Topic outline

  • General

    Teachers

    • Kalle Kytölä (lecturer)
    • David Adame-Carrillo (teaching assistant)


    Teaching times

    • Lectures: Mon 14-16 and Fri 12-14 in M3
    • Exercises: Fri 14-16 in M3


    Course online chat forum: https://tipcft.zulip.aalto.fi (login with Aalto credentials).


    Description

    This course is meant as a mathematical introduction to scaling limits of two-dimensional probabilistic models and conformal field theory (CFT). The broad conjecture of conformal invariance of scaling limits was made by physicists in the 1980's, and in the past two decades also there has also been remarkable progress in the rigorous mathematical understanding of it (Fields' medals were awarded to Wendelin Werner in 2006, Stanislav Smirnov in 2010, and Hugo Duminil-Copin in 2022, partly for their work on conformally invariant scaling limits, and conformal invariance results for correlations in the Ising model have been proved in landmark works of Dmitry Chelkak, Clément Hongler, and Konstantin Izyurov). A key ingredient has been discrete holomorphic and discrete harmonic observables in the probabilistic models. These feature in the proofs of existence and characterization of scaling limits, and they also underlie the emergence of the rich algebraic structures of CFT predicted by physicists.

    In this course we focus on basic cases illustrating the general ideas. The planned contents include

    • Conformal maps (in particular Riemann mapping theorem)
    • Discrete probabilistic models (in particular uniform spanning trees, discrete Gaussian free field, maybe also the Ising model)
    • Discrete harmonicity and discrete holomorphicity (in particular observables in models and convergence of the observables in the scaling limit)
    See Lectures tab for up-to-date content in more detail.


    Completing the course

    The course is graded on a pass/fail scale based on weekly exercises and a short (approximately 30min) presentation of a topic related to the course. There is no exam. The presentations are done in the end of the course on topics agreed with the lecturer during the course. The range of appropriate topics is wide; from complex analysis to probability to representation theory, and the topics are chosen with the students' own background and interests in mind (percolation, dimer model, weak convergence of probability measures, Loewner flows, regularity of weakly harmonic functions, central extension of the Witt algebra, ...).