Course info

Topics

Topological concepts provide the key to understanding various phenomena in contemporary condensed-matter physics research. This course gives an introduction to the mathematical basics of these concepts located at the intersection of algebraic topology, differential topology and differential geometry, as well as to their physical applications. It will cover the following topics:

• Topological spaces and smooth manifolds
• Real and complex projective spaces
• Homotopy theory, fundamental group
• Degree of a mapping, Hopf theorem
• Tangent vector fields, Poincaré-Hopf theorem
• Vector bundles, Euler number, Chern number
• Connection and curvature of vector bundles
• Localized field configurations characterized by topological invariants (domain walls, skyrmions, hopfions)
• Topology of the band structure in reciprocal space (integer quantum Hall effect, Chern insulators, topological insulators, Weyl semimetals)
• Berry connection, Berry curvature and their connection to topological invariants and observable phenomena
• Connection between topology and quantum statistics, fractional statistics, Majorana bound states, braiding

1. A presentation on the topological aspects in a certain physical problem based on a research paper. The list of papers will be made available at the beginning of the semester when each student has to select one. The presentation should consider aspects discussed during the lecture but not necessarily described in the paper itself. The lecturers offer consultations in preparation for the presentation.
2. The solution of certain problems using the methods learnt during the course. The list of problems will be made available during the semester as the course progresses. Students are encouraged to submit written solutions to these problems during the semester in moodle, where they will receive feedback from the lecturers. This will support the preparation for the oral exam.