morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

nonlinear superposition formulae of the ito equation and a model equation for shallow water waves nonlinear superposition formulae of the ito equation and a model equation for shallow water waves are proved under certain conditions some particular solutions of the ito equation are obtained as an illustrative application of the obtained nonlinear superposition formula

the decay mode solutions for the cylindrical kp equation abstract the decay mode solutions for the cylindrical kadomtsev petviashvili equation can be obtained by the backlund transformation and hirota method

poincare birkhoff theorems in random dynamics we propose a generalization of the poincare birkhoff theorem on area preserving twist maps to area preserving twist maps f that are random with respect to an ergodic probability measure in this direction we will prove several theorems concerning existence density and type of the fixed points to this end first we introduce a randomized version of generalized generating functions and verify the correspondence between its critical points and the fixed points of f a fact which we successively exploit in order to prove the theorems the study we carry out needs to combine probabilistic techniques with methods from nonlinear pde and from differential geometry notably moser s method and conley zehnder theory our stochastic model in the periodic case coincides with the classical setting of the poincare birkhoff theorem

property t for non unital c algebras abstract inspired by the recent work of bekka we study two reasonable analogues of property t for not necessarily unital c algebras the stronger one of the two is called property t and the weaker one is called property t e it is shown that all non unital c algebras do not have property t neither do their unitalizations moreover all non unital unital c algebras do not have property t e

on equivalence of floer s and quantum cohomology we show that the floer cohomology and quantum cohomology rings of the almost kahler manifoldm both defined over the novikov ring of the loop space m are isomorphic we do it using a brst trivial deformation of the topological a model the relevant aspect of noncompactness of the moduli of pseudoholomorphic instantons is discussed it is shown nonperturbatively that any brst trivial deformation of a model which does not change the dimensions of brst cohomology does not change the topological correlation functions either

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

rational solutions of integrable equations via nonlinear superposition formulae in this paper a method is developed to obtain rational solutions of integrable equations which include kdv boussinesq kp ito 1 1 dimensional cdgks ramani and 2 1 dimensional cdgks equations the key step that we use to find rational solutions of these equations is the use of nonlinear superposition formulae all these rational solutions obtained for each equation are connected by a backlund transformation which enables us to find other new solutions through the nonlinear superposition of rational solutions and other known solutions

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

property t c algebras with amenable tracial states abstract as a non unital analogue of the main result in 6 we show in this article that if a is a separable quasi central c algebra with property t and is nuclear then there is a sequence of positive integers n k k n such that the c 0 direct sum of the family m n k k n of matrix algebras is an ideal j a of a and that a j a has no tracial state in particular a separable c algebra a is a c 0 direct sum of matrix algebras if and only if a is quasi central is nuclear has property t and every unital simple quotient of a has a tracial state we also show that a separable property t c algebra a is residually finite dimensional if and only if a admits an amenable faithful tracial state and every ideal of a has non zero center

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

decay mode solution of the two dimensional kdv equation and the generalized backlund transformation we consider the generalization of the customary backlund transform bt for the two dimensional 2d kdv equation ut 6uux uxxx x 3 2uyy 0 with being constant a nonlinear superposition formula has been obtained and it is shown that the present generalized bt can produce multiple soliton multiple decay mode solutions

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

morse novikov theory and cohomology with forward supports we present a new approach to morse and novikov theories based on the derham federer theory of currents using the finite volume flow technique of harvey and lawson hl in the morse case we construct a noncompact analogue of the morse complex relating a morse function to the cohomology with compact forward supports of the manifold this complex is then used in novikov theory to obtain a geometric realization of the novikov complex as a complex of currents and a new characterization of novikov homology as cohomology with compact forward supports two natural backward forward dualities are also established a lambda duality over the novikov ring and a topological vector space duality over the reals

nonlinear superposition formulae for the differential difference analogue of the kdv equation and two dimensional toda equation in this paper nonlinear superposition formulae of the differential difference analogue of the kdv equation and two dimensional toda equation are proved rigorously some particular solutions of the differential difference analogue of the kdv equation are given as an illustrative application of the obtained result

unstable geodesics and topological field theory a topological field theory is used to study the cohomology of mapping space the cohomology is identified with the becchi rouet stora tyutin cohomology realizing the physical hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua the method is illustrated by a simple example

lagrangian intersections and the serre spectral sequence for a transversal pair of closed lagrangian submanifolds l lof a sym plectic manifold m such that 1 l 1 l 0 c1 2 m 2 m and for a generic almost complex structure j we construct an invariant with a high homotopical content which consists in the pages of order 2o f as pec tral sequence whose differentials provide an algebraic measure of the high dimensional moduli spaces of pseudo holomorpic strips of finite energy that join l and l when l and lare hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the serre spectral sequence of the path loop fibration l pl l and we deduce some applications

kirwan novikov inequalities on a manifold with boundary we extend the novikov morse type inequalities for closed 1 forms in 2 directions first we consider manifolds with boundary second we allow a very degenerate structure of the critical set of the form assuming only that the form is non degenerated in the sense of kirwan in particular we obtain a generalization of a result of floer about the usual morse inequalities on a manifold with boundary we also obtain an equivariant version of our inequalities our proof is based on an application of the witten deformation technique the main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end this leads to a considerable simplification of the local analysis in particular we obtain a new analytic proof of the morse bott inequalities on a closed manifold

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

nondegeneracy morse index and orbital stability of the kp i lump solution using backlund transformation we analyze the spectral property of the kp i lump solution it is proved that the lump is nondegenerate and its morse index equals one as a consequence we show that it is orbitally stable

diagrams of the linear backlund transformation of the cylindrical tow dimensional toda lattice a fundamental set of solutions to the linear backlund transformation lbt is obtained in a general form in the cylindrically symmetric toda lattice diagrams of solutions generated by the lbt are drawn and their local behaviours are entirely known once only one solution is given

kazhdan s property t and c algebras abstract kazhdan s property t has recently been imported to the c world by bekka our objective is to extend a well known fact to this realm we show that a nuclear c algebra with property t is finite dimensional for all intents and purposes though the result is not surprising the proof is a bit more complicated than the group case

a backlund transformation and nonlinear superposition formula of a modified korteweg de vries type bilinear equation a modified korteweg de vries mkdv type bilinear equation that has been shown to pass the three soliton solution and the painleve tests see j math phys 2094 28 1987 2572 31 1990 is considered herein in this article a backlund transformation and a nonlinear superposition formula are proven rigorously as an application of the obtained results n soliton solutions are obtained therefore the integrability of this mkdv type bilinear equation is confirmed