self duality equations for ginzburg landau and seiberg witten type functionals with 6th order potentials the abelian chern simons higgs model of hong kim pac and jackiw weinberg leads to a ginzburg landau type functional with a 6th order potential on a compact riemann surface we derive the existence of two solutions with different asymptotic behavior as the coupling parameter tends to 0 for any number of prescribed vortices we also introduce a seiberg witten type functional with a 6th order potential and again show the existence of two asymptotically different solutions on a compact kahler surface the analysis is based on maximum principle arguments and applies to a general class of scalar equations

second quantization of the elliptic calogero sutherland model we construct a quantum field theory model of anyons on a circle and at finite temperature we find an anyon hamiltonian providing a second quantization of the elliptic calogero sutherland model this allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic calogero sutherland hamiltonian

factorization homology i higher categories abstract we construct a pairing which we call factorization homology between framed manifolds and higher categories the essential geometric notion is that of a vari framing of a stratified manifold which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata our main result constructs labeling systems on disk stratified vari framed n manifolds from n categories these n categories in contrast with the literature to date are not required to have adjoints this allows the following conceptual definition the factorization homology m c of a framed n manifold m with coefficients in an n category c is the classifying space of c labeled disk stratifications over m the core calculation underlying our main result is the following for any disk stratified manifold the space of conically smooth diffeomorphisms which preserve a vari framing is discrete

associativity of quantum multiplication we proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold v for which c 1 a 1 for any effective class the same proof also works for any positive compact symplectic manifold with c 1 a 1

temperature states on loop groups boson fermion correspondence and a determinant identity the boson fermion correspondence for the representation of the car algebra over l2 s1 c defined by the kms state with chemical potential is considered and the non bijectivity shown using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of szego s theorem

dissolving four manifolds and positive scalar curvature we prove that many simply connected symplectic four manifolds dissolve after connected sum with only one copy of s2 s2 for any finite group g that acts freely on the three sphere we construct closed smooth four manifolds with fundamental group g which do not admit metrics of positive scalar curvature but whose universal covers do admit such metrics

fillings of unit cotangent bundles we study the topology of exact and stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface sigma g where g is at least 2 in particular we prove a uniqueness theorem asserting that any stein filling must be s cobordant rel boundary to the disk cotangent bundle of sigma g for exact fillings we show that the rational homology agrees with that of the disk cotangent bundle and that the integral homology takes on finitely many possible values including that of dt sigma g for example if g 1 is square free then any exact filling has the same integral homology and intersection form as dt sigma g

detecting non uniqueness of solutions to biharmonic integral equations through svd we consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation the integral operator and also the matrix depend on a real parameter which may also enter the singular values of the operator and the matrix when a singular value drops to zero for a certain critical value of the parameter the corresponding homogeneous integral equation or matrix equation has a nontrivial solution based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used it is therefore possible with a reasonable accuracy to detect such critical values for the integral operator simply through a singular value decomposition of the matrix derived by a quadrature and collocation

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

a stability property of symplectic packing we prove that for any closed symplectic 4 manifold m with h 2 m q there exists a number n 0 such that for every n n 0 m admits full symplectic packing by n equal balls we also indicate how to compute this n 0 our approach is based on donaldson s symplectic submanifold theorem and on tools from the framework of taubes theory of gromov invariants

symplectic topology and hamiltonian dynamics on etudie des applications symplectiques non lineaires capacites symplectiques construction d une capacite symplectique problemes de plongement problemes de rigidite

automorphisms of the canonical anticommutation relations and index theory let h be a complex hilbert space p an orthogonal projection on h and p the complementary projection if g is any symmetrically normed ideal in the ring of bounded operators on h then we consider the group of unitary operators on h such that p up and p u p lie in g when g is the hilbert schmidt class these unitaries define automorphisms of the c algebra b of the canonical anticommutation relations over h which are implementable in the representation of b determined by p we investigate the structure of the group u proving in particular that it has infinitely many connected components uk labelled by the fredholm index of p u p the connected component of the identity u0 is generated by unitaries of the form exp ia with a self adjoint and p a p in g finally we consider an application of these results to two dimensional field theory showing in particular that the charge and chiral charge quantum numbers arise as the fredholm indices of p up for certain unitary u on l2 r c2

