The of existence of framed manifolds of Kervaire invariant one
is one of the oldest unresolved problems in algebraic topology.
Important questions about smooth structures on spheres  and on the
homotopy groups of spheres depend on its solution.  In this talk I
will describe joint work with Mike Hill and Doug Ravenel which solves
this problem in all dimensions except 126.

I will sketch a recent approach to study the resonance spectrum of scattering Schroedinger operators, in cases where the trapped set of the corresponding classical dynamics (near some positive energy) is a fractal chaotic repeller. In that situation, we are interested in the distribution of resonances in the vicinity of the real axis, in the semiclassical limit. Our approach tends to mimic the PoincarÈ section method used to study the corresponding classical flows. Namely, we show that resonances can be defined through an implicit equation involving an appropriately defined quantized Poincare map. The subsequent study of this "quantum map" allows to recover known properties of the resonance spectrum (fractal bounds on the number of resonances in strips, spectral gap), and hopefully more.

This is joint work with J. Sjoestrand and M. Zworski.

The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. In this setting, a key objective is to find a suitable generalization of Ricci curvature, and to understand the associated ``quasi-Einstein'' metrics. Taking two different perspectives, Lott, Villani, Sturm and Chang, Gursky and Yang have found two distinct approaches to studying smooth metric measure spaces. While the formulations are different, they both introduce an extra dimensional parameter $m$ which, in the limit $m\to\infty$, recovers the curvatures that arise in Perelman's treatment of the Ricci flow. In this way it becomes interesting to see if the two approaches are related. As the quasi-Einstein metrics of these approaches include conformally Einstein metrics, the bases of Einstein warped products, and gradient Ricci solitons, finding a relation between them might also allow us to find interesting connections between these metrics.

In this talk, I will introduce what I call ``conformally warped manifolds'' as a way to unite the approaches of Lott-Villani-Sturm and Chang-Gursky-Yang. I will discuss three results which suggest that this notion is indeed the ``best'' approach. First, I will discuss the variational problem associated to quasi-Einstein metrics, which naturally relates the Yamabe constant to Perelman's shrinker entropy. Second, I will discuss a Liouville-type theorem which illustrates the usefulness of studying the limit $m\to\infty$ as a way to overcome difficulties in the $m=\infty$ comparison theory. Third, I will discuss a compactness theorem for compact quasi-Einstein metrics analogous to Anderson's theorem for Einstein metrics, which is independent of the parameter $m$.