David Radnell

David Radnell
Aalto University · Department of Mathematics and Systems Analysis

PhD

About

21
Publications
1,926
Reads
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161
Citations
Citations since 2016
10 Research Items
122 Citations
20162017201820192020202120220510152025
20162017201820192020202120220510152025
20162017201820192020202120220510152025
20162017201820192020202120220510152025
Introduction
Website: http://www.radnell.org/ I am interested in: The rigorous mathematical foundations and construction of conformal field theory, in particular the geometric and analytic structures. Quasiconformal Teichmueller theory, determinant line bundles, sewing / gluing of Riemann surfaces. Determinant line bundles and modular functors. Vertex operator algebras.
Additional affiliations
August 2015 - December 2016
Aalto University
Position
  • Professor
August 2006 - July 2015
American University of Sharjah
Position
  • Professor (Associate)
August 2004 - August 2005
Max Planck Institute for Mathematics
Position
  • PostDoc Position
Education
August 1997 - October 2003

Publications

Publications (21)
Article
Full-text available
Let $\Sigma$ be a Riemann surface of genus $g$ bordered by $n$ curves homeomorphic to the circle $\mathbb{S}^1$, and assume that $2g+2-n>0$. For such bordered Riemann surfaces, the authors have previously defined a Teichm\"uller space which is a Hilbert manifold and which is holomorphically included in the standard Teichm\"uller space. Based on thi...
Article
Full-text available
The restricted class of quasicircles sometimes called the "Weil-Petersson-class" has been a subject of interest in the last decade. In this paper we establish a Sokhotski-Plemelj jump formula for WP-class quasicircles, for boundary data in a certain conformally invariant Besov space. We show that this Besov space is precisely the set of traces on t...
Article
Full-text available
We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in the case that 2g+2-n>0, this refined Teichmueller space is a Hilbert manifold. The inclusion map from the refin...
Article
Full-text available
We show that the infinite-dimensional Teichmueller space of a Riemann surface whose boundary consists of n closed curves is a holomorphic fiber space over the Teichmueller space of n-punctured surfaces. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichmueller space. The local model of the fiber, t...
Article
Full-text available
One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces to...
Preprint
This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lat...
Preprint
We consider an operator associated to compact Riemann surfaces endowed with a conformal map, f , from the unit disk into the surface, which arises in conformal field theory. This operator projects holomorphic functions on the surface minus the image of the conformal map onto the set of functions h so that the Fourier series h f has only negative po...
Article
Full-text available
We consider Riemann surfaces Σ with n borders homeomorphic to S^1 and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an n-fold direct sum of Bergman spaces of the disk. We show that this period...
Data
The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant con...
Article
Full-text available
Let $\riem$ be a Riemann surface of genus $g$ bordered by $n$ curves homeomorphic to the circle $\mathbb{S}^1$. Consider quasiconformal maps $f:\riem \rightarrow \riem_1$ such that the restriction to each boundary curve is a Weil-Petersson class quasisymmetry. We show that any such $f$ is homotopic to a quasiconformal map whose Beltrami differentia...
Article
Consider a multiply-connected domain Σ in the sphere bounded by n non-intersecting quasicircles. We characterize the Dirichlet space of Σ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary v...
Article
Consider a multiply-connected domain Σ in the sphere bounded by n non-intersecting quasicircles. We characterize the Dirichlet space of Σ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary v...
Chapter
Full-text available
The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant con...
Article
We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of the dihedral group Dp, where p is prime.
Article
Full-text available
For a compact Riemann surface of genus [Formula: see text] with [Formula: see text] punctures, consider the class of [Formula: see text]-tuples of conformal mappings [Formula: see text] of the unit disk each taking [Formula: see text] to a puncture. Assume further that (1) these maps are quasiconformally extendible to [Formula: see text], (2) the p...
Article
Full-text available
We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller spa...
Article
We show the solvability of the Dirichlet problem on Weil-Petersson class quasidisks and establish a Sokhotski-Plemelj jump formula for the Weil-Petersson class quasicircles. Furthermore we show that the resulting Cauchy projections are bounded. In both cases the boundary data belongs to a certain conformally invariant Besov space. Moreover we show...
Article
Full-text available
Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show that this semigroup is a quotient of the Teichmüller space of doubly connected Riemann surfaces by a ℤ action...
Article
Full-text available
Let \Sigma be a compact Riemann surface with n distinguished points p_1,...,p_n. We prove that the set of n-tuples (\phi_1,...,\phi_n) of univalent mappings \phi_i from the open unit disc into \Sigma mapping 0 to p_i, with non-overlapping images and quasiconformal extensions to a neighbourhood of the closed unit disk, carries a natural complex Bana...
Article
Full-text available
The study of Riemann surfaces with parametrized boundary components was initiated in conformal field theory (CFT). Motivated by general principles from Teichmueller theory, and applications to the construction of CFT from vertex operator algebras, we generalize the parametrizations to quasisymmetric maps. For a precise mathematical definition of CF...
Article
Full-text available
"Graduate Program in Mathematics." Thesis (Ph. D.)--Rutgers University, 2003. The pure mathematical incarnation of conformal field theory was introduced by Segal and Kontsevich around 1987. Recently, Hu and Kriz further rigorized Segal's definition. Conformal field theory is intimately connected to vertex operator algebras and the complex geometr...

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Project (1)
Project
We are aiming to generalized the concept of the Grunsky coefficients which had been defined for univalent functions on the unit disc in the complex plane, to multiply connected subregions of compact Riemann surfaces of arbitrary genus. We could model these subregions as an open Riemann surface with $n$ boundary curves and the Grunsky coefficients as an infinite matrix, i.e. an operator. In our investigation, all the boundary curves are being considered as quasi-conformal.