Showing papers in "Illinois Journal of Mathematics in 2011"

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Abstract: Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

Homogeneous paracontact metric three-manifolds

Abstract: The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.

The weak Lefschetz property, monomial ideals, and lozenges

Abstract: We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function of each level Artinian monomial almost complete intersection in three variables is peaked strictly unimodal.

Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

Abstract: Let X be a (quasi-)Banach space. Let d = (dn)n 1 be an X-valued sequence of random variables adapted to a ltration ( Fn)n 1 on a probability space ( ;A;P), dene F1 := (Fn : n 1) and let e = (en)n 1 be aF1-conditionally independent sequence on ( ;A;P) such that L(dnjFn 1) =L(enjF1) for all n 1 (F0 =f ;?g). If there exists a p2 (0;1) and a constant Dp independent of d and e such that one has, for all n 1, X k=1 dk p D pE n X k=1 ek p

On weighted inequalities for fractional integrals of radial functions

Abstract: Fil: de Napoli, Pablo Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas "Luis A. Santalo"; Argentina

Irreducible vector-valued modular forms of dimension less than six

Abstract: An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincare series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert-Poincare series.

On actions of compact quantum groups

Abstract: We compare algebraic objects related to a compact quantum group action on a unital $C^∗$-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.

Tukey types of ultrafilters

Abstract: We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a $p$-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of $p$-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN.

Maps on noncommutative Orlicz spaces

Abstract: A generalization of the Pistone–Sempi argument, demonstrating the utility of noncommutative Orlicz spaces, is presented. In particular, regular quantum statistical systems are described. The question of lifting positive maps defined on von Neumann algebra to maps on corresponding noncommutative Orlicz spaces is discussed. In particular, we describe those Jordan ∗-morphisms on semifinite von Neumann algebras which in a canonical way induce quantum composition operators on noncommutative Orlicz spaces. Consequently, it is proved that the framework of noncommutative Orlicz spaces is well suited for an analysis of a large class of interesting noncommutative dynamical systems.

Asymptotic cones of Lie groups and cone equivalences

Abstract: We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Nontrivial examples appear in the context of Lie groups, and we thus prove that the study of asymptotic cones of connected Lie groups can be reduced to that of solvable Lie groups of a special form. We also focus on asymptotic cones of nilpotent groups.

Weakly controlled Moran constructions and iterated functions systems in metric spaces

Abstract: We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well-known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.

The space of commuting n-tuples in SU(2)

Abstract: Let $Y := \operatorname{Hom}(\mathbb{Z}^n, \operatorname{SU}(2))$ denote the space of commuting $n$-tuples in $\operatorname{SU}(2)$. We determine the homotopy type of the suspension $\Sigma Y$, and compute the integral cohomology groups of $Y$ for all positive integers $n$.

The number of representations of rationals as a sum of unit fractions

Abstract: For given positive integers $m$ and $n$, we consider the frequency of representations of $m/n$ as a sum of unit fractions.

Hypersurfaces with constant sectional curvature of $\mathbb {S}^{n}\times \mathbb {R}$ and $\mathbb {H}^{n}\times \mathbb {R}$

Abstract: We classify the hypersurfaces of $\mathbb {S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb {R}$ with constant sectional curvature and dimension $n\geq3$.

Backward iteration in the unit ball

Abstract: We will consider iteration of an analytic self-map $f$ of the unit ball in $\mathbb{C}^{N}$. Many facts were established about such dynamics in the 1-dimensional case (i.e., for self-maps of the unit disk), and we will generalize some of them in higher dimensions. In particular, in the case when $f$ is hyperbolic or elliptic, it will be shown that backward-iteration sequences with bounded hyperbolic step converge to a point on the boundary. These points will be called boundary repelling fixed points and will possess several nice properties. At each isolated boundary repelling fixed point, we will also construct a (semi) conjugation of $f$ to an automorphism via an analytic intertwining map. We will finish with some new examples.

HT90 and “simplest” number fields

Abstract: A standard formula (1.1) leads to a proof of HT90, but requires proving the existence of $\theta$ such that $\alpha e0$, so that $\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking $\theta=1$ makes $\alpha=0$. Taking $n=3$, we recover Shanks’s simplest cubic fields. The “simplest” number fields of degrees $3$ to $6$, Washington’s cyclic quartic fields, and a certain family of totally real cyclic extensions of $\mathbb{Q} (\cos(\pi/4m))$ all have defining polynomials whose zeroes satisfy (M). Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a $2$-parameter family of octic polynomials with “generic” Galois group ${}_{8}T_{11}$. Imposing an additional algebraic condition on these octics produces a new family of cyclic quartic extensions. This family includes the “simplest” quartic fields and Washington’s cyclic quartic fields as special cases. We obtain more detailed results on our octics when the parameters are algebraic integers in a number field. In particular, we identify certain sets of special units, including exceptional sequences of $3$ units, and give some of their properties.

Existence of regular neighborhoods for H-surfaces

Abstract: In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in $n$-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in certain $n$-manifolds.

