The main focus of this research is on two-dimensional random geometry and conformally invariant probability. This area is at the interface of several different fields of mathematics, including both discrete and continuous probability theory, complex analysis, differential equations, and geometry. The main questions have to do with understanding the large-scale behavior of random discrete models which arise from statistical mechanics. One concrete example is the so-called percolation model which was first considered in the mathematics literature by Broadbent and Hammersley in 1957 and serves to model how fluid flows through a porous medium.
In the 1980s, physicists derived a number of non-rigorous predictions about the behavior of critical percolation clusters (and many other related models) in two dimensions (in which case the graph is often taken to be the planar square grid). Establishing these predictions rigorously from a mathematical perspective remained elusive, however, until work initiated by Schramm in 1999. One of Schramm's key contributions was the introduction of the Schramm-Loewner evolution (SLE), which is a mathematical tool defined by combining ideas from probability and complex analysis. It is a family of random planar curves indexed by a parameter kappa > 0 which determines its roughness; when kappa is close to 0, SLE is close to being a smooth curve and as kappa gets larger the behavior of SLE is increasingly fractal in nature. SLE for different values of kappa was subsequently shown to describe the large-scale behavior of many discrete models in two dimensions. For example, the clusters in critical percolation were proved by Smirnov to correspond to SLE with kappa=6. SLE for other values of kappa has been shown to be the scaling limit of a number of other discrete models in works of Lawler, Sheffield, Schramm, Smirnov, Werner, and others.
The methods employed in the mathematics literature, including SLE, to verify the physics predictions are in many ways quite different from the way in which the physicists originally derived their predictions. One of the key insights employed by the physicists is that models such as percolation are more tractable if one studies them on a random planar map (RPM) rather than on a deterministic graph, such as the planar square grid. Recall that a planar map is a graph together with an embedding into the plane so that the nodes are points and the edges between nodes are drawn so that they do not cross. Commonly considered cases include triangulations or quadrangulations, which respectively mean that each face has three or four adjacent edges. The properties of RPMs is itself subject of intense interest in the combinatorics literature going back to work of Tutte in the 1960s in his attempt to prove the four color theorem as well as that of Mullin on spanning-tree decorated RPMs. The subject was revitalized by bijective techniques developed in the 1980s by Cori-Vauquelin, in the 1990s by Schaeffer, and tremendous progress has been made in starting around 2000 by Angel, Chassaing, Le Gall, Marckert, Miermont, Mokkadem, Schaeffer, Schramm, and others.
When one makes a change to the underlying geometry of the graph (i.e. going from the planar square grid to a RPM), the properties of discrete models such as percolation also change. The manner in which these properties change led physicists to conjecture that the continuous object which describes the large-scale behavior of RPMs is Liouville quantum gravity (LQG). LQG is a theory of random two-dimensional Riemannian manifolds which was developed by Polyakov in the physics literature in the 1980s in the context of string theory. As a mathematical object, LQG is ill-defined, however. The interpretation of LQG as the exponential of the Gaussian free field (GFF) was developed by Duplantier-Sheffield in the 2000s and this led to the construction of its volume form; it also turns out to be related to the theory of Gaussian multiplicative chaos (GMC) developed by Kahane in the 1980s.
It has become an intense topic of research in the probability community to make the family of ideas which underlies the methods used by physicists to derive predictions for two-dimensional discrete models mathematically precise. The overall objectives for this research are centered in this direction, including establishing new scaling limit results for RPMs and connections to LQG as well as making sense of LQG in a mathematically rigorous manner (understanding it as a metric space). Altogether, the aim is to put the ideas of the physicists onto firm mathematical ground in a way which will shed further light on these random geometric structures.
