Probability and Mathematical Physics Seminar

Probability and the City seminar

Speaker: Nina Holden and Ron Peled

Location: Columbia University, Mathematics Hall, 2990 Broadway 520

Date: Friday, September 16, 2022, 10 a.m.


The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The first meeting of Fall 2022 will feature:

  • Nina Holden (Courant Institute, NYU)
    Conformal welding in Liouville quantum gravity: recent results and applications


    Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origin in the physics literature. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface. The interfaces between the original LQG surfaces are typically described by variants of the random fractal curves known as Schramm-Loewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and their applications, which range from SLE and LQG to planar maps and random permutations. Based on joint works with Ang and Sun, with Lehmkuehler, and with Borga, Sun and Yu.
  • Ron Peled (Tel-Aviv University)
    Random packings and liquid crystals

    Let T be a subset of R^d, such as a ball, a cube or a cylinder, and consider all possibilities for packing translates of T, perhaps with its rotations, in some bounded domain in R^d. What does a typical packing of this sort look like? One mathematical formalization of this question is to fix the density of the packing and sample uniformly among all possible packings with this density. Discrete versions of the question may be formulated on lattice graphs.
    The question arises naturally in the sciences, where T may be thought of as a molecule and its packing is related to the spatial arrangement of molecules of a material under given conditions. In some cases, the material forms a liquid crystal - states of matter which are, in a sense, between liquids and crystals. 
    I will review ideas from this topic, mentioning some of the predictions and the mathematical progress. Time permitting, I will elaborate on a recent result, joint with Daniel Hadas, on the structure of high-density packings of 2x2 squares with centers on the square lattice.
    The talk is meant to be accessible to a general mathematical audience.