02.03.2018

Analysis on manifolds (2014)

Lectures

Wed 12-14 and Thu 10-12, room MaD380. The first lecture is Wed 24.09. at 12-14.

Lecturer: Mikko Salo. See course details in the Korppi system.

Lecture notes

Analysis on manifolds lecture notes (updated 10.12.) More or less complete version, though still rough.

Questions

Questions #1 (discussion 08.10.)

Questions #2 (discussion 22.10.)

Questions #3 (discussion 19.11.)

Exercises

Exercise set (return written answers by 27.11.)

Seminars

List of possible topics

Seminar schedule (seminars on 26.11. and 03.-04.12.)

Course description

This course is an introduction to analysis on manifolds. The topic may be viewed as an extension of multivariable calculus from the usual setting of Euclidean space to more general spaces, namely Riemannian manifolds. These spaces have enough structure so that they support a very rich theory for analysis and differential equations, and they also form a large class of nice metric spaces where distances are realized by geodesic curves.

The first half of the course will begin with a review of multivariable calculus in Euclidean space, and will then present corresponding notions on Riemannian manifolds. Geodesic curves, the Laplace operator and differential equations will also be covered.

The second half of the course intends to give a flavor of more advanced topics, such as Morse theory and Hodge theory (describe the topology of a space through analysis), conformal and quasiconformal mappings on manifolds, lower bounds for Ricci curvature and applications, inverse problems on manifolds (geodesic ray transform), Ricci flow and Perelman’s solution of the Poincaré conjecture.

Multivariable calculus and functional analysis are prerequisites for the course, and familiarity with smooth or Riemannian manifolds is helpful but not strictly necessary.