Linear Stability in an Ideal Incompressible Fluid

 We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 RID="*" ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. RID="**" ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084. Acknowledgements. The authors thank Susan Friedlander for useful discussions. Communicated by P. Constantin     

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     

Decay estimates for Schrödinger equations

We prove global existence and optimal decay estimates for classical solutions with small initial data for nonlinear nonlocal Schrödinger equations. The Laplacian in the Schrödinger equation can be replaced by an operator corresponding to a non-degenerate quadratic form of arbitrary signature. In particular, the Davey-Stewartson system is included in the the class of equations we discuss.Partially supported by NSF grant DMS-860-2031. Sloan Research Fellow     

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     

Schrödinger Operators with Few Bound States

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, H, with bounded positive ground states: Given a potential V, if both H±V are bounded from below by the ground-state energy of H, then V≡0.D. D. was supported in part by NSF grant DMS–0227289.R. K. was supported in part by NSF grant DMS–0401277.B. S. was supported in part by NSF grant DMS–0140592.     

Recovery of singularities for formally determined inverse problems

In this paper we considered several formally determined problems in two dimensions. There are no global identifiability results for these problems. However, we can recover an important feature of these functions, namely their singularities. More precisely, we prove that one can determine the location and strength of singularities of anL compactly supported potential by knowing the associated scattering amplitude at a fixed energy. Also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium.Partially supported by NSF grant DMS-9123742Partially supported by NSF grant DMS-9100178     

Entropy production by block variable summation and central limit theorems

We prove a strict lower bound on the entropy produced when independent random variables are summed and rescaled. Using this, we develop an approach to central limit theorems from a dynamical point of view in which the entropy is a Lyapunov functional governing approach to the Gaussian limit. This dynamical approach naturally extends to cover dependent variables, and leads to new results in pure probability theory as well as in statistical mechanics. It also provides a unified framework within which many previous results are easily derived.Partially supported by an NSF postdoctoral fellowshipPartially supported by NSF grant DMS-8905772; A. P. Sloan Fellow in Mathematics     

Why instantons are monopoles

It is shown that instantons are hyperbolic monopoles for the loop group with non-maximal symmetry breaking at infinity.Research supported in part by NSF grant DMS-8506130     

Topological Calculation of the Phase of the Determinant of a Non Self-Adjoint Elliptic Operator

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.The first author was partially supported by the Alfred P. Sloan foundation.The second author was partially supported by the NSF grant DMS-0204421.     

Precise Constants in Bosonization Formulas on Riemann Surfaces. I

A computation of the constant appearing in the spin-1 bosonization formula is given. This constant relates Faltings’ delta invariant to the zeta-regularized determinant of the Laplace operator with respect to the Arakelov metric. Research supported in part by NSF grant DMS-0505512.     

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     

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.     

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     

Holonomy and Skyrme's Model

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L²_loc connection; the local developing maps for such connections need not be continuous.The first author was partially supported by NSF grant DMS-0204651.The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670     

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.     

Moment maps at the quantum level

We introduce the notion of moment maps for quantum groups acting on their module algebras. When the module algebras are quantizations of Poisson manifolds, we prove that the construction at the quantum level is a quantization of that at the semi-classical level. We also prove that the corresponding smashed product algebras are quantizations of the semi-direct product Poisson structures.Research partially supported by NSF grant DMS-89-07710     

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.Acknowledgements We thank Sam Evens for many useful discussions. The first author was partially supported by NSF grant DMS-0072520. The second author was partially supported by NSF(USA) grants DMS-0105195 and DMS-0072551 and by the HHY Physical Sciences Fund at the University of Hong Kong.     

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.Partially supported by NSF grant DMS-0245358.     

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835