First, we will discuss sequences of closed minimal hypersurfaces (in closed Riemannian manifolds of dimension up to 7) that have uniformly bounded index and area. In particular, we explain a bubbling result which yields a bound on the total curvature along the sequence and, as a consequence, topological control in terms of index and area. We then specialise to minimal surfaces in ambient manifolds of dimension 3, where we use the bubbling analysis to obtain smooth multiplicity-one convergence under bounds on the index and genus. This is joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp

 The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics.  However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry.  “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many
problems.  We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras.  We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.

This will be a discussion of the paper https://arxiv.org/abs/1607.06978.

The “field with one element” is an interesting algebraic object that in some sense relates linear algebra with set theory. In a much deeper vein it is also expected to have a role in algebraic geometry that could potentially “lift" Deligne’s proof of the final Weil Conjecture for varieties over finite fields to a proof of the Riemann hypothesis for the Riemann zeta function. The only problem is that it doesn’t exist. In this highly speculative talk I will discuss some of these concepts, and focus mainly on zeta functions of algebraic varieties over finite fields. I will give a (very) brief sketch of how to interpret various zeta functions in a geometric context, and try to explain what goes wrong for the Riemann zeta function that makes this a difficult problem.