Geometric Analysis on Noncompact Manifolds Provider: Faculty of Science

Activity no.: 5572-17-07-31

Enrollment deadline: 17/09/2017

Place

Department of Mathematical Sciences Universitetsparken 5, 2100 København Ø

Date and time

09.10.2017, at: 09:00 - 13.10.2017, at: 16:00

Regular seats

50

ECTS credits

2.50

Contact person

Nina Weisse E-mail address: weisse@math.ku.dk

Enrolment Handling/Course Organiser

Niels Martin Møller E-mail address: nmoller@math.ku.dk

Written language

English

Teaching language

English

Block note

Duration: 5 days.

Exam form

Course participation

Exam details

Information will follow

Course workload

Course workload category

Hours

Preparation / Self-Study

30.00

Course hours

25.00

Sum

55.00

Formel requirements

The students should have a sound knowledge of the basic theory of linear elliptic PDEs corresponding to an advanced BSc or beginning level graduate class: Basic Sobolev theory (or basic Schauder/Hölder theory), basic Fourier Analysis, basic Functional Analysis (bounded operators on Hilbert/Banach spaces) and a similar level of command of Riemannian geometry. We will supply the participants with a list of topics, with corresponding resources, to brush up on before attending the course.

Learning outcome

By the end of the course, the participants will have acquired an up-to-date overview of some of the latest work in these areas and should be prepared for further study as well as embarking on research projects on their own.

Literature

Information will follow.

Target group

3. Outcome: While one important aspect of this class is creating a “high-level toolbox” in the transfer of as concrete advanced skills as possible to the students (for this we plan to include problem sessions for the students), we will also be framing and motivating the whole class via state-of-the-art examples (from the lecturers’ own recent research), as detailed in the above, in order to make contact with the forefront of current top-level international research in the field. Diligent students will after this be able to understand many of the major recent advances in the field and participate in such research themselves. Additionally, we believe that the networking potential of this PhD school is strong, and at least two-fold: Firstly via scientific connections made between the participants themselves, but also in the further branding of Copenhagen University as a place for activities evolving around “hot topics” in modern geometric analysis and mathematical physics.

From the course we will produce lecture notes, to be shared in .pdf format, via the open access database arXiv, which can then serve as reference and introduction for beginners in geometry/analysis/mathematical physics research, hence becoming a valuable resource for the scientific community as a whole.

Teaching and learning methods

Lectures and Exercise classes.

Lecturers

Claudio Arezzo, Professor, International Centre for Theoretical Physics, Trieste. Colin Guillarmou, Professor, CNRS/Univ. Paris Sud (Orsay), Paris, France. Klaus Kröncke, Assistant Professor, Hamburg Universität, Germany.

Niels Martin Møller, Assistant Professor, University of Copenhagen, Denmark.

Content

1. Scope of the Course: In the internationally thriving subjects of geometric analysis and mathematical physics, a core skill for researchers is that of solving (linear) elliptic partial differential equations, e.g. static Schrödinger equations, on Riemannian manifolds, with natural geometric content. While the basic theory in the compact case is classical and can be presented very cleanly (is indeed standard material in most advanced analysis programs), the noncompact case methods are harder to come by and can have a steep learning curve for beginners in the field.

In this PhD school, we thus wish to provide a solid beginner’s guide to the analysis of (linear) partial differential equations on noncompact Riemannian manifolds, as well as allow the time to touch on to major current research topics where these techniques are applied. We expect that a considerable number of PhD students (and postdocs) from not only Copenhagen University but many other departments, in Denmark and abroad, will be keenly interested in partaking in such a master class (f.ex. we expect visitors from Aarhus, Lund, London, Berlin, Hamburg, Regensburg, Münster, Paris and several other places).

The invited guest speakers, Prof. Claudio Arezzo (ICTP) and Prof. Colin Guillarmou (CNRS/École Polytechnique) are experts of highest international caliber in this currently very active field of research and are thrilled to have the opportunity to share their extensive knowledge and experience in this exciting field with the students. In case funding for the school is approved, we plan to invite 1-2 additional international experts as speakers, which QMATH (where Prof. Bergfinnur Durhuus is a PI) has kindly offered to support.

2. Course Content: 2.1. Participant Prerequisites: The students should have a sound knowledge of the basic theory of linear elliptic PDEs corresponding to an advanced BSc or beginning level graduate class: Basic Sobolev theory (or basic Schauder/Hölder theory), basic Fourier Analysis, basic Functional Analysis (bounded operators on Hilbert/Banach spaces) and a similar level of command of Riemannian geometry. We will supply the participants with a list of topics, with corresponding resources, to brush up on before attending the course.]

2.2 Detailed Syllabus: - Basic theory of weighted Schauder and weighted Sobolev spaces on noncompact Riemannian manifolds. - Basic examples of the influence of curvature on e.g. the presence of harmonic functions. - The bounded null space for Schrödinger operators, indicial roots and deficiency spaces (asymptotically flat manifolds; periodic coefficients; potentials with an isolated point singularity). - Advanced examples from modern research on gluing methods for nonlinear partial differential equations, in mean curvature equations and in complex geometry (applications of the asymptotically flat and periodic cases). [Arezzo]. - Basic theory of scattering, resolvents and resonances in (asymptotically hyperbolic geometry). - Advanced examples of modern research in analysis on asymptotically hyperbolic manifolds, e.g. conformally compact Poincare-Einstein manifolds (applications of the asympt. hyp. theory). [Guillarmou]

Remarks

Please visit the official website where you have to fill out an application. The application deadline is 17 September 2017.

Participants will be selected on the basis of the applications from the website. For PhD students, indicating the e-mail address of an academic supervisor is required, as well as a motivated explanation of the interest in the workshop.

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