Stochastic Quantisation of Yang-Mills


We report on recent progress on the problem of building a stochastic process that admits the still hypothetical 3D Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen. The talk will be a rather gentle introduction to this area which does not require familiarity with any of the above mentioned objects.

Nonlinear Harmonic Maps and the Energy Identity


We will begin this talk with a broad introduction to nonlinear harmonic maps with a discussion of some basic examples and background.  We will then focus our attention on a type of singularity formulation, resulting in so called defect measures.   Though not typically studied in such generality, these defect measures are a general construction for understanding the loss of energy in limits of H^1 functions.   In the case of nonlinear harmonic maps, there is a precise conjecture as to the form of these defect measures called the Energy Identity.  

This was recently proved together with Daniele Valorta, and a few words on the proof will be discussed.

Generalizations of the Bernstein Problem


Sergei Bernstein proved (in 1914) that an entire solution to the minimal surface equation on R^2 must be affine. This is a nonlinear version of the Liouville theorem for harmonic functions and turns out to have deep links with the regularity of minimal surfaces. I'll explain what happens in higher dimensions, as well as some natural generalizations.

Minimal Surfaces, Hyperbolic Surfaces, and Randomness


Minimal surfaces and hyperbolic surfaces are both "optimal 2d geometries" which are ubiquitous in differential geometry. The first kind is defined by an extrinsic condition (the mean curvature vanishes), while the second kind is defined by an intrinsic condition (the Gaussian curvature is equal to -1). I will discuss a surprising connection between the two geometries coming from randomness. The main statement is that there exists a sequence of closed minimal surfaces in Euclidean spheres, constructed from random permutations, which converges to the hyperbolic plane. This result came from my attempt to bridge minimal surfaces and unitary representations. I will introduce this circle of ideas and mention some general questions.