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Finite Geometries, Groups, and Computation
This volume is the proceedings of a conference on Finite Geometries, Groups, and Computation that took place on September 4-9, 2004, at Pingree Park, Colorado (a campus of Colorado State University). Not accidentally, the conference coincided with the 60th birthday of William Kantor, and the topics relate to his major research areas. Participants were encouraged to explore the deeper interplay between these fields. The survey papers by Kantor, O'Brien, and Penttila should serve to introduce both students and the broader mathematical community to these important topics and some of their connections while the volume as a whole gives an overview of current developments in these fields.
On Boundary Interpolation for Matrix Valued Schur Functions
A number of interpolation problems are considered in the Schur class of $p\times q$ matrix valued functions $S$ that are analytic and contractive in the open unit disk. The interpolation constraints are specified in terms of nontangential limits and angular derivatives at one or more (of a finite number of) boundary points. Necessary and sufficient conditions for existence of solutions to these problems and a description of all the solutions when these conditions are met is given.The analysis makes extensive use of a class of reproducing kernel Hilbert spaces ${\mathcal{H (S)$ that was introduced by de Branges and Rovnyak. The Stein equation that is associated with the interpolation problems under consideration is analyzed in detail. A lossless inverse scattering problem isalso considered.
On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups
The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie$(n)$, as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.
On Dynamical Poisson Groupoids I
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of (CDYBE) on simple Lie algebras as classified by the same authors.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
Contributes to the theory of Borel equivalence relations, considered up to Borel reducibility, and measures preserving group actions considered up to orbit equivalence. This title catalogs the actions of products of the free group and obtains additional rigidity theorems and relative ergodicity results in this context.
The Second Duals of Beurling Algebras
Let $A$ be a Banach algebra, with second dual space $A""$. We propose to study the space $A""$ as a Banach algebra. There are two Banach algebra products on $A""$, denoted by $\,\Box\,$ and $\,\Diamond\,$. The Banach algebra $A$ is Arens regular if the two products $\Box$ and $\Diamond$ coincide on $A""$.
Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls
Contents: A tree structure for the unit ball $mathbb B? n$ in $mathbb C'n$; Carleson measures; Pointwise multipliers; Interpolating sequences; An almost invariant holomorphic derivative; Besov spaces on trees; Holomorphic Besov spaces on Bergman trees; Completing the multiplier interpolation loop; Appendix; Bibliography
Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Caratheodory Spaces
The object of the present study is to characterize the traces of the Sobolev functions in a sub-Riemannian, or Carnot-Caratheodory space. Such traces are defined in terms of suitable Besov spaces with respect to a measure which is concentrated on a lower dimensional manifold, and which satisfies an Ahlfors type condition with respect to the standard Lebesgue measure. We also study the extension problem for the relevant Besov spaces. Various concrete applications to the setting of Carnot groups are analyzed in detail and an application to the solvability of the subelliptic Neumann problem is presented.
Conformal and Harmonic Measures on Laminations Associated with Rational Maps
This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $\mathcal A$ and the associated hyperbolic 3-lamination $\mathcal H$ endowed with an action of a discrete group of isomorphisms. This action is properly discontinuous on $\mathcal H$, which allows one to pass to the quotient hyperbolic lamination $\mathcal M$. Our work explores natural ``geometric'' measures on these laminations. We begin with a brief self-contained introduction to the measure...
Quasi-Ordinary Power Series and Their Zeta Functions
Intends to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, this title computes the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h, T)$ of a quasi-ordinary power series $h$ of arbitrary dimension
