Speaker: Jianping Jiang (New York University Shanghai)

Time: 9:30-11:30, March 24 and 25

Mini-course 3:

1. Random cluster models
   We introduce the random cluster models and show some useful inequalities.

2. Sharp phase transition
   We compute the critical point and show the phase transition is sharp.

3. Critical behavior
   We talk about the critical behavior of random cluster models. In particular, we prove Russo-Seymour-Welsh type estimates.

4. Conformal invariance
   We show conformal invariance for some critical random cluster models.

Speaker: Zhehua Li (New York University Shanghai)

Time: 14:30-16:30, March 24 and 25

Mini-course 4:

1. A probabilistic proof of index theorems
   We will give a detailed probabilistic proof of the Gauss-Bonnet-Chern theorem and use it as an example to overview the role of stochastic analysis in differential geometry.

2. Path integrals and supersymmetric proof of index theorems
   We will introduce path integrals and a physics proof of index theorems based on the supersymmetric quantum field. We will also mention several attempts to make it mathematically rigorous from the community of differential geometry.

3. Basic ideas of constructive field theory
   We will discuss some basic ideas of constructive quantum field theory related to path integrals, with the goal of motivating the informal expression for the Euclidean Yang-Mills measure.

4. Path integrals and Yang-Mills’ measure
   We will introduce the famous Yang-Mills’ measure. In particular, we will mention several attempts to make sense of its informal description.