Universität Wien

250093 VO Lattice Models (2022S)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

The course will be on site (hopefully), but I’m planning to do it on my IPad+projector and could link it to zoom. So it will be possible to follow the lectures online and there will be a recording + lecture notes.

  • Mittwoch 02.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 09.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 16.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 23.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 30.03. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 06.04. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 27.04. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 04.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 11.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 18.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 25.05. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 01.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 08.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 15.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 22.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Mittwoch 29.06. 13:15 - 14:45 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models: the material was modeled as a collection of particles on a regular lattice, interacting only with their nearest neighbors. In spite of the simplistic nature of this assumption, lattice models have proven to be a rich laboratory for the mathematical study of phase transitions. Since the revolutionary work of Schramm in 2000, the probabilistic approach to the study of these models has yielded a veritable explosion of new insights, with two Fields Medals being awarded to Smirnov and Werner for their breakthroughs.

In this course, we aim to familiarize the audience with a modern approach to some classical results from the probabilistic theory of lattice models, using Bernoulli percolation and the Random-Cluster model as our main examples. We then use these tools to discuss some very recent results on the study of random Lipschitz functions.

A particular focus will be given to the two-dimensional models, where even the simplest models lead to a dazzling array of different fractal behaviors. This is a consequence of the conformal invariance of these models, which is predicted for all the models discussed, but rigorously proved in very few cases. One of our goals is a presentation of Smirnov's proof of the conformal invariance of critical site percolation on the triangular lattice.

These models give a beautiful way to apply the material learnt in the Probability course. Quite a few ideas are of combinatorial nature and the field is connected to several other branches of mathematics: Mathematical Physics, Ergodic Theory, Complex Analysis, Conformal Geometry, Computer Science.

It is highly recommended to follow the exercise sessions.

Art der Leistungskontrolle und erlaubte Hilfsmittel

oral exam (possibly online)

Mindestanforderungen und Beurteilungsmaßstab

Thorough understanding and a working knowledge of the core part of the material presented in the lectures is required for passing the exam.

Prüfungsstoff

The material presented in the lectures; a more detailed description of what exactly is expected in the exam will be made available during the lectures

Literatur

Lecture notes of Hugo Duminil-Copin:
- https://www.ihes.fr/~duminil/publi/2017percolation.pdf
- https://arxiv.org/pdf/1707.00520.pdf

Grimmett "The Random-cluster model":
- http://www.statslab.cam.ac.uk/~grg/books/rcm1-1.pdf

Zuordnung im Vorlesungsverzeichnis

MSTV

Letzte Änderung: Mi 29.03.2023 09:28