312.266 (24S) Selected Topics in Optimization: Symmetries and Semidefinite Programming
Überblick
- Lehrende/r
- LV-Titel englisch Selected Topics in Optimization: Symmetries and Semidefinite Programming
- LV-Art Vorlesung
- LV-Modell Präsenzlehrveranstaltung
- Semesterstunde/n 2.0
- ECTS-Anrechnungspunkte 3.0
- Anmeldungen 17
- Organisationseinheit
- Unterrichtssprache Englisch
- LV-Beginn 06.03.2024
- eLearning zum Moodle-Kurs
Zeit und Ort
LV-Beschreibung
Intendierte Lernergebnisse
Students will be able to build semidefinite programming hierarchies for a wide variety of problems, and apply tools from representation theory and (real) algebraic geometry in practice.
Lehrmethodik
Lecture
Inhalt/e
Semidefinite programming, a generalization of linear programming, is a surprisingly rich and quickly growing sub-field of optimization. Initially introduced for combinatoric questions about graphs, the method has since found applications in (real) algebraic geometry, PDEs, discrete geometry, quantum information theory and much more. In this lecture the focus will be on a particularly interesting subclass of problems: high dimensional problems exhibiting symmetries. To be able to attack these problems computationally, results from representation theory, algebraic combinatorics and harmonic analysis are required.
Preliminary list of contents:
- Introduction to semidefinite programming and its duality theory: the theta-number of a graph.
- Goemanns-Williamson algorithm for Max-Cut.
- The Shannon-capacity of a graph: Bounding the independence number of an infinite graph.
- Error-correcting-codes: Symmetries and the Delsarte-LP bound.
- The regular representation of Matrix-*-Algebras and the crossing number of a graph.
- Energy-minimization and universal optimality, Bochner's theorem.
- Representation theory with focus on Artin-Wedderburn theory.
- Spherical harmonics: Bounds for the Kissing number (towards Maryna Viazovska's 2022 fields medal).
- Polynomial optimization and real algebraic geometry.
- Representation stability: Infinite dimensional and dimension-free optimization.
Erwartete Vorkenntnisse
Good understanding of linear algebra and group theory. No knowledge of linear optimization or representation theory is required.
Literatur
- M. Laurent and F. Vallentin: Semidefinite Optimization
- F. Vallentin: Semidefinite programs and harmonic analysis
- B. E. Sagan: The Symmetric Group
Relevant papers and additional references will be introduced in the lecture.
Prüfungsinformationen
Prüfungsmethode/n
Oral exam.
Prüfungsinhalt/e
Contents of the class.
Beurteilungskriterien/-maßstäbe
Knowledge of the contents of the class.
Beurteilungsschema
Note BenotungsschemaPosition im Curriculum
- Doktoratsprogramm Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(SKZ: ---, Version: 16W.1)
-
Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(Pflichtfach)
-
Modeling-Analysis - Optimization of discrete, continuous and stochastic systems (
0.0h XX / 0.0 ECTS)
- 312.266 Selected Topics in Optimization: Symmetries and Semidefinite Programming (2.0h VO / 3.0 ECTS)
-
Modeling-Analysis - Optimization of discrete, continuous and stochastic systems (
0.0h XX / 0.0 ECTS)
-
Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(Pflichtfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 18W.1)
-
Fach: Discrete Mathematics
(Wahlfach)
-
6.8 Selected Topics in Optimization (
2.0h VO / 3.0 ECTS)
- 312.266 Selected Topics in Optimization: Symmetries and Semidefinite Programming (2.0h VO / 3.0 ECTS)
-
6.8 Selected Topics in Optimization (
2.0h VO / 3.0 ECTS)
-
Fach: Discrete Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 18W.1)
-
Fach: Applied Mathematics
(Wahlfach)
-
Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
- 312.266 Selected Topics in Optimization: Symmetries and Semidefinite Programming (2.0h VO / 3.0 ECTS)
-
Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
-
Fach: Applied Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 22W.1)
-
Fach: Discrete Mathematics
(Wahlfach)
-
5.8 Selected Topics in Optimization (
2.0h VO / 3.0 ECTS)
- 312.266 Selected Topics in Optimization: Symmetries and Semidefinite Programming (2.0h VO / 3.0 ECTS) Absolvierung im 1., 2., 3. Semester empfohlen
-
5.8 Selected Topics in Optimization (
2.0h VO / 3.0 ECTS)
-
Fach: Discrete Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 22W.1)
-
Fach: Applied Mathematics
(Wahlfach)
-
7.1 Wahl von weiteren Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
- 312.266 Selected Topics in Optimization: Symmetries and Semidefinite Programming (2.0h VO / 3.0 ECTS) Absolvierung im 2., 3. Semester empfohlen
-
7.1 Wahl von weiteren Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
-
Fach: Applied Mathematics
(Wahlfach)