Induced operators on symmetry classes of tensors

Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and is a character of degree 1 on . Consider the symmetrizer on the tensor space

defined by and . The vector space

is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying

In this paper, several basic problems on induced operators are studied.

     

Representation formulae and inequalities for solutions of a class of second order partial differential equations

Let be a possibly degenerate second order differential operator and let be its fundamental solution at ; here is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of on to satisfy the representation formula

We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition

then (R) holds with no growth assumptions on .

     

On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod :

where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are -cores and discuss an intriguing relation between -cores and the Andrews-Garvan crank.

     

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)

.

     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

Some quotient Hopf algebras of the dual Steenrod algebra

Fix a prime , and let be the polynomial part of the dual Steenrod algebra. The Frobenius map on induces the Steenrod operation on cohomology, and in this paper, we investigate this operation. We point out that if , then for any element in the cohomology of , if one applies enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that ``enough times' should be ``once.'

The bulk of the paper is a study of some quotients of in which the Frobenius is an isomorphism of order . We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about . The dual complete Steenrod algebra makes an appearance.

     

Dynamical forcing of circular groups

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties.

As an application, we show that the set consisting of rotation numbers which can be forced by finitely presented groups is an infinitely generated -module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number is forced by a pair , where is a finitely presented group and is some element, if the set of rotation numbers of as varies over is precisely the set .

We show that the set of subsets of which are of the form

where varies over countable groups, are exactly the set of closed subsets which contain and are invariant under . Moreover, we show that every such subset can be approximated from above by for finitely presented .

As another application, we construct a finitely generated group which acts faithfully on the circle, but which does not admit any faithful action, thus answering in the negative a question of John Franks.

     

Sharp dimension estimates of holomorphic functions and rigidity

Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that

for all , with equality for some positive integer if and only if is holomorphically isometric to . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

     

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If is a complex hyperplane arrangement, with complement , we show that the Chen ranks of are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring , viewed as a module over the exterior algebra on :

   for

where is a field of characteristic 0. The Chen ranks conjecture asserts that, for sufficiently large, , where is the number of -dimensional components of the projective resonance variety . Our earlier work on the resolution of over and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of and a localization argument, we establish the inequality

   for

for arbitrary . Finally, we show that there is a polynomial of degree equal to the dimension of , such that , for all .

     

Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes

Let be the relativistic -stable process in , , , with infinitesimal generator . We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup for this process with generator , , locally bounded. We prove that if , then for every the operator is compact. We consider the class of potentials such that , and is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For in the class we show that the semigroup is IU if and only if . If this condition is satisfied we also obtain sharp estimates of the first eigenfunction for . In particular, when , , then the semigroup is IU if and only if . For the first eigenfunction is comparable to

     

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.

     

Lagrangian submanifolds and moment convexity

We consider a Hamiltonian torus action on a compact connected symplectic manifold and its associated momentum map . For certain Lagrangian submanifolds we show that is convex. The submanifolds arise as the fixed point set of an involutive diffeomorphism which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.

     

Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

A homotopy principle for maps with prescribed Thom-Boardman singularities

Let and be smooth manifolds of dimensions and ( ) respectively. Let denote an open subspace of which consists of all Boardman submanifolds of symbols with . An -regular map refers to a smooth map such that . We will prove what is called the homotopy principle for -regular maps on the existence level. Namely, a continuous section of over has an -regular map such that and are homotopic as sections.

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Let denote the ring of integers of an algebraic number field of degree which is totally and tamely ramified at the prime . Write , where . We evaluate the twisted Kloosterman sum

where and denote trace and norm, and where is a Dirichlet character (mod ). This extends results of Salié for and of Yangbo Ye for prime dividing Our method is based upon our evaluation of the Gauss sum

which extends results of Mauclaire for .

     

Brownian intersection local times: Exponential moments and law of large masses

Consider independent Brownian motions in , each running up to its first exit time from an open domain , and their intersection local time as a measure on . We give a sharp criterion for the finiteness of exponential moments,

where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.

     

Besov spaces with non-doubling measures

Suppose that is a Radon measure on which may be non-doubling. The only condition on is the growth condition, namely, there is a constant 0$"> such that for all and 0,$">

where In this paper, the authors establish a theory of Besov spaces for and , where 0$"> is a real number which depends on the non-doubling measure , , and . The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.