FMA290F

Temporary

Global Analysis

7.5

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

2025-06-16

Maria Sandsten

English

If sufficient demand

Global analysis provides essential theoretical foundations for advanced studies in differential geometry, partial differential equations, and theoretical physics. The theory of manifolds and differential forms offers a coordinate-free framework for formulating and analyzing geometric and analytical problems on curved spaces. The course introduces key concepts such as tangent/vector bundles, differential forms, the exterior derivative, integration on manifolds, and Stokes' theorem, enabling the formulation of global properties of solutions to differential equations and field theories. An important tool in this endeavour will be the powerful techniques of pseudodifferential operators that allow us to understand the geometric analysis of manifolds. These tools are indispensable in modern applications and of importance to students aiming to specialize in pure or applied mathematics with a geometric focus.

In this course we will study the geometry of manifolds by means of the differential operators thereon and their analytic properties. The course will focus on the view towards global results starting from a local perspective. The course introduces key geometric concepts such as tangent/vector bundles, differential forms, the exterior derivative, integration on manifolds, and Stokes' theorem. The course will also introduce the machinery of pseudodifferential operators as a method to study the elliptic differential operators arising on manifolds. The highlights from this include the de Rham theorem relating the exterior derivative to cohomology.

be able to present a coherent picture of the fundamentals of geometric analysis on manifolds and vector bundles via pseudodifferential calculus, the notion of ellipticity and parametrices of elliptic operators.

be able to justify functional analytic and spectral properties of elliptic operators as operators on Sobolev spaces as well as their connection to cohomology through de Rham’s theorem.

be able to calculate within the form formalism and wield the exterior derivative and Stokes’ theorem.

be able to perform symbolic computations with pseudodifferential operators arising from geometry.

be able to determine ellipticity in geometric differential operators.

Lectures

Oral exam

Written assignments

Failed, pass

Basic course on partial differential equations , e.g. FMAN55 Kontinuerliga system. Fourier analysis, basic functional analysis, distribution theory and differential geometry.

Hörmander, L.: Advanced differential calculus.
Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer, 2001. ISBN 9783540411956.
Taira, K.: Brownian motion and index formulas for the de Rham complex. Wiley, 1998. ISBN 3527401393.

FMA290F

2025-06-16

Maria Sandsten