Inhalt:
topological recursion, noncommutative geometry,
quantum Hall effect
Topological recursion is a universal structure invented
by Chekhov, Eynard and Orantin. It was discovered 15 years ago as a
recursive procedure to compute all-order large-N expansion of certain
matrix models. Once formulated abstractly in terms of algebraic geometry
of curves, it has found an increasing number of applications beyond
matrix models in enumerative geometry, mirror symmetry,
low-dimensional (topological, conformal, etc.) field theories, and
more recently deformation quantisation, free probability and
hyperbolic geometry.
The first seminar talks give an overview, the basic definitions and
main results. A few selected examples and applications will follow.
Techniques from noncommutative differential geometry play a major
role in mathematical physics. A prime example is the quantum Hall
effect, experimentally discovered in 1980, where the Hall conductance
in two-dimensional materials subjected to low temperatures and strong
magnetic fields is an integer (later also fractional)
multiple of e^2/h. A mathematical theory for this phenomenon which
takes into account (a) the independence of material and geometry,
(b) insensitivity to rational or irrational magnetic flux per unit cell
and (c) stability against disorder was developed by J. Bellissard (1986).
The necessary tools include the non-commutative Brillouin
zone, its relation to irrational rotation algebras, the K-theory
of these C^*-algebras, quantum calculus with Dixmier trace,
the noncommutative analogue of the Chern number, through Connes'
cyclic cohomology, and the index formula. The seminar talks cover the
main steps of this programme and give an outlook to the open problem
of the fractional quantum Hall effect.
Literatur:
J. Bellissard, A. van Est & H. Schulz-Baldes,
"The noncommutative geometry of the quantum Hall effect",
J. Math. Phys. 35 (1994) 5373
B. Eynard, "Counting Surfaces", Progress in Mathematical Physics 70,
Springer-Verlag 2016
Termine:
dienstags 12h30-14h00, N1
Beginn: 16.10.2018
Seminarvorträge
| 16.10. | Vorbesprechung | |
| 23.10. | Quantum Hall effect: Summary of last term | Jean Bellissard |
| 30.10. | The Integer Quantum Hall Effect (IQHE), I | Jean Bellissard |
| 06.11. | ------ | |
| 13.11. | The Integer Quantum Hall Effect (IQHE), II. The first Laughlin argument. Problems with localized states. The Kubo-Chern formula in the zero temperature limit and quantization of the transverse conductivity. Defining the localization length. | Jean Bellissard |
| 20.11. | Tools from noncommutative geometry: the four traces way, the Connes formulas in cyclic cohomology, the Dixmier trace and localization length, existence of the Index. | Jean Bellissard |
| 27.11. | An incomplete overview about the Kontsevich model, I. | Raimar Wulkenhaar |
| 04.12. | An incomplete overview about the Kontsevich model, II. | Raimar Wulkenhaar |
| 11.12. | Topological recursion: I. Overview | Carlos Pérez |
| 18.12. | Topological recursion: II. | Alexander Hock |
| 08.01. | --- | |
| 15.01. | Fractional Quantum Hall effect, I. Interactions: second quantization. Algebraic construction. Introducing the disorder. The Spehner construction of a Hilbert C*-bimodule. | |
| 22.01. | Fractional Quantum Hall effect, II. Open problems |