Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations

By a gluing construction, we produce steady Kahler-Ricci solitons on equivariant crepant resolutions of C^n/G, where G is a finite subgroup of SU(n), generalizing Cao's construction of such a soliton on a resolution of C^n/Z_n.  This is joint work with Olivier Biquard.

The celebrated Alexandrov-Bakelman-Pucci Maximum Principle (often abbreviated as ABP estimate) is a pointwise estimate for solutions of elliptic equations, which was introduced in the 1960s. It was motivated by beautiful geometric ideas and has been a fundamental tool in the study of non-divergent PDEs. More recently, this PDE technique also pays back to geometry - the ABP estimate and its extensions can be used to prove some optimal classical geometric inequalities such as the Isoperimetric and Minkowski inequalites.

How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from Z^d to Z^d. The cokernel of M is Z^d/Im(M). If det(M) is nonzero, then the cokernel is a finite abelian group of order det(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of Z^d.