the singular function expansion in time dependent scattering a class of linear initial boundary value problems relevant to time dependent scattering is solved exactly in terms of the complete orthogonal set of singular functions of the appropriate integral operator the method is primarily presented as a mathematically defensible alternative to the eigenmode expansion method the results extend the singularity expansion method to include the steady state response at source frequencies and provide a connection between resonance behavior in the steady state and low loss natural modes the method furthermore leads to an efficient automatic algorithm for the determination of natural modes and natural frequencies by observation of the singular values the conclusions are supported by numerical calculations

loop groups anyons and the calogero sutherland model the positive energy representations of the loop group of u 1 are used to construct a boson anyon correspondence we compute all the correlation functions of our anyon fields and study an anyonic w algebra of unbounded operators with a common dense domain this algebra contains an operator with peculiar exchange relations with the anyon fields this operator can be interpreted as a second quantized calogero sutherland cs hamiltonian and may be used to solve the cs model in particular we inductively construct all eigenfunctions of the cs model from anyon correlation functions for all particle numbers and positive couplings

return to thermal equilibrium by the solution of a quantum langevin equation a quantum mechanical treatment of the evolution of an anharmonic oscillator coupled to a heat bath is given it is shown that for a certain class of anharmonic potentials the heat bath drives the oscillator to an equilibrium state close to the quantum gibbs state associated to the potential thus a partial proof is provided for a conjecture of r benguria and m kac

the curve cone of almost complex 4 manifolds in this paper we study the curve cone of an almost complex 4 manifold which is tamed by a symplectic form in particular we prove the cone theorem as in mori theory for all such manifolds using the seiberg witten theory for small rational surfaces and minimal ruled surfaces we study the configuration of negative curves we define abstract configuration of negative curves which records the homological and intersection information of curves combinatorial blowdown is the main tool to study these configurations as an application of our investigation of the curve cone we prove the nakai moishezon type duality for all almost kahler structures on cp2 kcp2 with k 9 and minimal ruled surfaces with a negative curve this is proved using a version of gram schmidt orthogonalization process for the j tamed symplectic inflation

clustering of critical points in lefschetz fibrations and the symplectic szpiro inequality we prove upper bounds for the number of critical points in semi stable symplectic lefschetz fibrations we also obtain a new lower bound for the number of nonseparating vanishing cycles in lefschetz pencils and reprove the known lower bounds for the commutator lengths of dehn twists

zero sets of solutions to semilinear elliptic systems of first order consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n dimensional manifold assume the operator has the strong unique continuation property we show that the zero set of the solution is contained in a countable union of smooth n 2 dimensional submanifolds hence it is countably n 2 rectifiable and its hausdorff dimension is at most n 2 moreover it has locally finite n 2 dimensional hausdorff measure we show by example that every real number between 0 and n 2 actually occurs as the hausdorff dimension for a suitable choice of operator we also derive results for scalar elliptic equations of second order

many closed symplectic manifolds have infinite hofer zehnder capacity we exhibit many examples of closed symplectic manifolds on which there is an autonomous hamiltonian whose associated flow has no nonconstant periodic orbits the only previous explicit example in the literature was the torus t2n n 2 with an irrational symplectic structure the underlying smooth manifolds of our examples include for instance the k3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it infinitely many minimal four manifolds having any given finitely presented group as their fundamental group and simply connected minimal four manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3 the examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the neck in the symplectic sum have no closed characteristics we conjecture that any closed symplectic four manifold with b 1 admits symplectic forms with a similar property

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

almost kahler forms on rational 4 manifolds we study nakai moishezon type question and donaldson s tamed to compatible question for almost complex structures on rational four manifolds by extending taubes subvarieties current form technique to j nef genus 0 classes we give affirmative answers of these two questions for all tamed almost complex structures on s 2 bundles over s 2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds including the del pezzo ones

application of the singular function expansion to an integral equation for scattering the dirichlet scattering problem is solved exactly for a surface of arbitrary smooth shape the solution is given in terms of the complete orthonormal set of functions defined on the surface and arising as singular functions of the integral operator since the singular values do not have a point of accumulation at zero the method is of much greater practical value than the eigenmode expansion method eem since the terms in the infinite series are naturally ordered by size of singular value the dominance of one or several terms in the series may be established without explicit evaluation of the coupling coefficients extentions of the method to other boundary value problems and applications to the singularity expansion method sem are described

