312.266 (24S) Selected Topics in Optimization: Symmetries and Semidefinite Programming

Sommersemester 2024

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Erster Termin der LV
06.03.2024 15:15 - 16:45 N.2.01 On Campus
Nächster Termin:
26.06.2024 15:15 - 16:45 N.2.01 On Campus

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

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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

Relevant papers and additional references will be introduced in the lecture.

Prüfungsinformationen

Im Fall von online durchgeführten Prüfungen sind die Standards zu beachten, die die technischen Geräte der Studierenden erfüllen müssen, um an diesen Prüfungen teilnehmen zu können.

Prüfungsmethode/n

Oral exam.

Prüfungsinhalt/e

Contents of the class.

Beurteilungskriterien/-maßstäbe

Knowledge of the contents of the class.

Beurteilungsschema

Note Benotungsschema

Position 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)
  • 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)
  • 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)
  • 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
  • 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

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