Distributions supported in a hypersurface and local

We give a necessary condition for a distribution with compact support in a hypersurface to be in the local Hardy space . We apply this condition to prove a result distinguishing two types of Hardy spaces of distributions on a smooth domain .

     

An uncertainty inequality involving norms

We derive a sharp uncertainty inequality of the form

with . As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum of the functional , taken over all with (). Precisely, we obtain that which improves a previous bound of T. Gneiting.

     

The nonarchimedean theta correspondence for

In this paper we consider the theta correspondence between the sets and when is a nonarchimedean local field and . Our main theorem determines all the elements of that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of that occur in the theta correspondence between and . We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

     

Irreducible representations of the Cuntz algebra

In this paper, we establish formulas for the configuration of a special class of irreducible representations of the Cuntz algebra , . These irreducible representations arise as subrepresentations of naturally occurring representations of acting in and arise from consideration of multiresolution wavelet filters.

     

Subvarieties of -divisors in the Jacobian

We explore some of the interplay between Brill-Noether subvarieties of the moduli space of rank 2 bundles with canonical determinant on a smooth projective curve and -divisors, via the inclusion of the moduli space into , singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus , though no higher, bundles with sections are cut out by ; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space of curves of genus 3.

     

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

On an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Reducibility and nonreducibility between equivalence relations

We show that, for , the relation of -equivalence between infinite sequences of real numbers is Borel reducible to the relation of -equivalence (i.e., the Borel cardinality of the quotient is no larger than that of ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.

     

Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

Gap estimates of the spectrum of Hill's equation and action variables for

Consider the Schrödinger equation for a potential of period 1 in the weighted Sobolev space

where denote the Fourier coefficients of when considered as a function of period 1,

and where is the circle of length 1. Denote by the periodic eigenvalues of when considered on the interval with multiplicities and ordered so that We prove the following result. Theorem. For any bounded set there exist and so that for and , the eigenvalues are isolated pairs, satisfying (with

(i)

,

(ii)

.

     

On the image of the -adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Let be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic . It is shown that the Tate conjecture implies the surjectivity of the -adic Abel-Jacobi map, , for all and almost all . For a special class of threefolds the surjectivity of is proved without assuming any conjectures.

     

Non-existence of a curve over of genus 5 with 14 rational points

We show that an absolutely irreducible, smooth, projective curve of genus over with rational points cannot exist.

     

Weighted ergodic theorems for mean ergodic -contractions

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any -contraction with mean ergodic (ME) modulus, and for any positive contraction of with . We extend the return times theorem by proving that if is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any bounded measurable is a universally good weight for a.e. We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any -contraction with mean ergodic modulus converge in -norm. In order to produce weights, good for weighted ergodic theorems for -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of -contractions is the product of their moduli, and that the tensor product of positive quasi-ME -contractions is quasi-ME.

     

On the divergence of the means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double means of the Walsh-Fourier series () for functions in ( is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function we have a function such as and does not converge to a.e. (in the Pringsheim sense).

     

The general hyperplane section of a curve

In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of . We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of , arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in . For curves in , we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.