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FSF3674 Differential Geometry 7.5 credits

This is an overview course targeted at all graduate students in mathematics. The goal is to give an introduction to some of the methods and research areas of modern differential geometry.

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Headings with content from the Course syllabus FSF3674 (Spring 2019–) are denoted with an asterisk ( )

Content and learning outcomes

Course contents

Introduction:

  • Manifolds, semi-Riemannian metrics, curvature, submanifolds, hyperquadrics, geodesics, comparison theorems for positive/negative curvature.

Followed by a selection from the subjects:

  • Curvature in general, holonomy, characteristic classes
  • Lorentzian geometry, Hawking-Penrose singularity theorems
  • Lie groups, homogeneous spaces, symmetric spaces
  • Morse theory, De Rham cohomology
  • Elliptic operators, spectral geometry, index theory
  • Vector fields, distributions, foliations, differential systems,
  • Frobenius theorem
  • General relativity

Intended learning outcomes

After the course, the student should have a sufficiently wide and deep knowledge of differential geometry to be able to start reading research level texts in the area and be able to connect and apply methods and results of differential geometry to other areas of mathematics.

Course disposition

The course is given as a series of lectures (approximately 16 x 2h), possibly with presentations given by the participants.

Literature and preparations

Specific prerequisites

Prerequisite for the course is knowledge of differential geometry (differential manifolds, tensors, differential forms) corresponding for example to the advanced level course SF2722 “Differential geometry”. Further, the participants should have read advanced level courses on algebra, analysis, and topology.

Recommended prerequisites

No information inserted

Equipment

No information inserted

Literature

For the introduction:

  • Barrett O'Neill, "Semi-Riemannian Geometry"

  • Christian Bär, lecture notes

For the other subjects:

  • Hand outs of appropriate literature.

Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

Grading scale

P, F

Examination

  • HEM1 - Home assignments, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Completed homework assignments and oral test or presentation.

Other requirements for final grade

Homework assignments and oral test or presentation.

Opportunity to complete the requirements via supplementary examination

No information inserted

Opportunity to raise an approved grade via renewed examination

No information inserted

Examiner

Ethical approach

  • All members of a group are responsible for the group's work.
  • In any assessment, every student shall honestly disclose any help received and sources used.
  • In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

Further information

Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSF3674

Offered by

Main field of study

This course does not belong to any Main field of study.

Education cycle

Third cycle

Add-on studies

No information inserted

Contact

Hans Ringström (hansr@kth.se); Mattias Dahl (dahl@math.kth.se)

Postgraduate course

Postgraduate courses at SCI/Mathematics