The Holy Grail of Artificial Intelligence, true “deep” AI, has been 10 years away ever since the Dartmouth Conference in 1956, in the same way that fusion reactors have been 20 years out for the past 50 years. In the meantime, we’ve gone through 2 “AI Winters” and 3 “AI Summers,” as the expectations of investors get lowered to meet the rising actual capabilities of neural nets. The talk will be a brisk tour through the history of neural networks, with particular emphasis on the intuition of how neural networks work.

We will discuss the Makeenko--Migdal equation (MM equation) which relates variations of a "Wilson loop functional" (relative to the Euclidean Yang--Mills measure) in the neighborhood of a simple crossing to the associated Wilson loops on either side of the crossing. We will begin by introducing the 2d -- Yang-Mills measure and explaining the necessary background in order to understand the theorem. The goal is to describe the original heuristic argument of Makeenko and Migdal and then explain how these arguments can be made rigorous using stochastic calculus.

We discuss traveling front solutions $u(t,x) = U(x-ct)$ of reaction-diffusion equations $u_t = Lu + f(u)$ in 1d with ignition reactions $f$ and diffusion operators $L$ generated by symmetric Levy processes $X_t$. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of $X_1$ is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of $X_1$ is infinite.

Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years.
Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial $p$ and any matrix $A$,
$||p(A)|| <= 2 max(|p(z)|: z$ in $W(A))$, where the norm is the 2-norm and $W(A)$ is the field
of values (numerical range) of $A$, that is the set of points attained by $v*Av$ for some
vector $v$ of unit length. Remarkably, Crouzeix proved in 2007 that the inequality above
holds if 2 is replaced by 11.08. Furthermore, it is known that the conjecture holds in a
number of special cases, including $n=2$. We use nonsmooth optimization to investigate
the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the
quotient with numerator the right-hand side and denominator the left-hand side of the
conjectured inequality. We present numerical results that lead to some theorems and
further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers.
All the computations strongly support the truth of Crouzeix’s conjecture.

This is joint work with Anne Greenbaum and Adrian Lewis.

Work by McQuillan and Brunella demonstrates the existence of a
Mori theory for rank 1 foliations on surfaces. In this talk we will
discuss an extension of some of these results to the case of rank 2
foliations on threefolds, as well as indicating how a complete Mori theory
could be developed in this case.

We will discuss Lefschetz theorems on the homotopy groups of
hyperplane sections in the arithmetic setting, i.e. for a divisor, ample
in the Arakelov sense, over a projective scheme defined over the ring of
integers in a number field. An interesting corollary is that the integral
model of a generic complete intersection curve of big height, is a simply
connected arithmetic surface. Joint work with Michael McQuillan.