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.     

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     

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.     

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     

Thickenings and conformal gravity

A twistor correspondence is given for complex conformal space-times with vanishing Bach and Eastwood-Dighton tensors; when the Weyl curvature is algebraically general, these equations are precisely the conformal version of Einstein's vacuum equations with cosmological constant. This gives a fully curved version of the linearized correspondence of Baston and Mason [B-M].Research supported in part by NSF grant DMS-8704401     

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     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Computer calculation of Witten's 3-manifold invariant

Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, and a Presidential Young Investigators award. The second author is supported by NSF grant DMS-8902153     

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     

Rotation Sets of Billiards with One Obstacle

We investigate the rotation sets of billiards on the m-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.The first author was partially supported by NSF grant DMS 0456748.The second author was partially supported by NSF grant DMS 0456526.The third author was partially supported by NSF grant DMS 0457168.     

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.     

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE ₆ paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).Research partially supported by the U.S. NSF under grant DMS-01-04278.     

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     

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     

Supersymmetric Vertex Algebras

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Supported in part by NSF grants DMS-0201017 and DMS-0501395.     

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     

A newN= 6 superconformal algebra

In this paper we construct a newN = 6 superconformal algebra which extends the Virasoro algebra by theSO ₆ current algebra, by 6 odd primary fields of conformal weight 3/2 and by 10 odd primary fields of conformal weight 1/2. The commutation relations of this algebra, which we will refer to asCK ₆, are represented by short distance operator product expansions (OPE). We constructCK ₆, as a subalgebra of theSO(6) superconformal algebra K₆, thus giving it a natural representation as first order differential operators on the circle withN = 6 extended symmetry. We show thatCK ₆ has no nontrivial central extensions. Partially supported by NSC grant 85-2121-M-006-019 of the ROC. Partially supported by NSF grant DMS-9622870.     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Equivariant Differential Characters and Symplectic Reduction

We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction. Supported in part by NSF grant DMS-0603892. Supported in part by NSF grant DMS-0456714.