equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

kramers equation and supersymmetry hamilton s equations with noise and friction possess a hidden supersymmetry valid for time independent as well as periodically time dependent systems it is used to derive topological properties of critical points and periodic trajectories in an elementary way from a more practical point of view the formalism provides new tools to study the reaction paths in systems with separated time scales a reduced current which contains the relevant part of the phase space probability current is introduced together with strategies for its computation

integrability of the 2d lotka volterra system via polynomial first integrals and polynomial inverse integrating factors we present new first integrals of the two dimensional lotka volterra systems which have a polynomial inverse integrating factor moreover we characterize all the polynomial first integrals of the two dimensional lotka volterra systems

examples of summing integral and nuclear operators on the space c 0 1 x with values in c0 abstract we give necessary and sufficient conditions that some operators on the space c 0 1 x with values in c 0 be as in the title related questions are investigated

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

a method to tackle first order ordinary differential equations with liouvillian functions in the solution the usual prelle singer ps approach misses many first order ordinary differential equations presenting liouvillian functions in the solution lfoodes we point out why and propose a method extending the ps method to solve a class of these previously unsolved lfoodes although our method does not cover all the lfoodes it maintains the semi decision nature of the usual ps method

first integrals of autonomous systems of differential equations and the prelle singer procedure the prelle singer procedure for determining elementary first integrals of two dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed the application of this procedure in several dynamical systems is reported

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

solving second order ordinary differential equations by extending the prelle singer method we propose a method for solving second order ordinary differential equations odes which is based on the ideas behind the prelle singer ps procedure for first order odes while the ps procedure treats differential equations des of the form y p x y q x y with p and q polynomials whose coefficients lie in the field of complex numbers our method is applicable to des of the form y p x y y q x y y the key to our approach is to focus not on the final solution but on the first order invariants of the equation our method is an attempt to address algorithmically the solution of second order odes with solutions in terms of elementary functions

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

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

on the complete integrability and linearization of nonlinear ordinary differential equations ii third order equations we introduce a method for finding general solutions of third order nonlinear differential equations by extending the modified prelle singer method we describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals the method is illustrated with several examples further we propose a powerful method of identifying linearizing transformations the proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation in addition to the above we provide an algorithm to invert the non local linearizing transformation through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation

fermionic path integration and grassmann brownian motion a grassmann probability theory with anticommuting random variables and stochastic processes is developed using an extension of berezin integration to infinite dimensional spaces a kolmogorov type consistency condition allows integration on spaces of paths in anticommuting space one particular stochastic process grassmann brownian motion is described and the associated measure used to give a path integral formula for the kernel of the evolution operator in fermionic quantum mechanics the fourier mode expansion of grassmann brownian motion is derived

supersymmetry and the atiyah singer index theorem using a recently introduced index for supersymmetric theories we present a simple derivation of the atiyah singer index theorem for classical complexes and itsg index generalization using elementary properties of quantum mechanical supersymmetric systems

interplay of symmetries and other integrability quantifiers in finite dimensional integrable nonlinear dynamical systems in this work we establish a connection between the extended prelle singer procedure and other widely used analytical methods to identify integrable systems in the case of n th order nonlinear ordinary differential equations odes by synthesizing these methods we bring out the interlink between lie point symmetries contact symmetries symmetries adjoint symmetries null forms darboux polynomials integrating factors the jacobi last multiplier and generalized symmetries corresponding to the n th order odes we also prove these interlinks with suitable examples by exploiting these interconnections the characteristic quantities associated with different methods can be deduced without solving the associated determining equations

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

interplay of symmetries null forms darboux polynomials integrating factors and jacobi multipliers in integrable second order differential equations in this work we establish a connection between the extended prellesinger procedure with five other analytical methods which are widely used to identify integrable systems in the contemporary liter

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

2 absolutely summing operators on the space c t x abstract we give for some banach spaces x and y examples of linear and continuous operators u c t x y such that u as2 x y for each c t and u c t as2 x y is a 2 absolutely summing operator with respect to the 2 absolute norm on as2 x y but u is not 2 absolutely summing

operator valued operators that are associated to vector valued operators this paper is motivated by a long standing conjecture of dinculeanu from 1967 let x and y be banach spaces and let be a compact hausdorff space dinculeanu conjectured that there exist operators s l c l x y s l c l x y which are not associated to any u l c x y u l c x y we study this existence problem systematically on three possible levels of generality the classical case c x c x of continuous vector valued functions p continuous vector valued functions and tensor products on each level we establish necessary and sufficient conditions for an l x y l x y valued operator to be associated to a y valued operator among others we see that examples proving dinculeanu s conjecture come out on the all three levels of generality

integral mappings between banach spaces abstract we consider the classes of grothendieck integral g integral and pietsch integral p integral linear and multilinear operators see definitions below and we prove that a multilinear operator between banach spaces is g integral resp p integral if and only if its linearization is g integral resp p integral on the injective tensor product of the spaces together with some related results concerning certain canonically associated linear operators as an application we give a new proof of a result on the radon nikodym property of the dual of the injective tensor product of banach spaces moreover we give a simple proof of a characterization of the g integral operators on c k x spaces and we also give a partial characterization of p integral operators on c k x spaces

a nonlocal connection between certain linear and nonlinear ordinary differential equations part ii complex nonlinear oscillators in this paper we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions for this purpose we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations odes with complex linear odes thereby proving the integrability of the former we also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples

