tensor categories from quantum groups

Let be a semisimple Lie algebra and let be the ratio between the square of the lengths of a long and a short root. Moreover, let be the quotient category of the category of tilting modules of modulo the ideal of tilting modules with zero -dimension for . We show that for a sufficiently large integer, the morphisms of are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II factor for each object of .

     

The Arason invariant and mod 2 algebraic cycles

Let be a field, over a smooth variety with function field and a quadratic vector bundle over . Assuming that the generic fibre of is in , we compute the image of its Arason invariant

in by the differential of the Bloch-Ogus spectral sequence. This gives an obstruction to being a global cohomology class.

     

Lattice paths and Kazhdan-Lusztig polynomials

The purpose of this paper is to present a new non-recursive combinatorial formula for the Kazhdan-Lusztig polynomials of a Coxeter group . More precisely, we show that each directed path in the Bruhat graph of has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of is the sum, over all the lattice paths associated to all the paths going from to , of where , and are three natural statistics on the lattice path.

     

On an near an elliptic complex tangent

In this paper, we study the local biholomorphic property of a real -manifold near an elliptic complex tangent point . In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy of near , first considered in a paper of Bishop. When is a -smooth submanifold, using a result established by Kenig-Webster, we show that near , is a smooth Levi-flat -manifold with a neighborhood of in as part of its boundary. Moreover, near , is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to . This result was proved in the previous work of Kenig-Webster when . When is real analytic, we show that is real analytic with a neighborhood of in as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When or when but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author's joint work with Krantz.

     

L-series with nonzero central critical value

Given a cusp form of even integral weight and its associated -function , we expect that a positive proportion of the quadratic twists of will have nonzero central critical value. In this paper we give examples of weight two newforms whose associated -functions have the property that a positive proportion of its quadratic twists have nonzero central critical value.

     

Mean growth of Koenigs eigenfunctions

In 1884, G. Koenigs solved Schroeder's functional equation

in the following context: is a given holomorphic function mapping the open unit disk into itself and fixing a point , is holomorphic on , and is a complex scalar. Koenigs showed that if , then Schroeder's equation for has a unique holomorphic solution satisfying

moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of . We call the Koenigs eigenfunction of . Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For , we prove a sufficient condition for the Koenigs eigenfunction of to belong to the Hardy space and show that the condition is necessary when is analytic on the closed disk. For many mappings the condition may be expressed as a relationship between and derivatives of at points on that are fixed by some iterate of . Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space .

     

Finite-dimensional representations of quantum affine algebras at roots of unity

We describe explicitly the canonical map Spec Spec , where is a quantum loop algebra at an odd root of unity . Here is the center of and Spec stands for the set of all finite--dimensional irreducible representations of an algebra . We show that Spec is a Poisson proalgebraic group which is essentially the group of points of over the regular adeles concentrated at and . Our main result is that the image under of Spec is the subgroup of principal adeles.

     

Asymptotic properties of Banach spaces under renormings

It is shown that a separable Banach space can be given an equivalent norm with the following properties: If is relatively weakly compact and , then converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in is 1-equivalent to the unit vector basis of (respectively, ) implies that contains an isomorph of (respectively, ).

     

Local exactness in a class of differential complexes

The article studies the local exactness at level in the differential complex defined by commuting, linearly independent real-analytic complex vector fields in independent variables. Locally the system admits a first integral , i.e., a complex function such that and . The germs of the ``level sets' of , the sets , are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension , of these level sets is sufficient for local exactness at the level . The condition was already known to be necessary.

     

Local Rankin-Selberg convolutions for : explicit conductor formula

Let be a non-Archimedean local field and , positive integers. For , let and let be an irreducible supercuspidal representation of . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant to the and an additive character of . This object is of central importance in the study of the local Langlands conjecture. It takes the form

where is an integer. The irreducible supercuspidal representations of have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of . This paper gives an explicit formula for in terms of the inducing data for the . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in , developed by Bushnell and Kutzko.

     

The number of Reidemeister moves needed for unknotting

There is a positive constant such that for any diagram representing the unknot, there is a sequence of at most Reidemeister moves that will convert it to a trivial knot diagram, where is the number of crossings in . A similar result holds for elementary moves on a polygonal knot embedded in the 1-skeleton of the interior of a compact, orientable, triangulated 3-manifold . There is a positive constant such that for each , if consists of tetrahedra and is unknotted, then there is a sequence of at most elementary moves in which transforms to a triangle contained inside one tetrahedron of . We obtain explicit values for and .

     

The enumerative geometry of surfaces and modular forms

Let be a surface, and let be a holomorphic curve in representing a primitive homology class. We count the number of curves of geometric genus with nodes passing through generic points in in the linear system for any and satisfying .

When , this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus constrained to points are also given in terms of quasi-modular forms.     

A new proof of D. Popescu's theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu's theorem which says that if is a regular homomorphism of noetherian rings, then is a filtered inductive limit of smooth finite type -algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type -algebra , mapping to , has a desingularization which is smooth wherever possible (roughly speaking, above the smooth locus of ). Secondly, we give sufficient conditions for to be a filtered inductive limit of its smooth finite type -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.

     

Regularity of the free boundary for the porous medium equation

We study the regularity of the free boundary for solutions of the porous medium equation , , on , with initial data nonnegative and compactly supported. We show that, under certain assumptions on the initial data , the pressure will be smooth up to the interface , when , for some . As a consequence, the free-boundary is smooth.

     

Locally analytic distributions and

In this paper we study continuous representations of locally -analytic groups in locally convex -vector spaces, where is a finite extension of and is a spherically complete nonarchimedean extension field of . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of , along with interesting new objects such as the action of on global sections of equivariant vector bundles on -adic symmetric spaces. We introduce a restricted category of such representations that we call ``strongly admissible' and we show that, when is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of . As an application we prove the topological irreducibility of generic members of the -adic principal series for . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous -valued representations of locally -analytic groups.

     

Grothendieck's theorem on non-abelian and local-global principles

A theorem of Grothendieck asserts that over a perfect field of cohomological dimension one, all non-abelian -cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization - to the context of perfect fields of virtual cohomological dimension one - takes the form of a local-global principle for the -sets with respect to the orderings of the field. This principle asserts in particular that an element in is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of . Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over is also given.

     

A new proof of Federer's structure theorem for

We prove that Federer's structure theorem for -dimensional sets in follows from the special case of -dimensional sets in the plane, which was proved earlier by Besicovitch.

     

Phragmén-Lindelöf principles on algebraic varieties

Estimates of Phragmén-Lindelöf (PL) type for plurisubharmonic functions on algebraic varieties in have been of interest for a number of years because of their equivalence with certain properties of constant coefficient partial differential operators; e.g. surjectivity, continuation properties of solutions and existence of continuous linear right inverses. Besides intrinsic interest, their importance lies in the fact that, in many cases, verification of the relevant PL-condition is the only method to check whether a given operator has the property in question.

In the present paper the property which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical Phragmén-Lindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between and the PL-condition used by Hörmander to characterize the surjectivity of differential operators on real-analytic functions.

We prove the property for an algebraic variety implies that , the tangent cone of at infinity, also has this property. The converse implication fails in general. However, if is a manifold outside the origin, then satisfies if and only if the real points in have maximal dimension and if the distance of to is bounded by as tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy is known, nor any of the other PL-conditions alluded to above.

Besides these main results the paper contains several auxiliary necessary conditions and sufficient conditions which make it possible to treat interesting examples completely. Since it was submitted they have been applied by several authors to achieve further progress on questions left open here.

     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.