Lecture (2010)

Speaker Hiroyuki Tasaki (Tsukuba University)
Title Introduction to symmetric spaces (Second)
Date March 16 （Wed.）10:00-11:30 13:00-14:30
March 17 （Thu.）10:00-11:30 13:00-14:30 15:00-16:30
March 18 （Fri.） 10:00-11:30 13:00-14:30
Place March 16 （Wed.）：Dept. of Mathematics, Sci. Bldg., 3153
March 17 （Thu.）March 18 （Fri.）： Dept. of Mathematics, Sci. Bldg., 3040
Abstract 対称空間の構造を詳しく調べるために、対称空間の性質を等長変換群のLie環から構成される直交対称Lie代数の性質に帰着させます。直交対称Lie代数の性質を調べるためには、 複素半単純Lie環の構造に関する情報が重要になります。そこで、複素半単純Lie環のルート空間分解、分類、コンパクト実形等に関する準備を行います。 直交対称Lie代数に複素半単純Lie環の結果を適用して、直交対称Lie代数さらに対称空間のコンパクト型・非コンパクト型・Euclid型の定義、分解、双対性、既約分解、分類などを扱います。

　

Speaker Athanase Papadopoulos (Strasbourg)
Title Finsler structures with examples from Hilbert geometry and from Teichmuller spaces
Date January 13 （Thu.）13：00-15：00, 15：30-17：30
January 14 （Fri.）10：30-12：30, 14：00-16：00
Place Dept. of Mathematics, Sci. Bldg., 3040
Abstract A weak Finsler structure on a $C^1$ manifold $M$ is a given by a family of convex sets on the tangent bundle of $M$. From a weak Finsler structure one defines a weak (in general non-symmetric) metric on the manifold $M$. Finsler geometry can be considered as a generalization of the study of convex sets. In particular, we address the problem of symmetrization of Finsler metrics. We shall give examples and study them in some detail. The examples are the Funk metric, the Hilbert metric (which is a symmetrization of the Hilbert metric), and various metrics on Teichmuller spaces of surfaces of finite type. the metrics include Thurston's asymmetric metric, the length-spectrum metric (a symmetrization of Thurston's asymmetric metric), the Teichmuller metric, and there are several others. We study geodesics, triangles, isometries, perpendicularity, horocycles, and several other geometric notions for these examples. We also discuss rigidity properties.

　

Speaker Hiroyuki Tasaki (Tsukuba University)
Title Introduction to symmetric spaces
Date Nov. 11(Thu.) 10:00-11:30 13:00-14:30 15:00-16:30
Nov. 12(Fri.) 10:00-11:30 13:00-14:30 15:00-16:30
Nov. 13(Sat) 10:00-11:30
Place Dept. of Mathematics, Sci. Bldg., 3040
Abstract A Riemannian manifold having a symmetry at each point is called a Riemannian symmetric space. Riemannian symmetric spaces give a family of fundamental Riemannian manifolds, which contains spaces of constant curvature, Grassmann manifolds and compact Lie groups. In the lectures I explain fundamental properties of Riemannian symmetric spaces. The lectures start with some preliminaries about Lie groups, Riemannian manifolds and Riemannian homogeneous spaces.