2026 FALL ABSTRACTS

Sep 22    Penn: Reuben Drogin  (Yale)

Random Band Matrices and Random Permutations

Random band matrices are Hermitian matrices with random entries supported in a band of width W around the diagonal. The eigenfunctions of such matrices are expected to decay exponentially at the scale W^2 in dimension one, and exp(CW^2) in dimension two. Remarkably, the same scaling is expected for the cycle lengths in various models of random permutations where points are typically displaced by distances of order W. In this talk I will discuss some recent progress on these problems, some connections, and some open questions.

Sep 29    Penn: Dmitry Krachun  (Princeton)

Towards conformal invariance of critical planar lattice models

Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the proof of rotational invariance of the FK-percolation model and more recently the computation of critical exponents of planar FK model with q=4. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz. We will then explore the connection to the six-vertex model further to derive critical exponents of the FK model with q=4.

Based on joint works with Hong-Bin Chen, Hugo Duminil-Copin, Tiancheng He, François Jacopin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, Tatiana Tikhonovskaia, and Jiaming Xia.

Oct 20    Penn: Remy Mahfouf  (EPFL)

From the planar Ising model to quasiconformal mappings

We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate s-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations connecting the scaling limits of (near-)critical planar Ising models to Green kernels of uniformly elliptic operators and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graph. Moreover, we prove a connection between the planar Ising energy field in smooth limiting structure and (massive)-holomorphic fermions defined on surfaces of the Minkowski space R^{(2,1)}. These results highlight that the scaling limits of generic (near-)critical Ising models naturally live on a substantially richer conformal structure than the classical Euclidean one.

Oct 27    Penn: Hindy Drillick (NYU) 

Title and abstract to be announced.

Dec 01    Penn: Matthew Rosenzweig  (Carnegie Mellon)

Title and abstract to be announced.