2,243 research outputs found
Meromorphic quadratic differentials with half-plane structures
We prove the existence of "half-plane differentials" with prescribed local
data on any Riemann surface. These are meromorphic quadratic differentials with
higher-order poles which have an associated singular flat metric isometric to a
collection of euclidean half-planes glued by an interval-exchange map on their
boundaries. The local data is associated with the poles and consists of the
integer order, a non-negative real residue, and a positive real leading order
term. This generalizes a result of Strebel for differentials with double-order
poles, and associates metric spines with the Riemann surface.Comment: 46 pages, 23 figures. Some minor corrections in v2, and a
clarification added in section 1
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
A Guide to Stochastic Loewner Evolution and its Applications
This article is meant to serve as a guide to recent developments in the study
of the scaling limit of critical models. These new developments were made
possible through the definition of the Stochastic Loewner Evolution (SLE) by
Oded Schramm. This article opens with a discussion of Loewner's method,
explaining how this method can be used to describe families of random curves.
Then we define SLE and discuss some of its properties. We also explain how the
connection can be made between SLE and the discrete models whose scaling limits
it describes, or is believed to describe. Finally, we have included a
discussion of results that were obtained from SLE computations. Some explicit
proofs are presented as typical examples of such computations. To understand
SLE sufficient knowledge of conformal mapping theory and stochastic calculus is
required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to
the text and improved hyperref suppor
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