Welcome to my class on Topological phases and K-theory for Banach algebras in the winter term 2024/5! The class starts with a crash course in quantum mechanics. This motivates the introduction of K-theory as a tool to classify the different topological phases for a given symmetry type. The study of K-theory and its properties will then be the main topic of the course. Quantum mechanics plays a role as a motivation for the theory. It also suggests to highlight certain topics that are not always treated in a first course on K-theory. This includes the K-theory for real Banach algebras, possibly with a grading by the group Z/2, as introduced by van Daele. The computation of the K-theory for tori with suitable real structure will be what is most relevant for the physical applications.
Classes are scheduled on Tuesday and Friday 8-10, from October, 22nd, 2024 until February, 7th, 2025, in the Hörsaal Sitzungszimmer in the Mathematical Institute in Göttingen. The usual lecture stream does not work at the moment, the only option that works requires Stud.IP access, use the meeting module in the course. The recordings are listed below.
| Date |
Topic and Recording Link |
| 22.10.2024 |
General course introduction |
|
Physics crash course: Hilbert spaces, states, observables, unbounded selfadjoint operators |
| 25.10.2024 |
Functional calculus, spectral measure |
|
Representations generated by selfadjoint operators, Stone's Theorem |
| 29.10.2024 |
Dynamics, bound states and scattering states, wave packets |
|
Rotations and angular momentum, spin, projective representations, especially for SO(3) |
| 1.11.2024 |
Antiunitary symmetries of physical systems |
|
Different symmetry types and groups they form, single-particle approximation |
| 5.11.2024 |
Single particle approximation, Fermi energy |
|
Tight binding models, translation invariant Hamiltonians |
| 8.11.2024 |
Lattices, tight-binding Hamiltonians |
|
Spectrum of tight-binding Hamiltonians, energy bands, conductors and insulators |
| 12.11.2024 |
Lattices, Su-Schrieffer-Heeger model |
|
The graphene crystal and its symmetries and Hilbert space |
| 15.11.2024 |
Graphene model Hamiltonian |
|
Half-space restriction of the SSH-model Hamiltonian is not invertible |
| 19.11.2024 |
Fredholm index, motivation, compact perturbation invariance |
|
Index theory: additivity, compact perturbations of identity map |
| 22.11.2024 |
Index theory, following the Murphy's book |
|
Index theory, following the Murphy's book |
| 26.11.2024 |
The index theorem for Toeplitz operator |
|
The index theorem for Toeplitz operator, lifting invertible elements, definition of first K-group |
| 29.11.2024 |
Unifying different symmetry classes |
|
Towards van Daele's definition of K-theory for Z/2-graded Banach algebras |
| 3.12.2024 |
Definition of van Daele's K-theory |
|
Independence of van Daele's K-theory on e |
| 6.12.2024 |
Grothendieck construction, Roe's definition of K-theory |
|
K-theory for general Z/2-graded Banach algebras |
| 10.12.2024 |
Exact and additive functors |
|
Matrix-stable and homotopy invariant functors, homotopy invariance of van Daele's K-theory |
| 13.12.2024 |
Stability of van Daele K-theory, special homotopies by conjugation |
|
Homotopies of even invertibles, special representatives for van Daele K-theory classes |
| 17.12.2024 |
Functoriality of van Daele K-theory, matrix stability and continuity, exactness |
|
| 7.1.2024 |
The exactness of K-theory |
|
From exactness to the long exact sequence |
| 10.1.2024 |
The K-theory boundary map, the Toeplitz extension and quasi-homomorphisms |
| 14.1.2024 |
Computation of K-theory for Toeplitz algebras 1 |
| 17.1.2024 |
Computation for Toeplitz algebras finished |
|
More on Toeplitz algebras, computation for spheres, automatic homotopy invariance |
| 21.1.2024 |
The Pimsner-Voiculescu exact sequence |
| 24.1.2024 |
The K-theory of Cuntz algebras and graph C*-algebras Passcode: CiAQ4*x% |
|
Notes from the lecture |
| 28.1.2024 |
Clifford algebras |
| 31.1.2024 |
Classification of Clifford algebras |
| 4.2.2024 |
A glimpse of bivariant K-theory |
| 7.2.2024 |
K-Homology |