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 .

     

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.

     

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.

     

Invariant decomposition of functions with respect to commuting invertible transformations

Consider arbitrary elements. We characterize those functions that decompose into the sum of -periodic functions, i.e., with . We show that has such a decomposition if and only if for all partitions with consisting of commensurable elements with least common multiples one has .

Actually, we prove a more general result for periodic decompositions of functions defined on an Abelian group ; in fact, we even consider invariant decompositions of functions with respect to commuting, invertible self-mappings of some abstract set .

We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

     

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 .

     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

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 .

     

Every

It is known that congruence lattices of algebras in -permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest.

We show here that every -permutable variety satisfies the well-known and well-studied congruence identity . We also get a new condition equivalent to -permutability.

     

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 .

     

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.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

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 .

     

Projective normality of ruled surfaces

In this article we study normal generation of irrational ruled surfaces. When is a smooth curve of genus , Green and Lazarsfeld proved that a very ample line bundle    Pic with deg   Cliff is normally generated where Cliff denotes the Clifford index of the curve (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over .

     

Pseudocompact totally dense subgroups

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group admits a proper totally dense pseudocompact subgroup, then cannot have a torsion closed -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup of that contains an -bounded dense subgroup of (such an is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119-1129).

     

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.

     

Mean value of mixed exponential sums

For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

     

A polarized partition relation for cardinals of countable cofinality

We prove that if and , then

for all . This polarized partition relation holds if for every partition either there are and with or there are and with .

     

The sharp weighted bound for the Riesz transforms

We establish the best possible bound on the norm of the Riesz transforms as operators in the weighted space for in terms of the classical characteristic of the weight.

     

Characterizing algebraic stacks

We extend the notion of algebraic stack to an arbitrary subcanonical site . If the topology on is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to representable presheaves of groupoids whose domain map is a cover. This leads naturally to a definition of algebraic -stacks. We also compare different sites naturally associated to a stack.

     

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.