Nonlinear Cauchy-Riemann operators in

This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

     

Small prime solutions of quadratic equations II

Let be non-zero integers and any integer. Suppose that and for . In this paper we prove that (i) if the are not all of the same sign, then the above quadratic equation has prime solutions satisfying and (ii) if all the are positive and , then the quadratic equation is soluble in primes Our previous results are and in place of and above, respectively.

     

Seiberg-Witten invariants, orbifolds, and circle actions

The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point-free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include the fact that } 1$"> -manifolds with fixed-point-free circle actions are simple type and a new proof of the equality . An infinite number of -manifolds with whose Seiberg-Witten invariants are still diffeomorphism invariants is constructed and studied.

     

Class numbers of real cyclotomic fields of prime conductor

The class numbers of the real cyclotomic fields are notoriously hard to compute. Indeed, the number is not known for a single prime . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields for the primes . It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue --on the basis of the Cohen-Lenstra heuristics-- that the probability that our table is actually a table of class numbers , is at least .

     

Subgroup growth in some pro- groups

For a group let be the number of subgroups of index and let be the number of normal subgroups of index . We show that for 2$">. If and does not divide or if and or , we show that for all sufficiently large . On the other hand if and divides , then is not even bounded as a function of .

     

Quantum cohomology of Hilb and enumerative applications

We compute the small quantum cohomology of Hilb and determine recursively most of the big quantum cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in . This extends the results obtained by Graber (2001) for Hilb and hyperelliptic curves in .

     

On standardized models of isogenous elliptic curves

Let be isogenous elliptic curves over given by standardized Weierstrass models. We show that (in the obvious notation)

and, moreover, that there are integers such that

where .

     

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.

     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

A Pieri-type formula for the -theory of a flag manifold

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form or the form , and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the -Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.

     

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 correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree if and only if it is a polynomial of degree on every geometric sequence of knots. Here a sequence with is called geometric if the knots coincide outside a ball , inside of which they satisfy for all and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in that can be distinguished by -invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over a universal Vassiliev invariant of degree for knots in .

     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

Dimension of families of determinantal schemes

A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

     

Solution of the quadratically hyponormal completion problem

For , let be a collection of () positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on to guarantee the existence of a quadratically hyponormal unilateral weighted shift with as the initial segment of weights. We prove that admits a quadratically hyponormal completion if and only if the self-adjoint matrix

is positive and invertible, where , , , , , and, for notational convenience, . As a particular case, this result shows that a collection of four positive numbers always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Minimal degrees which are but not

We give a short proof of the existence of minimal Turing degrees which are but not .

     

Sharp estimates for the Bochner-Riesz operator of negative order in

The Bochner-Riesz operator on of order is defined by

where denotes the Fourier transform and if , and if . We determine all pairs such that on of negative order is bounded from to . To be more precise, we prove that for the estimate holds if and only if , where

We also obtain some weak-type results for .

     

A partition relation using strongly compact cardinals

If If is strongly compact and \kappa $"> and is regular (or alternatively cf , then holds for .