A survey of geometric structure in geometric analysis
On the first page, i.e., page 325, the author says ``This article is based on lectures given at the university of California at Los Angeles in April 2007''.NEWLINENEWLINENEWLINENEWLINEThe article has three parts. In the whole article, the author speaks about geometric structures. NEWLINENEWLINENEWLINENEWLINE In the first part, the author first mentiones holonomy groups, then complex and Kähler structures. Then he mentiones stability, he says ``The idea is that stability arose from actions of noncompact groups and based on this, I proposed the following point of view. If there is a noncompact group acting behind a system of nonlinear differential equations, the existence question of such system will be related to the question of the stability of some algebraic structure that defines this system of nonlinear equations.'' Then the author speaks about the moduli spaces of Riemann surfaces. For example, he mentiones the proof of the Mumford conjecture.NEWLINENEWLINENEWLINENEWLINEIn the second part, the author speaks mostly about Calabi-Yau manifolds and mirror symmetry. NEWLINENEWLINENEWLINENEWLINE In the third part, he speaks about the Ricci flow, Perelman's proof of the Poincare conjecture and Thurston's geometrization conjecture. He says ``The accumulated works of Hamilton-Perelman are spectacular. Today, 5 years after the first preprint of Perelman was available, several groups of...; at the same time, other experts are still working diligently ... of this century old conjecture.'' Later, he mentiones that ``For \(\text{dim}_{\mathbb C} \geq 3\), every almost complex manifold admits an integrable complex structure.'' This is very interesting, since it is opposite to many claims that there is no complex structure on \(S^6\). He also mentiones the Clemens-Friedman(-Tian, one of his student, -Lu) construction and Reids fantasy on Calabi-Yau manifolds.NEWLINENEWLINEIn the last paragraph, the author mentiones a few sentences by Einstein: ``Pure logical thinking cannot yield us any knowledge of the empirical world....''NEWLINENEWLINEFor the entire collection see [Zbl 1230.53007].
scientific article; zbMATH DE number 6085996