Local and global -regularity

A bounded domain is called -regular if the plurisubharmonic envelope of every continuous function on extends continuously to . We show using Gauthier's Fusion Lemma that a domain is locally -regular if and only if it is -regular.

     

A solution to the Baer splitting problem

Let be a commutative domain. We prove that an -module is projective if and only if for any torsion module . This answers in the affirmative a question raised by Kaplansky in 1962.

     

Invariant subspaces for Banach space operators with an annular spectral set

Consider an annulus for some , and let be a bounded invertible linear operator on a Banach space whose spectrum contains . Assume there exists a constant such that and for all polynomials . Then there exists a nontrivial common invariant subspace for and .

     

The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups and exceptional Lie groups

For a fixed prime , we compute the Brown-Peterson cohomologies of classifying spaces of and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that and are even dimensionally generated.

     

Higher order PDE's and iterated processes

We introduce a class of stochastic processes based on symmetric -stable processes, for . These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric -stable process. We call them -time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of -time processes to some higher order PDE's for rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

     

Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Let be a Riemannian compact -manifold. We know that for any , there exists such that for any , , being the smallest constant possible such that the inequality remains true for any . We call the ``first best constant'. We prove in this paper that it is possible to choose and keep a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.

     

On a Ramanujan equation connected with the median of the gamma distribution

In this paper, we consider the sequence solving the Ramanujan equation

The three main achievements are the following. We introduce a continuous-time extension of and show its close connections with the medians of the distributions and the Charlier polynomials. We give upper and lower bounds for both and , in particular for , which are sharper than other known estimates. Finally, we show (and at the same time complete) two conjectures by Chen and Rubin referring to the sequence of medians .

     

Properly embedded least area planes in Gromov hyperbolic -spaces

Let be a Gromov hyperbolic -space with cocompact metric, and the sphere at infinity of . We show that for any simple closed curve in , there exists a properly embedded least area plane in spanning . This gives a positive answer to Gabai's conjecture from 1997. Soma has already proven this conjecture in 2004. Our technique here is simpler and more general, and it can be applied to many similar settings.

     

Homogeneous Hilbert scheme

Let be a locally noetherian scheme and an -graded -algebra of finite type. We say that is a homogeneous variety over . In this paper we prove that the functor

is representable by an -scheme that is a disjoint union of locally projective schemes over . The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective -scheme.

     

Rational functions with linear relations

We find all polynomials over a field such that and are linear and . We also solve the same problem for rational functions , in case the field is algebraically closed.

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Convex solids with planar midsurfaces

We show that the boundary of an -dimensional closed convex set , possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of lie in a hyperplane. To prove this statement, we show that the boundary of is a convex quadric surface if and only if there is a point such that all sections of by 2-dimensional planes through are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.

     

Amenability of Banach and C-algebras generated by unitary representations

We study the amenability of a locally compact group in relation to the amenability properties of a variety of -algebras and (quantized/dual) Banach algebras naturally associated to a unitary representation of .

     

On the Pytkeev property in spaces of continuous functions

Answering a question of Sakai, we show that the minimal cardinality of a set of reals such that does not have the Pytkeev property is equal to the pseudo-intersection number . Our approach leads to a natural characterization of the Pytkeev property of by means of a covering property of , and to a similar result for the Reznichenko property of .

     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

Hypersurfaces whose tangent geodesics do not cover the ambient space

Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .

     

On sums of powers of inverse complete quotients

For an irrational number , let denote its -th continued fraction inverse complete quotient, obtained by deleting the first partial quotients. For any positive real number , we establish the optimal linear bound on the sum of the -th powers of the first complete quotients. That is, we find the smallest constants such that for all and all irrationals .

     

Strong integrality of quantum invariants of 3-manifolds

We prove that the quantum -invariant of an arbitrary 3-manifold is always an algebraic integer if the order of the quantum parameter is co-prime with the order of the torsion part of . An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we also generalize Habiro's result to all rational homology 3-spheres.

     

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let be the blow-up of a projective scheme along the ideal sheaf of . It is known that there are embeddings for , where denotes the maximal generating degree of , and that there exists a Cohen-Macaulay ring of the form (which gives an arithmetic Macaulayfication of ) if and only if , for , and is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants and such that is Cohen-Macaulay for all d(I)e + \varepsilon$"> and e_0$">, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form . If has negative -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if , for 0$">, and is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of for all d(I)e + \varepsilon$"> and e_0$">.