on the orders of periodic diffeomorphisms of 4 manifolds this paper initiated an investigation on the following question sup pose a smooth 4 manifold does not admit any smooth circle actions does there exist a constant c 0 such that the manifold support no smooth zp actions of prime order for p c we gave affirmative results to this question for the case of holomorphic and symplectic actions with an interesting finding that the constant c in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold

gromov s compactness of pseudo holomorphic curves and symplectic geometry on utilise certaines idees de schoerr et uhlenbeck sur les equations aux derivees partielles pour demontrer la compacite de gromov des courbes pseudoholomorphes

continuous solutions of nonlinear cauchy riemann equations and pseudoholomorphic curves in normal coordinates we establish elliptic regularity for nonlinear inhomogeneous cauchy riemann equations under weak assumptions and give a counterexample in a borderline case in some cases where the inhomogeneous term has a separable factorization the solution set can be explicitly calculated the methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures

unitary implementations of one parameter squeezing groups for the case of infinitely many photon boson modes we investigate the unitary implementability of a class of symplectic one parameter groups more exactly of the associated groups of bogoliubov automorphisms on the ccr algebra in the fock representation and in representations of the ccr algebra which are symplectically related and inequivalent to fock furthermore the existence of the associated squeezing quadratic hamiltonians is discussed finally applications in the theory of quantization in qed and in the luttinger model are pointed out

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

infinite dimensional groups and riemann surface field theories we show how to obtain positive energy representations of the groupg of smooth maps from a union of circles tou n from geometric data associated with a riemann surface having these circles as boundary using covering spaces we can reduce to the case wheren 1 then our main result shows that mackey induction may be applied and yields representations of the connected component of the identity ofg which have the form of a fock representation of an infinite dimensional heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs we give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum or cyclic vector this gives a geometric construction of correlation functions which had been obtained earlier using tau functions by choosing particular functions ing with non zero winding numbers on the boundary we obtain analogues of vertex operators described by segal in the genus zero case these special elements ofg which have a simple interpretation in terms of function theory on theriemann surface approximate fermion or clifford algebra operators they enable a rigorous derivation of a form of boson fermion correspondence in the sense that we construct generators of a clifford algebra from the unitaries representing these elements ofg

bounded cohomology and non uniform perfection of mapping class groups using the existence of certain symplectic submanifolds in symplectic 4 manifolds we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal lefschetz fibrations over surfaces of positive genus this estimate is then used to deduce that mapping class groups are not uniformly perfect and that the map from their second bounded cohomology to ordinary cohomology is not injective

symplectic 4 manifolds with kodaira dimension zero we discuss the notion of the kodaira dimension for symplectic manifolds in dimension 4 in particular we propose and partially verify betti number bounds for symplectic 4 manifolds with kodaira dimension zero

smooth s cobordisms of elliptic 3 manifolds the main result of this paper states that a symplectic s cobordism of elliptic 3 manifolds is diffeomorphic to a product assuming a canonical contact structure on the boundary based on this theorem we conjecture that a smooth s cobordism of elliptic 3 manifolds is smoothly a product if its universal cover is smoothly a product we explain how the conjecture fits naturally into the program of taubes of constructing symplectic structures on an oriented smooth 4 manifold with b 1 from generic self dual harmonic forms the paper also contains an auxiliary result of independent interest which generalizes taubes theorem sw gr to the case of symplectic 4 orbifolds

on nodal sets for dirac and laplace operators we prove that the nodal set zero set of a solution of a generalized dirac equation on a riemannian manifold has codimension 2 at least if the underlying manifold is a surface then the nodal set is discrete we obtain a quick proof of the fact that the nodal set of an eigenfunction for the laplace beltrami operator on a riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension we also see that the nodal set of a harmonic differential form on a closed manifold has codimension 2 at least a fact which is not true if the manifold is not closed examples show that all bounds are optimal

standard surfaces and nodal curves in symplectic 4 manifolds continuing the program of ds and u1 we introduce refinements of the donaldson smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components in cases where a corresponding analogue of the gromov taubes invariant is easy to define our invariants agree with those analogues we also prove a vanishing result for some of the invariants that count nodal curves

