Laplace’s invariants of Monge–Ampre equations Kushner A. G. (Russia) Astrakhan State University, Astrakhan kushnera@mail.ru Classical Laplace’s invariants are defined for linear hyperbolic equations [1]. We construct generalized Laplace’s invariants for Monge-Ampre equations Avxx + 2Bvxy + Cvyy + D(vxxvyy - vxy) + E = 0, (1) Here A, B, C, D, E are smooth functions on x, y, v, vx, vy. Such equations can be considered as real (in hyperbolic case) [4] or complex (in elliptic case) [2] almost product structures on the space J1R2 of 1-jets of smooth functions on the plane R2(x, y). Corresponding decomposition of the de Rham complex on J1R2 generates tensor differential invariants of Monge-Ampre equations [3]. Generalized Laplace’s invariants 1 and for Monge-Ampre equations are differential 2-forms on J1R2 (in contrast to classical case, where Laplace’s invariants are functions). Stress that classical Laplace’s invariants are defined only for hyperbolic equations.

Generalized Laplace’s invariants allow to solve the problem of contact linearization of Monge-Ampre equations [3]. For example, if 1 = 2 = 0, then equation (1) is contact equivalent to the linear wave equation vxx - vyy = 0 (in hyperbolic case) or to the Laplace equation vxx + vyy = (in elliptic case).

References [1] Forsyth A.R. Theory of Differential Equations. Partial Differential Equations (part 4, vol.6) // Cambridge University Press, 596 pp., (1906) [2] Kushner A., Lychagin V., Rubtsov V., Contact Geometry and Nonlinear Differential Equations // Cambridge University Press, pp., (2007) [3] Kushner A. Almost Product Structures and Monge-Ampre Equations // Lobachevskii Journal of Mathematics, http://ljm.ksu.ru, vol. 23, pp. 151–181., (2006) [4] Lychagin V. Lectures on Geometry of Differential Equations (part 2) // ”La Sapienza”, Rome (1993) Representations and quantum integrability.

Recent development Lebedev D. R. (Russia) ITEP lebedev@mpim-bonn.mpg.de I shall review resent results of our group (A. Gerasimov, S. Kharchev, S. Oblezin and myself) on a direct connection between such phenomenons arising in integrable systems as separation of variables (quantum and classical) and Baxter Q-operator with representation theory of universal enveloping algebras and its quantum deformations. I shall explain connection of separated variables appear in integrable systems with moduli space of monopoles.

On some problems of symplectic topology arising in Hamiltonian dynamics Lerman L. M. (Russia) The University of Nizhny Novgorod, Research Institute for Applied Math. & Cybernetics lermanl@mm.unn.ru Hamiltonian dynamics is a rich source of problems in symplectic topology. In fact, symplectic topology itself has appeared as a geometrization of several problems in Hamiltonian dynamics. I intend to discuss several problems which I faced with during my work under the problems in Hamiltonian dynamics, both integrable and nonintegrable. Among them interrelations between the topology of the ambient manifold and the topology of the degeneracy set for an integrable Hamiltonian system, the presence of degenerations of any order in “generic” integrable systems and bifurcations within the class of integrable systems [1], counting of intersection points for tori in scattering problems arising in homoclinic dynamics [2, 3, 4].

References [1] Lerman L.M. Isoenergetical structure of integrable Hamiltonian system in extended neighborhood of a simple singular point: Three degrees of freedom //Methods of Qualitative Theory of Differential Equations and Related Topics /Eds. L.Lerman, G.Polotovsky, L.Shilnikov), Supplement, AMS Translations, Ser. 2, V.200, Adv.

in Math. Sci., AMS, Providence, R.I., 2000, pp. 219-242.

[2] Koltsova O.Yu., Lerman L.M. Transverse Poincare homoclinic orbits in 2N-dimensional Hamiltonian Systems Close to the System with a Loop to a Saddle-Center // Int. J. Bifurcation & Chaos, V.6, No.(1996), 991-1006.

[3] Koltsova O.Yu., Lerman L.M., Delshams A., Gutierrez P. Homoclinic orbits to invariant tori near a homoclinic orbit to center-centersaddle equilibrium // Physica D 201 (2005), 268-290.

[4] Pushkar P.E. Lagrangian intersections in a symplectic space // Func.

Analysis Appl., v.34 (2000), No. 4, 288-292.

On the spectrum of the Sturm–Liouville operator with regular boundary conditions Makin A. S. (Russia) Moscow State University of Instrument-Making and Informatics alexmakin@yandex.ru This paper deals with the eigenvalue problem for the Sturm–Liouville equation u - q(x)u + u = 0 (1) considered on the interval (0, ) with the two-point boundary conditions determined by linearly independent forms with arbitrary complex-valued coefficients:

ai1u(0) + ai2u() + ai3u(0) + ai4u() = 0 (2) (i = 1, 2). The function q(x) is an arbitrary complex-valued function of the class L1(0, ). We denote Aij = a1ia2j - a2ia1j.