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

new aspects of integrability of force free duffing van der pol oscillator and related nonlinear systems in this paper we show that the force free duffing van der pol oscillator is completely integrable for a specific parametric choice we derive a general solution for this parametric choice further we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature the procedure is shown to have a wider applicability through additional examples we also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example

on the complete integrability and linearization of certain second order nonlinear ordinary differential equations a method for finding general solutions of second order nonlinear ordinary differential equations by extending the prelle singer ps method is briefly discussed we explore integrating factors integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the ps method we also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equations we illustrate the theory with several potentially important examples and show that our procedure is widely applicable

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

global analytic first integrals for the real planar lotka volterra system we provide the complete classification of all lotka volterra systems of the form x x ax by c and y y ax by c in r2 having a global analytic first integral

asymptotic expansion of the witten deformation of the analytic torsion given a compact riemannian manifold md g a finite dimensional representation 1 m gl v of the fundamental group 1 m on a vector spacevof dimensionland a hermitian structure on the flat vector bundle formula associated to ray singer rs have introduced the analytic torsiont t m g 0 witten s deformationdq t of the exterior derivativedq dq t e htdqeht withh m ra smooth morse function can be used to define a deformationt h t 0 of the analytic torsiontwitht h 0 t the main results of this paper are to provide assuming that grad ghis morse smale an asymptotic expansion for log t h t fort of the form formula and to present two different formulae fora0 as an application we obtain a shorter derivation of results due to ray singer rs cheeger ch muller mu1 2 which in increasing generality concern the equality for odd dimensional manifolds of the analytic torsion with the average of the reidemeister torsion corresponding to the triangulation t h g and the dual triangulation td d h g

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

first integrals and reduction of a class of nonlinear higher order ordinary differential equations abstract we propose a method for constructing first integrals of higher order ordinary differential equations in particular third fourth and fifth order equations of the form x n h x x n 1 x are considered the relation of the proposed method to local and nonlocal symmetries are discussed

discontinuous galerkin hp adaptive methods for multiscale chemical reactors quiescent reactors abstract we present a class of chemical reactor systems modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems a mixed form discontinuous galerkin method is presented with implicit and explicit imex timestepping strategies coupled to dioristic entropy schemes for h p adaptivity of the solution where the h and p are adapted based on an l 1 stability result finally we provide some numerical studies on the convergence behavior adaptation and asymptotics of the system applied to a pair of equilibrium problems as well as to general three dimensional nonlinear lotka volterra chemical systems

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

on α nuclear operators with applications to vector valued function spaces abstract let x be a banach space and let y be a closed subspace of a banach space z let be a tensor norm our main result is as follows assume that x or z has the approximation property if there is a bounded linear extension operator from y to z then any bounded linear operator t x y is nuclear whenever t is nuclear from x to z using this result we characterize the space c p x respectively uc p x of continuous functions from a compact hausdorff space into a banach space x whose range is p compact respectively unconditionally p compact

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

factorization technique and isochronous condition for coupled quadratic and mixed lienard type nonlinear systems presented a procedure to factorize coupled quadratic lienard equation derived the isochronous condition and discussed the isochronous properties illustrated the procedure for a specific examples of physical interest discussed the factorization of coupled mixed lienard type equation in this paper we discuss a systematic and self consistent procedure to factorize a rather general class of coupled nonlinear ordinary differential equations odes namely coupled quadratic and mixed lienard type equations which include various physical and mathematical models the procedure is broadly divided into two parts in the first part we consider a general factorized form for the equation under consideration in terms of some unknown functions and identify the determining equations for them in the second part we systematically solve the determining equations and identify the compatible factorizing form for this class of equations in addition we also discuss the problem of identification of isochronous dynamical systems belonging to the above class of equations in particular we deduce an isochronicity condition for the coupled quadratic lienard equation we also present specific examples of physical interest

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the bartle dunford schwartz and the dinculeanu singer theorems revisited abstract let x and y be banach spaces and let be a compact hausdorff space by the classical bartle dunford schwartz theorem any operator s l c y admits an integral representation with respect to a y valued measure by the dinculeanu singer theorem each operator u l c x y admits an integral representation with respect to an l x y valued measure we establish an integral representation for any operator s l c l x y with respect to an l x y valued measure this far reaching extension of the bartle dunford schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper in particular it is an efficient tool that enables us to give an alternative simple proof to the dinculeanu singer theorem the latter theorem is proved in a more general context of x valued continuous functions with p compact range among others useful formulas which connect different vector measures are deduced

