阪大-阪市大‐神戸大-九大合同幾何学セミナー (第7回) 


David Brander教授 集中講義


開催日 平成24年7月28日(土)
場所 大阪市立大学 理学部棟3階 3040室(数学講究室)

Wayne Rossman(神戸大学)


援助 大阪市立大学 特色となる教育体制への支援事業
平成24年度教育推進本部経費 「学生の自主的研究活動支援とオープンソース数学ソフトウエアによる新たな数物系教育」
(代表 大仁田 義裕,理:数学)
後援 大阪市立大学数学研究所(OCAMI)

Professor David Brander (Technical University of Denmark & Kobe University)
Mr. Mason Pember (doctoral student, University of Bath, UK & Kobe University)

Mason Pember (University of Bath, UK & Kobe University)
10:00-11:00 "Lie Sphere Geometry"
We will give a basic introduction to Lie sphere geometry, the study of oriented spheres and their oriented contact in 3-dimensional Riemannian and Lorentzian space forms. We will see how this can be used to study the differential geometry of surfaces, and more generally fronts, in these space forms. For simplicity's sake, we will mainly focus on S^3.
David Brander (Technical University of Denmark & Kobe University)
"DPW-type Methods in Surface Theory"
11:30-12:30 Lecture (I)
14:30-15:30 Lecture (II) 
16:00-17:00 Lecture (III)
Over the last 20 years loop group methods have been used to study many "classical" classes of surfaces in space forms; for example, constant mean or Gauss curvature surfaces. These types of surfaces are sometimes called "integrable surfaces", and the relevant property is that the surfaces are characterized by the existence of a special type of map (which is easy to describe) into a subgroup of the group of loops in a complex semisimple Lie group. There are at least three different basic ideas which can be used to generate solutions: dressing, the AKS-type theory for finite type solutions, and the "DPW"-type method.

In these talks I will describe the last mentioned of these. This method has the property that it produces ALL solutions for the problem at hand from simpler data, in analogue to the classical Weierstrass representation for minimal surfaces. The challenge, however, is that it is difficult to read geometric information about the surface in the generalized Weierstrass data, due to a loop group splitting involved in passing between the two equivalent pieces of information. A basic idea which can be exploited is that the Weierstrass data associated to a given surface is not unique, and therefore one may try to choose this in such a way as to make this issue easier. I will illustrate this with two examples: one is the solution of the generalization of Bjorling's problem to other surface classes; the other is the use of special potentials to analyze singularities of surfaces. These ideas can be implemented both for surfaces associated to Riemannian harmonic maps, which have a generalized Weierstrass representation in terms of holomorphic functions, and for those associated to Lorentzian harmonic maps, which have a generalized d'Alembert type solution.

リンク等 案内

大仁田 義裕: ohnita (at)
Wayne Rossman: wayne (at)
製作 のだ & ボス Last updated on 22/July/2012