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Conformal Maps, Bergman Spaces, and Random Growth Models
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and five research papers on topics related to conformal mapping, the Loewner equation and its applications, and Bergman-type spaces of holomorphic functions. The first two papers are devoted to the study of integral means of derivatives of conformal mappings. In Paper I, we present improved upper estimates of the universal means spectrum of conformal mappingsof the unit disk. These estimates rely on inequalities  obtained by Hedenmalm and Shimorin using Bergman space techniques, and on computer calculations. Paper II is a survey of recent results on the universal means spectrum, with particular emphasis on Bergman spacetechniques.Paper III concerns Bergman-type spaces of holomorphic functions in subsets of $\textbf{C}^d$ and their reproducing kernel functions. By expanding the norm of a function in a Bergman space along the zero variety of a polynomial, we obtain a series expansion of reproducing kernel functions in terms of kernels associated with lower-dimensionalspaces of holomorphic functions. We show how this general approach can be used to explicitly compute kernel functions for certain weighted Bergman and Bargmann-Fock spaces defined in domains in $\textbf{C}^2$.The last two papers contribute to the theory of Loewner chains and theirapplications in the analysis of planar random growth model defined in terms of compositions of conformal maps.In Paper IV, we study Loewner chains generated by unimodular L\'evy processes.We first establish the existence of a capacity scaling limit for the associated growing hulls in terms of whole-plane Loewner chains driven by a time-reversed process. We then analyze the properties of Loewner chains associated with a class of two-parameter compound Poisson processes, and we describe the dependence of the geometric properties of the hulls on the parameters of the driving process. In Paper V, we consider a variation of the Hastings-Levitov growth model, with anisotropic growth. We again establish results concerning scaling limits, when the number of compositions increases and the basic conformal mappings tends to the identity. We show that the resulting limit sets can be associated with solutions to the Loewner equation.We also prove that, in the limit, the evolution of harmonic measure on the boundary is deterministic and is determined by the flow associated with an ordinary differential equation, and we give a description of the fluctuations around this deterministic limit flow.

Place, publisher, year, edition, pages
Stockholm: KTH , 2010. , vi, 46 p.
Series
TRITA-MAT. MA, ISSN 1401-2278 ; 10:03
Keyword [en]
Conformal maps, Bergman kernels, planar growth models
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-12364ISBN: 978-91-7415-593-8 OAI: oai:DiVA.org:kth-12364DiVA: diva2:309999
Public defence
2010-05-04, F3, Lindstedsv. 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20100414

Available from: 2010-04-14 Created: 2010-04-12 Last updated: 2017-02-23Bibliographically approved
List of papers
1. An estimate of the universal means spectrum of conformal mappings
Open this publication in new window or tab >>An estimate of the universal means spectrum of conformal mappings
2006 (English)In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 6, no 2, 423-436 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7045 (URN)
Note
QC 20010414Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2017-12-14Bibliographically approved
2. Norm expansion along a zero variety
Open this publication in new window or tab >>Norm expansion along a zero variety
2008 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 254, 1601-1625 p.Article in journal (Refereed) Published
Abstract [en]

The reproducing kernel function of a weighted Bergman space over domains in C-d is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d = 2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z(1) = z(2) as subvariety) and the ball (with z(2) = 0 as subvariety), as well as for a weighted Bargmann-Fock space over C-2 (with the diagonal z(1) = z(2) as subvariety).

Keyword
norm expansion; Bergman kernel expansion
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7046 (URN)10.1016/j.jfa.2007.09.011 (DOI)000254488800006 ()2-s2.0-39149131739 (Scopus ID)
Note

QC 20100414

Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2017-06-14Bibliographically approved
3. Spectral notions for conformal maps: a survey
Open this publication in new window or tab >>Spectral notions for conformal maps: a survey
2008 (English)In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 8, no 2, 447-474 p.Article in journal (Refereed) Published
Abstract [en]

The universal means spectrum of conformal mappingshas been studied extensively in recent years. In some situations,sharp results are available, in others, only upper and lower estimateshave been obtained so far. We review some of the classicalresults before discussing the recent work of Hedenmalm andShimorin on estimates of the universal means spectrum near theorigin. It is our ambition to explain how their method works andwhat its limitations are. We then move on to the recent studyof the universal means spectrum of bounded functions near thepoint two conducted by Baranov and Hedenmalm. A number ofopen problems related to these topics are pointed out together withsome auxilliary results which are interesting in their own right.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7044 (URN)
Note
QC 20010414Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2017-12-14Bibliographically approved
4. Rescaled Levy-Loewner hulls and random growth
Open this publication in new window or tab >>Rescaled Levy-Loewner hulls and random growth
2009 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 133, no 3, 238-256 p.Article in journal (Refereed) Published
Keyword
Loewner differential equation, Rescaling, Levy process, Compound Poisson process, Growth models, LAPLACIAN GROWTH
Identifiers
urn:nbn:se:kth:diva-12379 (URN)10.1016/j.bulsci.2008.12.006 (DOI)000265475000003 ()2-s2.0-62849087227 (Scopus ID)
Note
QC 20100414Available from: 2010-04-14 Created: 2010-04-14 Last updated: 2017-12-12Bibliographically approved
5. Scaling limits of anisotropic Hastings-Levitov clusters
Open this publication in new window or tab >>Scaling limits of anisotropic Hastings-Levitov clusters
2012 (English)In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 48, no 1, 235-257 p.Article in journal (Refereed) Published
Abstract [en]

We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.

Keyword
Anisotropic growth models, Scaling limits, Loewner differential equation, Boundary flow
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-12363 (URN)10.1214/10-AIHP395 (DOI)000300338800009 ()
Note
QC 20100618. Updated from submitted to published 20120327Available from: 2010-04-12 Created: 2010-04-12 Last updated: 2017-12-12Bibliographically approved

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