第一届云师数学前沿讲习班

In this talk, I will survey the definition of curves and surfaces in Euclidean spaces. I will review the concepts of Frenet frame, curvature and torsion of curves in 3-space, and then move on to some classical theorems on curves, including Hopf vertex theorem, rotation number theorem and Fary-Milnor theorem. On the surface part, I will give intuitive definitions of principal curvatures and Gauss curvature and introduce the Gauss-Bonnet theorem as well as some of its applications.

This course provides a brief introduction to the basic concepts in differential geometry and Riemannian geometry. The study of curved spaces naturally arises in both mathematics and physics. A curved space that locally resembles Euclidean space is known as a manifold. In lower dimensions, manifolds correspond to the familiar notions of curves and surfaces. The study of manifolds can be divided into four main topics: topology, geometry, analysis and physics. The first four lectures will cover the notions of manifolds and Riemannian metrics. The last lecture will include discussions on the Riemann curvature tensor. Participants are expected to have a basic knowledge of calculus and linear algebra.
Lecture 1: The notion of manifolds
Lecture 2: The tangent bundle and the cotangent bundle
Lecture 3: Submanifolds
Lecture 4: Differential forms
Lecture 5: Riemannian manifolds

This lecture begins by recalling the study of the Schrödinger equation in the Euclidean setting. We will explain how the dispersive property and the duality argument can be used to establish the Strichartz inequality, and give its applications to global well-posedness and scattering for the nonlinear problem. Next, we review the basic structure of hyperbolic spaces; we will consider those of rank one mainly formed by semisimple Lie groups and explain how to use harmonic analysis (spherical functions, Harish-Chandra transform, Kunze-Stein phenomenon, etc.) in this context. Using these tools, we derive the pointwise kernel estimates in hyperbolic spaces and then we derive different properties of the dispersion and the Strichartz inequality. Some generalizations to so-called non-compact symmetric spaces of higher rank will be discussed at the end.