Let boundary conditions (2) be regular but not strengthened regular which is equivalent to the conditions A12 = 0, A14 + A23 = 0, A14 + A23 = (-1)+1(A13 + A24), where = 0, 1.

It is known [1] that the eigenvalues of problem (1)+(2) form two series:

0 = µ2, n,j = (2n + o(1))2 (3) if = 0, and n,j = (2n - 1 + o(1))2 (4) if = 1. Here, in both cases, j = 1, 2 and n = 1, 2,.... We introduce the notation µn,j = n,j = 2n - + o(1). It follows from [2] that asymptotic formulas (3) and (4) can be refined. Specifically, µn,j = 2n - + O(n-1/2).

For convenience we denote < q >= q(x) dx.

The main objective of the paper is obtaining essentially more precise asymptotic formulas for the eigenvalues of problem (1)+(2) and computation the regularized trace of first order.

Theorem 1. The numbers µn,j defined above satisfy the asymptotic relations < q > Aµn,j = 2n - + - + 2(2n - ) (2n - )(A14 + A23) fn rn,j +(-1)j +, (5) n 2n - where fn = o(1), rn,j = o(1), and if q(x) L2(0, ), then {fn} l4, {rn,j} l2.

If A14 = A23, A34 = 0, then formula (5) can be reduced to the form < q > ((-1)j - 1)Aµn,j = 2n - + + 2(2n - ) (A14 + A23)(2n - ) (-1)jA- q(t) cos[2(2n - )t] dt+ (A14 + A23)(2n - ) (-1)jA+ 2A34(A14 + A23)(2n - ) dn,j q(t)e2i(2n-)t dt q(t)e-2i(2n-)t dt +, n0 where dn,j = o(1), and if q(x) L2(0, ), then {dn,j} l2.

[2] Naimark M.A., Linear Differential Operators (Ungar, New York, 1967; Nauka, Moscow, 1969).

Modular strata of unimodal singularities Martin B. (Germany) Brandenburgische Technische Universitt Cottbus martin@math.tu-cottbus.de We find and describe unexpected isomorphisms between two very different objects associated to hypersurface singularities. One object is the Milnor algebra of a function, while the other object associated to a singularity is the local ring of the flatness stratum of the singular locus in a miniversal deformation, an invariant of the contact class of a defining function. Such isomorphisms exist for unimodal hypersurface singularities. However, for the moment it is badly understood, which principle causes these isomorphisms and how far this observation generalises. Here we also provide an algorithmic approach for checking algebra isomorphy.

References [1] V.I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps Vol. I, Birkhuser, (1985).

[2] C. Hertling, Frobenius manifolds and moduli spaces for sigularities, Camb. Univ. Press (2002).

[3] T. Hirsch, B. Martin, Modular strata of deformation functors, Computational Commutative and Non-commutative Algebraic Geometry, IOS Press, Amsterdam, (2005), 156-166.

[4] B. Martin, Algorithmic computation of flattenings and of modular deformations, J. Symbolic Computation 34(3) (2002), 199-212.

[5] B. Martin, Modular deformation and Space Curve Singularities, Rev.

Mat. Iberoamericana 19(2) (2003), 613-621.

[6] B. Martin, Modular Lines for Singularities of the T -series, Real and Complex Singularites, Birkhuser Verlag, Basel, (2006), 219-228.

On rational cuspidal plane curves, open surfaces and local singularities Melle-Hernndez A. (Spain) University Complutense of Madrid amelle@mat.ucm.es Let C be an irreducile projevtive plane curve in the complex projective plane P2. The classification of such curves, up to the action of the automorphism group of the plane, is a very difficult problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this talk we are mainly interested in the case when C is a rational curve.

Evenmore, the classification problem of the rational cuspidal projective plane curves is quite misterious. That is, to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of unibranch singularities of given topological type. One of the integers which help in the classification problem is the logarithmic Kodaira dimension of open surface P2 \ C. The classification of curves with (P2\C) < 2 has been recently finished by Miyanishi and Sugie, Tsunoda.

and Tono.

This remarkable problem of classification is not only important for its own sake, but it is also connected with crucial properties, problems and joint work with J. Fernndez de Bobadilla, I. Luengo and A. Nmethi conjectures in the theory of open surfaces, and in the classical algebraic geometry:

• Coolidge and Nagata problem. It predicts that every rational cuspidal curve can be transformed by a Cremona transformation into a line, (it is verified in all known cases).

• Orevkov’s conjecture which formulates an inequality involving the degree d and numerical invariants of local singularities. In a different formulation, this is equivalent with the positivity of the virtual dimension of the space of curves with fixed degree and certain local type of singularities which can be geometrically realized.

