapest University of Technology and Economics, Department of Mechanics, Materials and Structures vpeter@mit.bme.hu Static equulibia of rigid bodies belong to the classical chapters of mechanics. Nevertheless, there are still interesting open mathematical questions, in particular, homogeneous, convex objects with just one stable equilibrium (called monostatic) appear to be an intriguing subject.

It is easy to show [1] that in 2D (e.g. among convex homogeneous slabs, rolling in their own plane) no monostatic bodies exist, this statement is analogous to the Four-Vertex-Theorem. In 3D the situation is different: monostatic bodies do exist, even among polyhedra, here the minimal number of faces is an interesting challenge. Conway and Guy [2] showed a monostatic polyhedron with 19 faces and up to now this appears to be the lowest number. If we go to higher dimensions, monostatic bodies appear to be less and less exotic. Even monostatic simplices have been identified by Dawson, Finbow and Mak [3,4] for suffeiciently high (D7) dimensions.

Arnold approached the problem from a different angle and asked [5] whether in 3D one could find a monostatic object with just one unstable equilibrium. Due to the Poincare-Hopf Theorem, if such a body exists, it will have no further (saddle) equilibrium point, hence we will refer to this type of object as mono-monostatic.

This idea not ony yields immediately (via the Poincare-Hopf Theorem) a nice claasification for 3D objects, one can also show that the existence of 3D bodies in an arbitrary equilibrium class can be deduced from the existence of a mono-monostatic body.

We constructed such a mono-monostatic object [6] and showed [7] that these geometric forms are particularly sensitive to small perturbations.

Statistical experiments with pebbles on the sea coast confirmed this statement. We also noticed that our mono-monostatic object [8] appeared to be similar to some turtle species. This visual similarity was confirmed by systematic measurements.

References [1] Ivanov I.I. Method of theory of functions at the boundary problems on the plate. // Diff. equations.– 1997. N8. p.1069 – 1075.

[2] Domokos G., Papadopoulos J. and Ruina A. Static equilibria of planar, rigid bodies: is there anything new // J. Elasticity. – 1994.

Vol36, p.59–66.

[3] Conway J.H., Guy R. Stability of polyhedra // SIAM Rev.–1969.Vol 11 p. 78—82.

[4] Dawson J., Finbow W., Mak P. Monostatic simplexes II. // Geometriae Dedicata – 1998. Vol 70 p.209–[5] Dawson J., Finbow W. Monostatic simplexes III. // Geometriae Dedicata – 2001. Vol 84 p.101–[6] Domokos G. My lunch with Arnold // Mathematical Intelligencer – 2006. Vol 28. No 4. p.31-[7] Vrkonyi P.L., Domokos G. Static equilibria of rigid bodies: dice, pebbles and the Poincare-Hopf Theorem // J. Nonlinear Sci. –2006.

Vol 16., p255–281.

[8] Vrkonyi P.L., Domokos G. Mono-monostatic bodies: the answer to Arnold’s question // Mathematical Intelligencer –2006. Vol 28.

No.4. p34–38.

[9] www.gomboc.eu On Uhlenbeck’s manifold Dymarskii Yakov (Ukraine) Lugansk National Pedagogical University dymarsky@lep.lg.ua We consider a family -y + p(x)y = y, y| = of Dirihlet eigenfunctions problems in which a real potential p C3() is a 2 functional parameter. Let W2 () be Sobolev space, S = {y W2 () :

y| = 0, y2 dx = 1}. K. Uhlenbeck [1] showed that the set Q = {q = (, y, p) R S C3() : -y + p(x)y = y} is smooth manifold with C3() as the model space. We equip a point q = (, y, p) the number n and multiplicity m of the eigenvalue and the obtain the stratification Q = Q(n, m). The topological properties of n,m Q and the stratification will be describe.

References [1] Uhlenbeck K. Generic properties of eigenfunctions // Amer. J.

Math. – 1976. v. 98. N 4. p. 1059–1078.

Identities for Beltrami differential parameter Dzhumadil’daev A. S. (Kazakhstan) Institute of Mathematics, Alma-Ata askar56@hotmail.com An algebraic structure on functions space of n-dimensional manifold defined by Beltrami differential parameter (f, g) = (grad f, grad g) is considered. Polynomial identities are found.

Poincar series and monodromy of the simple and unimodal boundary singularities Ebeling Wolfgang (Germany) Leibniz Universitat ebeling@math.uni-hannover.de A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities.

We show that the characteristic polynomial of the monodromy is related to the Poincar series of the coordinate ring of the ambient singularity Densities of topological invariants of subanalytic quasiperiodic sets.

Esterov A. I. (Canada) University of Toronto esterov@mccme.ru The composition of a linear mapping l : Rn RN and the projection RN RN/ZN is called a winding of the torus RN/ZN, if its image is dense in RN/ZN. The preimage L Rn of a subanalytic set M RN/ZN under a winding is called a quasiperiodic subanalytic set. The density of a numerical topological invariant i of L is the limit of the ratio i(L B)/Vol(B) where B is a ball in Rn, and its radius tends to infinity.

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