A von Neumann type inequality for certain domains in

Strict contractions on a Hilbert space have a functional calculus with functions that are analytic in the unit disc of the complex plane; an estimate of the norm is then provided by von Neumann's inequality. We consider functions that satisfy related inequalities with respect to multioperators connected to certain domains in ; a representation formula and a Nevanlinna-Pick type theorem are obtained.

     

Resultants and Chow forms via exterior syzygies

Given a sheaf on a projective space , we define a sequence of canonical and effectively computable Chow complexes on the Grassmannians of planes in , generalizing the well-known Beilinson monad on . If the sheaf has dimension , then the Chow form of the associated -cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension . Using the theory of vector bundles and the canonical nature of the complexes, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to a polynomial formula for the resultant of five homogeneous forms of degree eight in five variables.

     

Sharp maximal estimates for doubly oscillatory integrals

We study doubly oscillatory integrals

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .

     

An analogue of Hardy's theorem for semi-simple Lie groups

A well known theorem of Hardy on Fourier transform pairs says that a function on and its Fourier transform cannot both be ``very rapidly decreasing'. We prove here an analogue of this result in the case of semi-simple Lie groups.

     

Strongly singular convolution operators on the Heisenberg group

We consider the mapping properties of a model class of strongly singular integral operators on the Heisenberg group ; these are convolution operators on whose kernels are too singular at the origin to be of Calderón-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation.

Our results are obtained by utilizing the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

Bounding the number of cycles of O.D.E.s in

Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the differential equation in . The special case of an annulus is further considered, and the criteria are used to deduce sufficient conditions for the differential equation to have at most one cycle. A bound on the number of cycles on surfaces of higher connectivity is given by similar conditions.

     

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

Given two densities on with the same total mass, the Monge transport problem is to find a Borel map rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.

     

Subvarieties of general type on a general projective hypersurface

We study subvarieties of a general projective degree hypersurface . Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface , for and , is of general type.

     

Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian

We show how for every integer one can explicitly construct distinct plane quartics and one hyperelliptic curve over all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in whose minimal polynomials over can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over , each with a pair of commuting involutions, are isomorphic to one another.

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

Exotic smooth structures on

We construct exotic and as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic using the surgery techniques of R. Fintushel and R. J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

Sebestyén moment problem: The multi-dimensional case

Given a family of vectors in a Hilbert space we characterize the existence of a family of commuting contractions on having regular dilation and such that

The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .

     

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

The multiplier ideals of a sum of ideals

We prove that if , are nonzero sheaves of ideals on a complex smooth variety , then for every we have the following relation between the multiplier ideals of , and :

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.

     

Exotic smooth structures on , Part II

We construct exotic and using the surgery techniques of R. Fintushel and R.J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

On the local smoothing for the Schrödinger equation

We prove a family of identities that involve the solution to the following Cauchy problem:

and the -norm of the initial datum . As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate proved by Constantin and Saut (1989), Sjölin (1987) and Vega (1988) and a uniqueness criterion for the solutions to the Schrödinger equation.

     

The range of traces on quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product -algebra. This enables us to show that all tracial states on induce the same homomorphism on , whose range is the group .