Completely integrable gradient flows

In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.Supported in part by NSF Grant DMS-90-02136, NSF PYI Grant DMS-9157556, and a Seed Grant from Ohio State UniversitySupported in part by AFOSR grant AFOSR-96-0197, by U.S. Army Research Office grant DAAL03-86-K-0171 and by NSF grant CDR-85-00108Supported in part by NSF Grant DMS-8922699     

Quantum Riemann surfaces III. the exceptional cases

We discuss quantum deformations of Riemann surfaces whose fundamental groups are Abelian (the exceptional Riemann surfaces). We prove uniformization theorems, state deformation estimates, and study the dependence of quantum Riemann surfaces on the deformation parameter.Supported in part by NSF grant DMS-9206936.Supported in part by DOE grant DE-FG02-88ER25065.     

The existence of generalised self-dual Chern-Simons vortices

We establish the existence and uniqueness of radially symmetric self-dual topological vortices in thep=2 members of the hierarchies of (generalised) Chern-Simons Higgs and Abelian Higgs models. We also obtain all possible symmetric nontopological vortices in the Chern-Simons model characterised by an additional parameter governing the decay rates of the fields.Supported in part by CEC under grant HCM-ERBCHRXCT930362.Supported in part by NSF under grant DMS-9400243.     

Projective embeddings of complex supermanifolds

We generalize the Kodaira Embedding Theorem and Chow's Theorem to the context of families of complex supermanifolds. In particular, we show that every family of super Riemann surfaces is a family of projective superalgebraic varieties.Research supported in part by NSF grant DMS-8704401Research supported in part by NSF grant DMS-4253943Research also supported in part by NSF grant DMS-4253943     

A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation

We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in hyperbolic coordinates to the study of an ODE. Similar arguments extend to higher dimensions and other long range type nonlinear problems. Mathematics Subject Classifications (2000): 35L15, 74J30, 76B15 ★ Part of this work was done while H.L. was a Member of the Institute for Advanced Study, Princeton, supported by the NSF grant DMS-0111298 to the Institute. H.L. was also partially supported by the NSF Grant DMS-0200226. † Also a member of the Institute of Advanced Study, Princeton. Supported in part by NSF grant DMS-0100490.     

Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation

It is shown that for smooth initial data solutions of the Robinson-Trautman equation (also known as the two-dimensional Calabi equation) exist for all positive “times,” and asymptotically converge to a constant curvature metric. Supported in part by NSF grant DMS-885773 to the Courant Institute and by the Polish Ministry of Science Research grant RPBP 01.3     

Moduli of super Riemann surfaces

The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.Supported in part by NSF Grant No. DMS-8704401Supported in part by NSF Grants No. DMS-8501783 and No. DMS-86107301(1)     

Adiabatic theorems for dense point spectra

We prove adiabatic theorems in situations where the Hamiltonian has dense point spectrum. The gap condition of the standard adiabatic theorems is replaced by an appropriate condition on the ineffectiveness of resonances.Supported under NSF grant number DMS-8801548 and DMS-8416049     

Conformal blocks of minimal models on a Riemann surface

We give explicit integral representations for conformal blocks of minimal models on arbitrary compact Riemann surfaces.Supported by NSF under grant DMS-8505550     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

On the Averages of Characteristic Polynomials From Classical Groups

We provide an elementary and self-contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to Weyl and Littlewood. The first author was supported in part by the NSF grant FRG DMS-0354662. The second author was supported in part by the NSF postdoctoral fellowship and by the NSF grant DMS-0501245.     

Perturbation theory for the decay rate of eigenfunctions in the generalizedN-body problem

Simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe. We develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights. We show that generically (in a Baire category sense) eigenfunction decay rates are governed by the lowest threshold.Supported in part by NSF grant DMS-8807816     

Ann-dimensional Borg-Levinson theorem

We show that the potentialq is uniquely determined by the spectrum, and boundary values of the normal derivatives of the eigenfunctions of the Schrödinger operator –+q with Dirichlet boundary conditions on a bounded domain in ⁿ . This and related results can be viewed as a direct generalization of the theorem in the title, which states that the spectrum and the norming constants determine the potential in the one dimensional case.Supported by NSF grant DMS-8602033Supported by NSF grant DMS-8600797Supported by NSF grant DMS-8601118 and an Alfred P. Sloan Research Fellowship     

Gluing and Wormholes for the Einstein Constraint Equations

 We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes. Received: 4 October 2001 / Accepted: 26 July 2002 Published online: 29 October 2002 RID="*" ID="*" Supported by the NSF under Grant PHY-0099373 RID="**" ID="**" Supported by the NSF under Grant DMS-9971975 and at MSRI by NSF grant DMS-9701755 RID="***" ID="***" Supported by the NSF under Grant DMS-9704515     

On algebraic equations satisfied by hypergeometric correlators in WZW models. I

It is proven that integral expressions for conformal correlators insl(2) WZW model found in [SV] satisfy certain natural algebraic equations. This implies that the above integrals really take their values in spaces of conformal blocks.The second author was supported in part by the NSF grant DMS-9202280. The third author was supported in part by the NSF grant DMS-9203939     

The existence of non-minimal solutions of the Yang-Mills-Higgs equations overR 3 with arbitrary positive coupling constant

This paper proves the existence of a non-trivial critical point of theSU(2) Yang-Mills-Higgs functional onR ³ with arbitrary positive coupling constant. The critical point lies in the zero monopole class but has action bounded strictly away from zero.Supported in part by NSF Grant DMS-9200576.Supported in part by NSF Grant DMS-9109491.     

Stability of moving fronts in the Ginzburg-Landau equation

We use Renormalization Group ideas to study stability of moving fronts in the Ginzburg-Landau equation in one spatial dimension. In particular, we prove stability of the real fronts under complex perturbations. This extends the results of Aronson and Weinberger to situations where the maximum principle is inapplicable and constitutes a step in proving the general marginal stability hypothesis for the Ginzburg-Landau equation.Supported by EC grant SC1-CT91-0695Supported by NSF grant DMS-8903041     

On operad structures of moduli spaces and string theory

We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.To the memory of Ansgar SchnizerResearch supported by an NSF Postdoctoral Research FellowshipResearch supported in part by NSF grant DMS-9206929 and a Research and Study Leave from the University of North Carolina-Chapel HillResearch supported in part by NSF grant DMS-9108269.A03     

Characters and fusion rules forW-algebras via quantized Drinfeld-Sokolov reduction

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.Supported in part by Junior Fellowship from Harvard Society of FellowsSupported in part by NSF grants DMS-8802489 and DMS-9103792Supported in part by RIMS-91 ProjectCommunicated by A. Jaffe     

Pfaffian bundles and the Ising model

An infinite volume Pfaffian formalism is developed for the Ising model.Research supported in part by NSF grant DMS-8421289