In this talk, I'll describe conjectures and recent work concerning field invariants and their complexity, relating to quadratic forms, Galois cohomology, and similar concepts.

A configuration in the 1-2 model is a subgraph of the hexagonal lattice, in which each vertex is incident to 1 or 2 edges. By assigning weights to configurations at each vertex, we can define a family of probability measures on the space of these configurations, such that the probability of a configuration is proportional to the product of weights of configurations at vertices.

We study the phase transition of the model by investigating the probability measures with varying weights. We explicitly identify the critical weights, in the sense that the edge-edge correlation decays to 0 exponentially in the subcritical case, and converges to a non-zero constant in the supercritical case, under the limit measure obtained from torus approximation. These results are obtained by a novel measure-preserving correspondence between configurations in the 1-2 model and perfect matchings on a decorated graph, which appears to be a more efficient way to solve the model, compared to the holographic algorithm used by computer scientists to study the model. 

When the weights are uniform, we prove a weak mixing property for the finite-volume measures - this implies the uniqueness of the infinite-volume measure and the fast mixing of a Markov chain Monte Carlo sampling. The major difficulty here is the absence of stochastic monotonicity.

We present a general framework for approximating combinatorial assemblies when both the size $n$ and the number of components $k$ is specified.  The approach is an extension of the usual saddle point approximation, and we demonstrate near-universal behavior when the rank $r := n-k$ is small relative to $n$ (hence the name `low rank’).  

In particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha \in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$L_1$ of the largest component converges in probability to $\ell+2$.  When $r \sim t\, n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, $\P(L_1 \in \{\ell+1, \ell+2\}) \to 1$.  We also obtain as a corollary bounds on the number of such combinatorial assemblies, which in the special case of set partitions fills in a countable number of gaps in the asymptotic analysis of Louchard for Stirling numbers of the second kind. 

This is joint work with Richard Arratia.

The P-prime operator is a CR invariant operator on CR pluriharmonic functions and is closely related to a sharp Moser--Trudinger-type inequality in CR manifolds.  I will describe some analytic and geometric properties of this operator, and in particular use it to solve a nonlinear PDE of critical order which is the CR analogue of the Q-curvature prescription problem.  This talk is based on joint works with Paul Yang and Chin-Yu Hsiao.

Suppose one wants to calculate the eigenvalues of a large, non-normal matrix.  For example, consider the matrix which is 0 in most places except above the diagonal, where it is 1.  The eigenvalues of this matrix are all 0.  Similarly, if one conjugates this matrix, in exact arithmetic one would get all eigenvalues equal to 0.  However, when one makes floating point errors, the eigenvalues of this matrix are dramatically different.  One can model these errors as performing a small, random perturbation to the matrix.  And, far from being random, the eigenvalues of this perturbed matrix nearly exactly equidistribute on the unit circle.  This talk will give a probabilistic explanation of why this happens and discuss the general question: how does one predict the eigenvalues of a large, non-normal, randomly perturbed matrix?