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

ising model and n 2 supersymmetric theories we establish a direct link between massive ising model and arbitrary massiven 2 supersymmetric qft s in two dimensions this explains why the equations which appear in the computation of spin correlations in the non critical ising model are the same as those describing the geometry of vacua inn 2 theories the tau function appearing in the ising model i e the spin correlation function is reinterpreted in then 2 context as a new index in special cases this new index is related to the ray singer analytic torsion and can be viewed as a generalization of that to the loop space of kahler manifolds

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

n 2 supersymmetric quantum mechanics on riemann surfaces with meromorphic superpotentials we constructn 2 supersymmetric quantum hamiltonians with meromorphic superpotentials on compact riemann surfaces and investigate the topological properties of these hamiltonians l2 cohomology groups for supercharge a deformed bar partial operator are considered and the witten index for the supersymmetric hamiltonian with meromorphic superpotential is calculated in terms of euler characteristic of the riemann surface and the degree of a divisor of poles for the differential of the superpotential

on the semi classical analysis of the ground state energy of the dirichlet pauli operator we complete and improve the results of a recent paper by ekholm kova ik and portmann in connection with a question of c guillarmou about the semiclassical expansion of the lowest eigenvalue of the pauli operator with dirichlet conditions we exhibit connections with the properties of the torsion function in mechanics the exit time of a brownian motion and the analysis of the low eigenvalues of some witten laplacian

proof of character valued index theorems a character valued index is a generalization of the ordinary dirac index to manifolds with nontrivial automorphism groups a simple proof of the corresponding fixed point theorem is presented which uses the techniques of supersymmetric quantum mechanics this theorem relates the character valued index to a topological integral of curvature forms on the fixed point space of the automorphism in question

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

linearization of third order ordinary differential equations by point and contact transformations we present here the solution of the problem on linearization of third order ordinary differential equations by means of point and contact transformations we provide in explicit forms the necessary and sufficient conditions for linearization the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

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

conditions for linearization of a projectable system of two second order ordinary differential equations a new method for linearizing a system of ordinary differential equations is introduced the method is applied to a system of two second order ordinary differential equations it is shown that for a particular class of equations the method gives more general linearization criteria than linearization via point transformations examples of systems of equations which are not linearizable via point transformations but linearizable by the new method are given

p summing operators on injective tensor products of spaces the final version of this paper appears in proceedings of the royal society of edinburgh section b 120a 1992 283 296 print

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

supersymmetry and the formal loop space abstract for any algebraic super manifold m we define the super ind scheme lm of formal loops and study the transgression map radon transform on differential forms in this context applying this to the super manifold m s x the spectrum of the de rham complex of a manifold x we obtain in particular that the transgression map for x is a quasi isomorphism between the 2 3 truncated de rham complex of x and the additive part of the 1 2 truncated de rham complex of l x the proof uses the super manifold ss x and the action of the lie super algebra sl 1 2 on this manifold this quasi isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space

nonlocal invariants in index theory in its original form the atiyah singer index theorem equates two global quantities of a closed manifold one analytic the index of an elliptic operator and one topological a characteristic number because it relates invariants from different branches of mathematics the index theorem has many applications and extensions to differential geometry k theory mathematical physics and other fields this report focuses on advances in geometric aspects of index theory for operators naturally associated to a riemannian metric on a closed manifold the topological side of the index theorem can often be expressed as the integral of local i e pointwise curvature expression we will first discuss these local refinements in 1 which arise naturally in heat equation proofs of the index theorem in 2 3 we discuss further developments in index theory which lead to spectral invariants the eta invariant and the determinant of an elliptic operator that are definitely nonlocal finally in 4 we point out some recent connections among these nonlocal invariants and classical index theory

on the complete integrability and linearization of nonlinear ordinary differential equations v linearization of coupled second order equations linearization of coupled second order nonlinear ordinary differential equations snodes is one of the open and challenging problems in the theory of differential equations in this paper we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled snodes our procedure gives several new types of linearizing transformations of both invertible and non invertible kinds in both cases we provide algorithms to derive the general solution of the given snode we illustrate the theory with potentially important examples

a rigorous path integral for supersymmetic quantum mechanics and the heat kernel in a rigorous construction of the path integral for supersymmetric quantum mechanics on a riemann manifold based on bar and pfaffle s use of piecewise geodesic paths the kernel of the time evolution operator is the heat kernel for the laplacian on forms the path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of mathai and quillen s thom form of a bundle over this space in the case of closed paths the bundle is the tangent space to the space of geodesic paths and the integral of this form passes in the limit to the supertrace of the heat kernel

convergent expansions for tunneling a new method to compute effects of tunneling in one dimensional multiple well is developed a tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling these splittings are given by convergent series in term of this tunneling parameter for a wide class of double well