Conformally invariant random processes near their critical point
The random geometry of many planar lattice models (random walks, uniform spanning trees, domino tilings, percolation, the Ising model of magnetization) exhibits some magical symmetry near the phase transition points (think of the critical temperature above which metal loses its magnetizability). This magical symmetry is called conformal invariance, which turns out to be extremely helpful in understanding the critical systems, e.g., to compute the fractal dimension of interfaces, using the celebrated Schramm-Loewner Evolution. I will give an overview of what kind of things can be proved, with a small emphasis on my own work that concerns what happens when we start perturbing the critical system, e.g., by moving slowly away from the critical point.
You can watch the lecture here: