# Marston Morse Lectures

### On the Ising perceptron model

Nike Sun

(This lecture will be self-contained.) In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem...

### Probabilistic analysis of random CSPs

Nike Sun

(This lecture is related to the preceding lecture, but I will try to make it self-contained as much as possible.) In this lecture I will elaborate on some of the existing mathematical approaches to the study of random CSPs, particularly involving...

### Statistical physics of random CSPs

Nike Sun

I will describe recent progress in determination of asymptotic behavior in random constraint satisfaction problems, including the independent set problem on random graphs, random regular NAE-SAT, and random SAT. The results include sharp phase...

### Fluctuations look like white noise

Laure Saint-Raymond
At leading order, the fluctuations around the typical dynamics are described by the second cumulant. They actually satisfy a stochastic PDE with time-space white noise. Can we say more using higher order cumulants?

### Space-time correlations at equilibrium

Laure Saint-Raymond
Although the distribution of hard spheres remains essentially chaotic in this regime, collisions give birth to small correlations. The structure of these dynamical correlations is amazing, going through all scales. How combinatorial techniques...

### Disorder increases almost surely.

Laure Saint-Raymond
Consider a system of small hard spheres, which are initially (almost) independent and identically distributed. Then, in the low density limit, their empirical measure $\frac1N \sum_{i=1}^N \delta_{x_i(t), v_i(t)}$ converges almost surely to a...

### Exceptional holonomy and related geometric structures: Dimension reduction and boundary value problems.

Simon Donaldson

By imposing symmetry on manifolds of exceptional holonomy we get a variety of differential geometric questions in lower dimensions. Related to that, one can consider “adiabatic limits”, where the manifold has a fibration and the fibre size is scaled...

### Exceptional holonomy and related geometric structures: Examples and moduli theory.

Simon Donaldson

We will discuss the constructions of compact manifolds with exceptional holonomy (in fact, holonomy $G_{2}$), due to Joyce and Kovalev. These both use “gluing constructions”. The first involves de-singularising quotient spaces and the second...

### Exceptional holonomy and related geometric structures: Basic theory.

Simon Donaldson

In this lecture we will review the notion of the holonomy group of a Riemannian manifold and the Berger classification. We will discuss special algebraic structures in dimensions 6, 7 and 8, emphasising exterior algebra, and then go on to...

### Folding papers and turbulent flows

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it...