3 dimensional seiberg witten invariants and non kahlerian geometry the paper is concerned with seiberg witten invariants of closed 3 manifolds and their relation to the 4 dimensional invariants the main points are 1 the introduction of parameter dependent seiberg witten invariants for 3 manifolds with b1 0 these invariants depend on a complicated chamber structure defined by a differential geometric brill noether locus in the parameter space 2 a comparison theorem which shows that the seiberg witten invariants of 3 manifolds m with b 1 m 1 are simply the restrictions of the 4 dimensional invariants of the product s 1 m to the set of pull back spinc structures as an immediate consequence one gets the wall crossing formula for the invariants of 3 dimensional manifolds with b1 1 from the 4 dimensional wall crossing formula for manifolds with b 1 3 a novel method to compute the seiberg witten invariants of 3 manifolds m using non kahlerian complex geometry on s 1 m provided this product has such a structure which is compatible with a product metric and orientation induced from m this method relies essentially on a kobayashi hitchin correspondence which is new in the non kahler case 4 a complete description of those 3 manifolds to which our method applies these are riemannian manifolds which admit an oriented riemannian foliation by geodesics such manifolds are always diffeomorphic to seifert manifolds but the foliation has not to be seifert it can have non closed leaves 5 explicit computations of seiberg witten invariants for the 3 sphere s 3 using non seifert riemannian foliations we show that the chamber structure induced by the brill noether locus is very complicated there exist countably many dif ferent chambers and the seiberg witten invariant takes on all integer values

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

the maslov class of the lagrange surfaces and gromov s pseudo holomorphic curves for an immersed lagrange submanifold w c t x one can define a nonnegative integer topologic invariant m w such that the image of h1 w z under the maslov class is equal to m w z in this paper the value of m w is calculated for the case of a two dimensional oriented manifold x with the universal cover homeomorphic to r2 and an embedded lagrange torus w it is proved that if x t2 and w is homologic to the zero section then m w 0 in all the other cases m w 2 the last result is true also for a wide class of oriented properly embedded lagrange surfaces in t r the proof is based on the gromov s theory of pseudo holomorphic curves some applications to the hamiltonian mechanics are mentioned

on the holomorphicity of genus two lefschetz fibrations we prove that any genus 2 lefschetz fibration without reducible fibers and with transitive monodromy is holomorphic the latter condition comprises all cases where the number of singular fibers 10n is not congruent to 0 modulo 40 this proves a conjecture of the authors in siti1 an auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in s 2 bundles over s 2 of relative degree 7 over the base and of symplectic surfaces in cp 2 of degree 17

the corrected induced surface current for arbitrary conducting objects at resonance frequencies it is well known that using the method of moments in conjunction with either the electric or magnetic field surface integral equations efie or mfie produces inaccurate surface currents on conducting bodies at resonance frequencies a new technique based on singular value decomposition is developed to correct the computed current by adding a correction factor term this term is seen to be the resonant mode current obtained by employing the power method in the moment method matrix multiplied by an unknown complex factor applying the condition of vanishing field inside the conducting object results in obtaining the unknown complex factor therefore this technique is hereafter referred to as correction factor technique cft when the computed surface current on a conducting sphere proposed technique is compared with the exact one the numerical results show excellent agreement

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

small lefschetz fibrations and exotic 4 manifolds we explicitly construct genus 2 lefschetz fibrations whose total spaces are minimal symplectic 4 manifolds homeomorphic to complex rational surfaces mathbb cp 2 p overline mathbb cp 2 for p 7 8 9 and to 3 mathbb cp 2 q overline mathbb cp 2 for q 12 ldots 19 complementarily we prove that there are no minimal genus 2 lefschetz fibrations whose total spaces are homeomorphic to any other simply connected 4 manifold with b le 3 with one possible exception when b 3 meanwhile we produce positive dehn twist factorizations for several new genus 2 lefschetz fibrations with small number of critical points including the smallest possible example which follow from a reverse engineering procedure we introduce for this setting we also derive exotic minimal symplectic 4 manifolds in the homeomorphism classes of mathbb cp 2 4 overline mathbb cp 2 and 3 mathbb cp 2 6 overline mathbb cp 2 from small lefschetz fibrations over surfaces of higher genera

