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401-4475-71L 6 Credits DR , MSC D-MATH

Microlocal Analysis

Lecturers & Examiners: Dr. Peter Hintz
VVZ CR n/a

Last Updated: 2026-02-05 15:47:36

Abstract

Microlocal analysis is the analysis of partial differential equations in phase space. The first half of the course introduces basic notions such as pseudodifferential operators, wave front sets of distributions, and elliptic parametrices. The second half develops modern tools for the study of nonelliptic equations, with applications to wave equations arising in general relativity.

Objective

Students will be able to analyze linear partial differential operators (with smooth coefficients) and their solutions in phase space, i.e. in the cotangent bundle. For various classes of operators including, but not limited to, elliptic and hyperbolic operators, they will be able to prove existence and uniqueness (possibly up to finite-dimensional obstructions) of solutions, and study the precise regularity properties of solutions. The first goal is to construct and apply parametrices (approximate inverses) or approximate solutions of PDEs using suitable calculi of pseudodifferential operators (ps.d.o.s). This requires defining ps.d.o.s and the associated symbol calculus on Euclidean space, proving the coordinate invariance of ps.d.o.s, and defining a ps.d.o. calculus on manifolds (including mapping properties on Sobolev spaces). The second goal is to analyze distributions and operations on them (such as: products, restrictions to submanifolds) using information about their wave front sets or other microlocal regularity information. Students will in particular be able to compute the wave front set of distributions. The third goal is to infer microlocal properties (in the sense of wave front sets) of solutions of general linear PDEs, with a focus on elliptic, hyperbolic and certain degenerate hyperbolic PDE. For hyperbolic operators, this includes proving the Duistermaat-Hörmander theorem on the propagation of singularities. For certain degenerate hyperbolic operators, students will apply positive commutator methods to prove results on the propagation of microlocal regularity at critical or invariant sets for the Hamiltonian vector field of the principal symbol of the partial differential operator under study.

Content

Tempered distributions, Sobolev spaces, Schwartz kernel theorem. Symbols, asymptotic summation. Pseudodifferential operators on Euclidean space: composition, principal symbols and the symbol calculus, elliptic parametrix construction, boundedness on Sobolev spaces. Pseudodifferential operators on manifolds, elliptic operators on compact manifolds and Fredholm theory, basic symplectic geometry. Microlocalization: wave front set, characteristic set; pairings, products, restrictions of distributions. Hyperbolic evolution equations: existence and uniqueness of solutions, Egorov's theorem. Propagation of singularities: the Duistermaat-Hörmander theorem, microlocal estimates at radial sets. Applications to general relativity: asymptotic behavior of waves on de Sitter space.

Resources

Lecture Notes

Lecture notes will be made available on the course website.

Literature

Lars Hörmander, "The Analysis of Linear Partial Differential Operators", Volumes I and III. Alain Grigis and Johannes Sjöstrand, "Microlocal Analysis for differential operators: an introduction".

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes

Course Components

Type Title Time & Place Hours
lecture with exercise Microlocal Analysis
Online lecture: This lecture will take place online. Reserved rooms will remain reserved on campus for students to follow the course from there.
  • Tue 10:15-12:00 (LFW E 13)
  • Thu 08:15-09:00 (HG E 7)
3 h weekly

Offered In

  • Mathematics Master
    • Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
  • Doctoral Department of Mathematics (More Information at: The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. WARNING: Do not mistake ECTS credits for credit points for doctoral studies!)