• Rigidity conjecture of Flenner and Zaidenberg. Fix one of ‘minimal logarithmic compactifications’ (V, D) of P2 \ C, that is V is a smooth projective surface with a normal crossing divisor D, such that P2 \ C = V \ D, and (V, D) is minimal with these properties. The sheaf of the logarithmic tangent vectors V D controls the deformation theory of the pair (V, D), The rigidity conjecture asserts that every Q-acyclic affine surfaces P2 \ C with logarithmic Kodaira dimension (P2 \ C) = 2 is rigid and has unobstructed deformations. Note that the open surface P2 \ C is Q-acyclic if and only if C is a rational cuspidal curve.

The aim of the propose talk is to present some of these conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors in [1]:

• ‘Compatibility property’ is a sequence of inequalities, conjecturally satisfied by the degree and local invariants of the singularities of a rational cuspidal curve.

Consider a collection (C, pi) of locally irreducible plane curve sini=gularities, let i(t) be the characteristic polynomial of the monodromy action associated with (C, pi), and (t) := i(t), with deg (t) = i 2 (C, pi). Then (t) can be written as 1 + (t - 1) + (t - 1)2Q(t) for some polynomial Q(t). Let cl be the coefficient of t(d-3-l)d in Q(t) for any l = 0,..., d - 3.

Conjecture CP Let (C, pi) be a collection of local plane curve sini=gularities, all of them locally irreducible, such that 2 = (d - 1)(d - 2) for some integer d. If (C, pi) can be realized as the local singularities of a i=degree d (automatically rational and cuspidal) projective plane curve then cl (l + 1)(l + 2)/2 for all l = 0,..., d - 3. (1) Theorem [1] If (P2 \ C) is 1, then Conjecture CP is true (in fact with nl = 0).

References [1] J. Fernndez de Bobadilla, I. Luengo, A. Melle-Hernndez, and A. Nmethi, On rational cuspidal projective plane curves, Proc.

London Math. Soc. (3) 92 (2006), no. 1, 99–138.

Submanifolds in Pseudo-Euclidean spaces, associativity equations, and Frobenius manifolds Mokhov O. I. (Russia) Centre for Nonlinear Studies, L. D. Landau Institute for Theoretical Physics RAS;

Department of Geometry and Topology, Moscow State University mokhov@mi.ras.ru We prove that the associativity equations of two-dimensional topological quantum field theories (the Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations, see [1]) for a function (a potential or prepotential) = (u1,..., uN), N N 3 kl = uiujuk ulumun k=1 l=N N 3 = kl, (1) uiumuk ulujun k=1 l=where ij is an arbitrary constant nondegenerate symmetric matrix, ij = ji, ij = const, det(ij) = 0, are very natural reductions of the fun damental nonlinear equations of the theory of submanifolds in pseudoEuclidean spaces (namely, the Gauss equations, the Peterson–Codazzi– Mainardi equations, and the Ricci equations) and determine a natural class of potential flat submanifolds without torsion. We prove that for this class of submanifolds the Peterson–Codazzi–Mainardi equations are fulfilled automatically, the Gauss equations coincide with the Ricci equations and both of them coincide with the associativity equations of twodimensional topological quantum field theories for a potential (u). We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces possess natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds).

Recall that each solution (u1,..., uN) of the associativity equations (1) gives N-parametric deformations of Frobenius algebras, i.e., commutative associative algebras equipped by nondegenerate invariant symmetric bilinear forms, and that locally the tangent space at every point of any Frobenius manifold bears the structure of Frobenius algebra, which is determined by a solution of the associativity equations (1) and smoothly depends on the point (see [1]). We prove that each N-dimensional Frobenius manifold can locally be represented as a potential flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system, which is a natural generalization of the associativity equations (1), namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method. The connection of the construction with integrable hierarchies, nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, Poisson pencils and recursion operators can be found in [2].

References [1] Dubrovin B. Geometry of 2D topological field theories // In: Integrable Systems and Quantum Groups, Lecture Notes in Math., Vol. 1620, Springer-Verlag, Berlin, 1996, pp. 120–348;

http://arXiv.org/hep-th/9407018 (1994).

[2] Mokhov O. I. Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations // Funkts. Analiz i Ego Prilozh. – 2006. – Vol. 40, No. 1. – P. 14–29; English translation in: Functional Analysis and its Applications. – 2006. – Vol. 40, No. 1. – P. 11–23;

http://arXiv.org/math.DG/0406292 (2004).

[3] Mokhov O. I. Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds // Proceedings of the Workshop “Nonlinear Physics. Theory and Experiment. IV”, Gallipoli (Lecce), Italy, 22 June – 1 July, 2006; Preprint MPIM2006-152. Max-Planck-Institut fr Mathematik. Bonn, Germany. 2006; http://arXiv.org/math.DG/0610933 (2006) (will be published in “Theoretical and Mathematical Physics”, 2007).

Singularities and noncommutative Frobenius manifolds Natanzon S. M. (Russia) Moscow State University; Independent University of Moscow;

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