# 401-4475-71L  Microlocal Analysis

 Semester Autumn Semester 2021 Lecturers P. Hintz Periodicity non-recurring course Language of instruction English

AbstractMicrolocal 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.
Learning objectiveStudents 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.
ContentTempered 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.
Lecture notesLecture notes will be made available on the course website.
LiteratureLars Hörmander, "The Analysis of Linear Partial Differential Operators", Volumes I and III.

Alain Grigis and Johannes Sjöstrand, "Microlocal Analysis for differential operators: an introduction".
Prerequisites / NoticeStudents are expected to have a good understanding of functional analysis. Familiarity with distribution theory, the Fourier transform, and analysis on manifolds is useful but not strictly necessary; the relevant notions will be recalled in the course.
Competencies
 Subject-specific Competencies Concepts and Theories assessed Techniques and Technologies fostered Method-specific Competencies Analytical Competencies assessed Decision-making assessed Media and Digital Technologies fostered Problem-solving assessed Project Management fostered Social Competencies Communication assessed Cooperation and Teamwork fostered Customer Orientation fostered Leadership and Responsibility fostered Self-presentation and Social Influence fostered Sensitivity to Diversity fostered Negotiation fostered Personal Competencies Adaptability and Flexibility fostered Creative Thinking assessed Critical Thinking assessed Integrity and Work Ethics fostered Self-awareness and Self-reflection fostered Self-direction and Self-management fostered