This talk will be an introduction to first order analysis on non-smooth spaces using the approach of upper gradients. The talk will be accessible to anyone with knowledge of measure theory, but an introductory knowledge of classical Sobolev spaces in Euclidean domains would be helpful.

I will discuss the role of Brownian loop measure in the study of Schramm-Loewner evolution. This powerful perspective allows us to apply intuition from discrete models (inparticular, the λ-SAW model) to the study of SLE while simultaneously reducing many SLE computations to problems of stochastic calculus. I will discuss recent work on multiple radial SLE that employs this method, including the construction of global multiple radial SLE and its links to locally independent SLE and Dyson Brownian motion. (Joint work with Gregory F. Lawler.)

Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with loop-erased random walk), proved by Lawler, Schramm and Werner. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (Schramm-Sheffield 2009), and Ising model and FK-Ising model (Chelkak-Smirnov et al 2012). In this talk, we focus on crossing probabilities of these critical lattice models in polygons with alternating boundary conditions.

The talk has two parts. In the first part, we consider critical Ising model and give crossing probabilities of multiple interfaces in the critical Ising model in polygon with alternating boundary conditions. Similar formulas also hold for other models, for instance level lines of Gaussian free field and Bernoulli percolation. However, the situation is different when one considers uniform spanning tree. In the second part, we discuss uniform spanning tree and explain the corresponding results.

We will review several settings of such spaces, their geometry and corresponding stochastic analysis: Hilbert-Schmidt groups as natural infinite-dimensional analogues of matrix groups, loop groups, and the homogeneous space Diff(S^1)/S^1 associated with the Virasoro algebra.