Given a foliated manifold $M$, one feels that a leafwise smoothing operator on M ought to be represented by a smooth function on the subset of $M\times M$ consisting of pairs of points lying on the same leaf. However, this subset may fail to be a manifold due to an interesting and important phenomenon known as $\textbf{holonomy}$. The $\textbf{holonomy groupoid}$ of a foliation is, roughly speaking, what results when one attempts to smooth out this singular set, somewhat in the spirit of blowups in algebraic geometry. In this talk, I will attempt to explain the concept of holonomy and sketch some work I have done on the groupoids and algebras associated to singular foliations, based on work of Androulidakis and Skandalis.

A good way to put two subspaces or two matrices into general position is to randomly rotate one by an orthogonal or unitary matrix, randomly selected according to Haar measure. Then in order to find expected values, one has to compute matrix integrals. Adolf Hurwitz gave a construction of Haar measure on the $N \times N$ unitary group in 1897; many now consider his paper to be the beginning of random matrix theory. In 1978, Don Weingarten gave a new method based on Schur-Weil duality; Weingarten's method is now known as the \textit{Weingarten calculus}. While exact formulas are possible, they can be quite complicated. If one is willing to settle for an approximate answer, there is an expansion in $1/N$ which is much simpler. The Weingarten function has become central to free probability. I will explain how an extension of free probability, called infinitesimal freeness, gives new insight on the Weingarten calculus.