Mikko Salo

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Topics in differential geometry, spring 2022

This topics course is an introduction to analysis on manifolds. In particular we will explain how to compute derivatives, integrate, and solve differential equations (mostly Laplace and heat equation) on Riemannian manifolds. If time permits we will also use analytic methods to discuss certain landmark results in geometry, such as the uniformization theorem for Riemann surfaces, the Hodge theorem, the Weyl law for eigenvalues of the Laplacian, or the Gauss-Bonnet theorem. The course will be given in English.

Instructors

The lectures are given by Mikko Salo, and exercise sessions by Pu-Zhao Kow and Suman Kumar Sahoo.

Schedule

The lectures are Tuesdays and Thursdays at 14.15-16.00 in room MaD381, with the first lecture on 22 March. Exercise sessions are on Tuesdays at 12.15-14.00 in MaD355 starting on 29 March. The teaching is planned to take place in person.

Material

Lecture notes (version of 03.05.2022)

Exercises 1 (22.03.2022, return by 12.04.2022)

Exercises 2 (07.04.2022, return by 28.04.2022)

Exercises 3 (03.05.2022, return by 31.05.2022)

Prerequisites

Multivariable calculus and functional analysis (partial differential equations are also helpful). The course is suitable as a continuation for Riemannian geometry in period 3, but it is also suitable for advanced students, PhD students and postdocs in analysis, geometry or PDEs (the required geometric background will be reviewed in the beginning).

Completion

The course can be taken for credit (5 cr) by attending the lectures and by returning written solutions to at least 50 % of the exercises in each exercise set. Please register via Sisu.