18.966 - Geometry of Manifolds II (Spring 2022)

Course Information:

Form and shape can be described by differential equations. Many of these equations originate in various branches of science and engineering. They are fundamental and, in a sense, canonical. The fact that they make sense geometrically means that they are relevant everywhere and have fundamental properties that appear over and over again in many settings. Understanding them requires simultaneous insight into analysis and geometry and the interplay between these.

In this class we will discuss a number of different ideas and estimates that have a wide range of applications to many fields including geometry, analysis, probability and applied mathematics. Common for them all is that they originate in geometry.

Topics will include (but not restricted to):

• Continues and discrete Laplacian.
• Drift Laplacian and weighted inequalities.
• Gradient estimates.
• Harnack inequalities.
• Sharp gradient estimate and monotonicity.
• $L^2$ eigenfunctions.
• Ornstein-Uhlenbeck operator and Hermite polynomials.
• Li-Yau differential Harnack inequality.
• Hamilton's matrix maximum principle.
• Perelman's monotonicity.

Prerequisites: The class will be suitable for anyone who has taken 18.965 or some advanced undergraduate geometry class.