On the moduli spaces of semi-stable plane sheaves of dimension one and multiplicity five

Abstract: We find locally free resolutions of length one for all semi-stable sheaves supported on curves of multiplicity five in the complex projective plane. In some cases we also find geometric descriptions of these sheaves by means of extensions. We give natural stratifications for their moduli spaces and we describe the strata as certain quotients modulo linear algebraic groups. In most cases we give concrete descriptions of these quotients as fibre bundles.

Rings of low rank with a standard involution

Abstract: We consider the problem of classifying (possibly noncom- mutative) R-algebras of low rank over an arbitrary base ring R. We rst classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard involution. We then investigate a class of exceptional rings of degree 2 which occur in every rank n 1 and show that they essentially characterize all algebras of degree 2 and rank 3. Let R be a commutative ring (with 1). Let B be an algebra over R, an associative ring with 1 equipped with an embeddingR ,! B of rings (mapping 12 R to 12 B) whose image lies in the center of B; we identify R with its image inB. Assume further thatB is a nitely generated, faithfully projective R-module of constant rank. The problem of classifying algebras B of low rank has an extensive history. The identication

On the spectrum of Banach algebra-valued entire functions

Abstract: In this paper, we investigate a notion of spectrum $\sigma(f)$ for Banach algebra-valued holomorphic functions on $\mathbb{C}^{n}$. We prove that the resolvent $\sigma^{c}(f)$ is a disjoint union of domains of holomorphy when $\mathcal{B}$ is a $C^{\ast}$-algebra or is reflexive as a Banach space. Further, we study the topology of the resolvent via consideration of the $\mathcal{B}$-valued Maurer–Cartan type $1$-form $f(z)^{-1}\,df(z)$. As an example, we explicitly compute the spectrum of a linear function associated with the tuple of standard unitary generators in a free group factor von Neumann algebra.

Higgs bundles for the non-compact dual of the unitary group

Abstract: Using Morse-theoretic techniques, we show that the moduli space of $U^{∗}(2n)$-Higgs bundles over a compact Riemann surface is connected.

Higgs bundles for the Lorentz group

Abstract: Using the Morse-theoretic methods introduced by Hitchin, we prove that the moduli space of $\operatorname{SO}_{0}(1,n)$-Higgs bundles when $n$ is odd has two connected components.

Almost Cohen–Macaulay algebras in mixed characteristic via Fontaine rings

Abstract: In the present paper, it is proved that any complete local domain of mixed charac- teristic has a weakly almost Cohen-Macaulay algebra B in the sense that a system of parameters is a weakly almost regular sequence in B, which is a notion defined via a valuation. In fact, the central idea of this result originates from the main statement obtained by Heitmann to prove the Monomial Conjecture in dimension 3. A weakly almost Cohen-Macaulay algebra is constructed over the absolute integral closure of a complete local domain by applying the meth- ods of Fontaine rings and Witt vectors. A connection of the main theorem with the Monomial Conjecture is also discussed.

Flexible suspensions with a hexagonal equator

Abstract: We construct a flexible (non-embedded) suspension with a hexagonal equator in Euclidean 3-space. It is known that the volume bounded by such a suspension is well defined and constant during the flex. We study its properties related to the Strong Bellows Conjecture which reads as follows: if a, possibly singular, polyhedron $\mathcal P$ in Euclidean 3-space is obtained from another, possibly singular, polyhedron $\mathcal Q$ by a continuous flex, then $\mathcal P$ and $\mathcal Q$ have the same Dehn invariants. It is well known that if $\mathcal P$ and $\mathcal Q$ are embedded, with the same volume and the same Dehn invariant, then they are scissors congruent.

The depth formula for modules with reducible complexity

Abstract: We prove that the depth formula holds for Tor-independent modules in certain cases over a Cohen-Macaulay local ring, provided one of the modules has reducible complexity.

Multiple operator integrals and spectral shift

Abstract: Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.

A new triangulated category for rational surface singularities

Abstract: In this paper, we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a stable quotient of the Frobenius category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.

Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem

Abstract: We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$. These results extend and provide a partial converse to a theorem of Ryabykh.

Multiple ergodic averages for flows and an application

Abstract: We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemeredi’s Theorem. For each average, we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the collection of polynomials has “low complexity”, then for every set $E$ of real numbers with positive density and for every $\delta>0$, the set of polynomial return times for the “$\delta$-thickened” set $E_{\delta}$ has bounded gaps. We give bounds for the flow average complexity and show that in some cases the flow average complexity is strictly less than the discrete average complexity.

Stable symmetric polynomials and the Schur–Agler class

Abstract: We call a multivariable polynomial an Agler denominator if it is the denominator of a rational inner function in the Schur-Agler class, an important subclass of the bounded analytic functions on the polydisk. We give a necessary and sufficient condition for a multi-affine, symmetric, and stable polynomial to be an Agler denominator and prove several consequences. We also sharpen a result due to Kummert related to three variable, multi-affine, stable polynomials.