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

hurwitz type bound knot surgery and smooth mathbb s 1 four manifolds in this paper we prove several related results concerning smooth mathbb z p or mathbb s 1 actions on 4 manifolds we show that there exists an infinite sequence of smooth 4 manifolds x n n ge 2 which have the same integral homology and intersection form and the same seiberg witten invariant such that each x n supports no smooth mathbb s 1 actions but admits a smooth mathbb z n action in order to construct such manifolds we devise a method for annihilating smooth mathbb s 1 actions on 4 manifolds using fintushel stern knot surgery and apply it to the kodaira thurston manifold in an equivariant setting finally the method for annihilating smooth mathbb s 1 actions relies on a new obstruction we derived in this paper for existence of smooth mathbb s 1 actions on a 4 manifold the fundamental group of a smooth mathbb s 1 four manifold with nonzero seiberg witten invariant must have infinite center we also include a discussion on various analogous or related results in the literature including locally linear actions or smooth actions in dimensions other than four

a note on the boson fermion correspondence and infinite dimensional groups we show how to construct irreducible projective representations of the infinite dimensional lie group map s1 mathbb t by embedding it into the group of bogoliubov automorphisms of the car using techniques of g segal for extending certain representations of map s1 su 2 we show that our representations extend to give representations of a certain infinite dimensional superalgebra we relate our work to the well known boson fermion correspondence which exists in 1 1 dimensions

norms on the cohomology of hyperbolic 3 manifolds we study the relationship between two norms on the first cohomology of a hyperbolic 3 manifold the purely topological thurston norm and the more geometric harmonic norm refining recent results of bergeron engun and venkatesh as well as older work of kronheimer and mrowka we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3 manifold itself moreover we give families of examples showing that some but not all qualitative aspects of our estimates are sharp finally we exhibit closed hyperbolic 3 manifolds where the thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

temperature states on loop groups theta functions and the luttinger model abstract we consider projective representations of the loop group of u n suggested by statistical mechanics these representations are defined by certain positive definite functions called temperature states on the universal central extension of the loop group we find that the boson fermion correspondence known to exist in a mathematically precise form at zero temperature persists at non zero temperature by expressing the temperature state for u 1 in two different ways we obtain identities between elliptic functions we apply these ideas to a simple model in statistical physics the luttinger model establishing the existence of a projective representation of the loop group of u 1 u 1 in the presence of interaction and at all temperatures a rigorous derivation of the correlation functions for this model is given using the boson fermion correspondence

integration on loop group iii asymptotic peter weyl orthogonality using the wiener measure on the continuous loop group l g of a compact connected group g whose properties were studied in integration on loop group i we construct for g simply laced a stochastic central extension l g of l g despite the fact that this object has a multiplication defined almost everywhere we are able to define for it a convolution algebra

hamiltonian collapsing of irrational lagrangian submanifolds with small first betti number among the main symplectic invariants of a closed lagrange submanifoldl of the cotangent of n is the tubular radiusr l defined as the smallest tube d r n 1 of n t n in whichl can be pushed by an hamiltonian diffeotopy of n we show here using pseudo holomorphic techniques that such a submanifold cannot collapse if the first betti number ofl is smaller than 3 and if the maslov class ofl does not vanish in other words r l is then strictly positive and one can actually give an explicit lower bound in terms of the liouville and maslov classes ofl

on the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane it is demonstrated by specific examples that eigenvalues of the electric field integral equation efie operator for finite extent objects in lossless media can have branch points in the complex frequency s plane specifically it is shown from the analysis of the spheroidal wave equation in the s plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid similarly it is concluded from the analysis of the mathieu s differential equation that branch points besides that associated with the infinite extent of the object exist in the s plane behavior of the eigenvalues of the efie operator for a perfectly conducting infinite elliptic cylinder a proof is given that branch point singularities cannot occur in the eigenvalues of efie operators for structures in which geometrical symmetry completely determines the eigenfunctions and hence they are frequency independent it is conjectured that branch points may always be present when sufficient object symmetry is lacking this conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one an analogous phenomenon has been observed in circuit and transmission line problems for example it is shown that when the taper parameter of an exponential transmission line goes to zero the line becomes uniform and thus a symmetrical structure the branch points in the eigenvalues of the impedance matrix move away to infinity