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Global Analysis I
The lecture Global Analysis I is an introduction to analysis on manifolds. In principle, it studies the interplay between geometry and (partial) differential equations. The global structure of the manifold often has astonishing effects on the solution behaviour of the respective differential equation.
A number of mathematical disciplines, such as geometry, topology and partial differential equations, come together in global analysis. It is also of great importance for mathematical physics. The main topics of the lecture are
- Differential forms and Stokes' theorem
- Vector fields and related topics
- Vector bundles and differential operators
- De Rham cohomology and Hodge theory
Prior knowledge
The lecture is aimed at students from the fifth semester onwards. The prerequisites are the lectures Linear Algebra I to II and Analysis I to III. Basic knowledge of differential topology and functional analysis is advantageous but not a prerequisite. The lecture is accompanied by exercises to consolidate the material.
As an introduction to the theory of manifolds we recommend:
- Abraham, Marsden, Ratiu: Manifolds, tensor analysis, and applications, Springer
- Guillemin, Pollack: Differential Topology
- Chern, S.S.; Chen, W.H.; Lam, K.S.: Lectures on differential geometry, 1999.
- Jänich K.: Vector analysis. Spinger, 1992.
- Milnor, J.W.: Topology from a differential viewpoint. Princeton, 1997.
- Bröcker T., Jänich K.: Introduction to Differential Topology, Springer, 1973.
- O'Neill B., Semi-Riemannian Geometry, Academic Press, 1983.
Global Analysis II
We will deal with the local and global behaviour of linear elliptic equations on manifolds.
The essential methodology for this has become known as "microlocal analysis".
The lecture is aimed at students with an interest in global analysis and/or partial differential equations.
Topics
- Distributions and Fourier transforms
- Oscillatory integrals and the stationary phase method
- Fourier integral operators, in particular pseudo-differential operators
- Regularity theory of elliptic equations
- Cauchy problem for hyperbolic equations and applications (optional)
- References to theoretical mechanics/symplectic geometry. (optional)
Prior knowledge
The basic lectures as well as the essential contents of the lecture Global Analysis I are required.
Literature
- M. Shubin: Pseudodifferential operators and spectral theory. Springer, 1978
- A. Grigis, J. Sjöstrand: Microlocal analysis for differential operators. An introduction. London Mathematical Society Lecture Grade Series No. 196, Cambridge University Press, 1994
- L. Hörmander. The analysis of linear partial differential operators, volumes 1-4, Springer, 1983.
Geometry and topology
The lecture offers an insight into the topics of modern geometry and topology. Possible contents include
- Topological and metric spaces, continuity, compactness, context,
- fundamental group, superposition theory,
- manifolds, Stokes' theorem,
- Gauss-Bonnet theorem,
- Morse theory.
Supplementary literature
- Jänich "Topology",
- B.v. Querenburg "Set-theoretic topology",
- Steen and Seebach "Counterexamples in Topology",
Special functions in mathematical physics
The seminar deals with concrete problems from global analysis. We study typical differential equations of mathematical physics and their solutions - the special functions. Important examples are the harmonic oscillator, the Schrödinger equation for rotationally symmetric potentials and the quantum mechanical description of the hydrogen atom. Another typical application is the development of spherical surface functions in quantum mechanics.
The ordinary differential equations associated with these physical problems include Legendre's, Hermite's, Bessel's and hypergeometric differential equations. In this seminar, some of these differential equations and the corresponding special functions will be analysed and applied in more detail.
Literature:
- [BGV] Berline, Getzler, Vergne "Heat Kernels and Dirac Operators ",
- [Hel] S. Helgason "Groups and Geometric Analysis ",
- [Leb] N. N. Lebedev "Special Functions and their Applications ".
- [Tay] M. Taylor "Partial Differential Equations II ",
- [Tri] H. Triebel "Höhere Analysis ",
- [Wei] J. Weidmann "Linear Operators in Hilbert Spaces, Part I".
Differential topology and quantum field theory
The 20th century was the scene of an intense and fruitful interaction between mathematics and physics. New physical theories of the first half of the 20th century, such as the theory of relativity and quantum mechanics, were only made possible by fundamental findings in differential geometry and functional analysis. The later quantum field theory also requires an intensive examination of mathematical methods and in turn raises interesting mathematical questions.
The aim of the seminar is first to discuss some mathematical foundations for quantum field theory, in particular vector bundles, characteristic classes, index and spectral theory of elliptic operators, algebraic varieties and much more. We then apply these tools in the discussion of Yang-Mills theory, as well as conformal and topological quantum field theory.
Charles Nash's "Differential Topology and Quantum Field Theory" is the lead work.