Extended Systems with Deterministic Local Dynamics and Random Jumps

We consider particle systems on lattices with internal dynamics at each site and random jumps between sites. Models with simple chaotic local dynamics, namely expanding circle maps, are considered. Results on mean drift rates, central limit theorems and dependences on jump parameters are proved. A version of most of the results in this paper is contained in this author’s Ph.D. thesis [K]. This research is partially supported by a grant from the NSF.     

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     

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the papers style. This research was partially supported by NSF grant DMS-0305048.     

Poisson Quasi-Nijenhuis Manifolds

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.     

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.     

The Beltrami spectrum for incompressible fluid flows

Recently V. Yakhot, S. Orszag, and their co-workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows. A mathematical framework is developed here to study organized Beltrami hierarchies in a systematic fashion. This framework is applied to several important classes of examples with universal Beltrami hierarchies. An analysis of the persistence of such Beltrami hierarchies is also presented for general solutions of the Navier-Stokes equations.Research partially supported by grant NSF DMS-860-2031. Sloan research fellowship gratefully acknowledgedThis research was partially supported by grants NSF DMS 86-11110 and DARPA — ONR N00014-86-K-0759     

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     

Constructing Locally Connected Non-Computable Julia Sets

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.     

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     

Freely Generated Vertex Algebras and Non–Linear Lie Conformal Algebras

We introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.Acknowledgement. We would like to thank M. Artin, B. Bakalov, A. Dandrea and P. Etingof for useful discussions. This research was conducted by A. De Sole for the Clay Mathematics Institute. The paper was partially supported by the NSF grant DMS0201017.     

Spin Gromov-Witten Invariants

We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r^th Gelfand-Dickey hierarchy (or, equivalently, the A_r−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397     

Invariant states of a thermally conducting barrier

For an infinite one-dimensional system representing a thermally conducting barrier and two semi-infinite reservoirs which it separates, we prove the existence of a unique stationary probability distribution, to which essentially any initial distribution converges for large times.Research partially supported by NSF grant No. PHY-8003298.Research partially supported by NSF grant No. PHY-8201708.     

Uniqueness Results for Ill-Posed Characteristic Problems in Curved Space-Times

We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski space-times, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern ill-posed Cauchy problems on bifurcate, characteristic hypersurfaces. In the case of the Kerr space-time, the hypersurface is precisely the event horizon of the black hole. The uniqueness theorem in this case, based on two Carleman estimates, is intimately connected to our strategy to prove uniqueness of the Kerr black holes among smooth, stationary solutions of the Einstein-vacuum equations, as formulated in [14]. The first author was supported in part by an NSF grant and a Packard Fellowship. The second author was supported by NSF grant DMS 0070696.     

Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations

We prove uniqueness of solutions to the Maxwell-Dirac system in the energy space, namely . We also give a proof for uniqueness of finite energy solutions to the Maxwell-Klein-Gordon equations, which is simpler than that given in [16]. The first author is partially supported by an NSF grant and an Alfred Sloan fellowship. The second author is supported by JSPS Postdoctoral Fellowships for Research Abroad (2001–2003)     

Geometry of Batalin-Vilkovisky quantization

The geometry ofP-manifolds (odd symplectic manifolds) andSP-manifolds (P-manifolds provided with a volume element) is studied. A complete classification of these manifolds is given. This classification is used to prove some results about Batalin-Vilkovisky procedure of quantization, in particular to obtain a very general result about gauge independence of this procedure.Research supported in part by NSF grant No. DMS-9201366     

Some Geometric Calculations on Wasserstein Space

We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. This research was partially supported by NSF grant DMS-0604829.     

Setting the Quantum Integrand of M-Theory

In anomaly-free quantum field theories the integrand in the bosonic functional integral—the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand'. In the low-energy approximation to M-theory the E₈-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders. The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949     

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.     

On Ulam-von Neumann transformations

We define and study Ulam-von Neumann transformations which are certain interval mappings and conjugate toq(x)=1–2x ² on [–1,1]. We use a singular metric on [–1,1] to study a Ulam-von Neumann transformation. This singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point. We give the smooth classification of Ulam-von Neumann transformations by their eigenvalues at periodic points and exponents and asymmetries.The author is partially supported by a PSC-CUNY grant and a NSF grant.