Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mⁿ_i, of riemannian manifolds with special holonomy. We show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension 4. Additional results on the the detailed structure of the singular set support our main conjecture that if the Mⁿ_i are compact and a certain characteristic number, C(Mⁿ_i), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.The first author was partially supported by NSF Grant DMS 0104128 and the second by NSF Grant DMS 0302744.     

Regularity of harmonic maps with prescribed singularities

In this paper, we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds. This is motivated from one of Hawking's conjectures on the uniqueness of Kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum Einstein equation in general relativity.Research partially supported by a NSF grant DMS-8907849.Research partially supported by a NSF grant     

Grad ø perturbations of massless Gaussian fields

We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.Research partially supported by NSF Grant DMS-8802912Research partially supported by NSF Grant DMS-8601978 and DMS-8806731     

Classical intertwiner space and quantization

Given two symplectic realizations, a symplectic manifold called the classical intertwiner space is introduced as a classical analogue of an intertwiner space of representations of an associative algebra. We describe explicitly how a quantum data on realizations induces a quantum data on their classical intertwiner space.Research partially supported by NSF Grant DMS92-03398 and at MSRI supported by NSF Grant DMS90-22140     

Non-self-dual Yang-Mills connections with quadrupole symmetry

We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.Research partially supported by NSF Grant DMS-8806731Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918     

Stark effect in multielectron systems: Non-existence of bound states

We prove non-existence of bound states for a class ofN-body systems in homogeneous electric fields. This class includes atoms and Born—Oppenheimer molecules. This result in conjunction with a stability result of [HS] implies existence of resonances for such systems.Research partially supported by NSERC under Grant No. A7901 and NSF under Grant No. DMS8507040     

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     

A symmetric family of Yang-Mills fields

We examine a family of finite energySO(3) Yang-Mills connections overS ⁴, indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies ofRP ². These singular YM connections interpolate between the smooth solutions. Depending on the parameters, the curvature may be self-dual, anti-self-dual, or neither. For the (anti)self-dual connections, we compute the formal dimension of the moduli space. For the non-self-dual connections we examine the second variation of the Yang-Mills functional, and count the negative and zero eigenvalues. Each component of the non-self-dual moduli space appears to consist only of conformal copies of a single solution.This work was partially supported by an NSF Mathematical Sciences Postdoctoral Fellowship     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems IV

We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method ofinfrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.Research partially supported by US National Science Foundation under Grant MPS-75-11864Research partially supported by Canadian National Research Council under Grant A4015Research partially supported by US National Science Foundation under Grant MCS-75-21684-A01     

Classical and quantum mechanical systems of Toda-lattice type

Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a large Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups.Research partially supported by NSF Grant MCS 83-01582Research partially supported by NSF Grant MCS 79-03153     

The semiclassical limit forSU(2) andSO(3) gauge theory on the torus

We prove that forSU(2) andSO(3) quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure converge, asT0, to integrals with respect to a symplectic volume measure µ₀ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.Research supported in part by LEQSF Grant RD-A-08, and NSF Grant DMS 9400961.     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.Research supported in part by NSF Grant No. MCS-76-05857Research supported in part by NSF Grant No. MCS-74-07313-A02     

A Penrose-Like Inequality for General Initial Data Sets

We establish a Penrose-Like Inequality for general (not necessarily time symmetric) initial data sets of the Einstein equations which satisfy the dominant energy condition. More precisely, it is shown that the ADM energy is bounded below by an expression which is proportional to the square root of the area of the outermost future (or past) apparent horizon. The author is partially supported by NSF Grant DMS-0707086 and a Sloan Research Fellowship.     

A mean field spin glass with short-range interactions

We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044Research supported by the NSF under Grant No. DMR-8503544Research supported by the NSF under Grant No. DMR-8319301     

On the spectral problem for anyons

We consider the spectral problem resulting from the Schrödinger equation for a quantum system ofn2 indistinguishable, spinless, hard-core particles on a domain in two dimensional Euclidian space. For particles obeying fractional statistics, and interacting via a repulsive hard core potential, we provide a rigorous framework for analysing the spectral problem with its multi-valued wave functions.Partially supported by the Mathematical Sciences Research Institute, Berkeley California, under NSF Grant # DMS 8505550Partially supported under NSF Grant no. DMR-9101542     

The short distance behavior of (φ 4)3

We consider the ₃ ⁴ quantum field theory on a torus and study the short distance behavior. We reproduce the standard result that the singularities can be removed by a simple mass renormalization. For the resulting model we give anL _p bound on the short distance regularity of the correlation functions. To obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields.Research supported by NSF Grant DMS 9102564Research supported by NSF Grant PHY9200278.Research supported by the Natural Sciences and Engineering Research Council of Canada.     

Quasi-Spherical Metrics and Applications

In this paper, using the idea of Bartnik [B2] on quasi-spherical metrics we continue our study on the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Unlike the previous work [ST] of the authors and the work of Liu-Yau [LY], we only assume each boundary component has nonnegative curvature which is not identically zero. We also study the case that the boundary is embedded in the quotient of the infinity of the Euclidean space over a finite group. The regularity of the black hole boundary condition of quasi-spherical metrics is also discussed.Research partially supported by NSF of China, Projects 10001001 and 10231010.Research partially supported by Earmarked Grant of Hong Kong #CUHK4032/02P.     

Mean field upper bound on the transition temperature in multicomponent ferromagnets

Using local Ward identities and a new correlation inequality, we prove that the mean field transition temperature is an upper bound on the true transition temperature in multicomponent Heisenberg-type classical ferromagnets.Research partially supported by the NSF under Grant MCS-78-01885.