In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings.[-]

I will discuss some projective differential geometric invariants of properly convex domains arising from affine dfferential geometry. Consider a properly convex domain $\Omega $ in $R^n\subset RP^n$, and the cone $C$ over $\Omega $ in $R^{n+1}$. Then Cheng-Yau have shown that there is a unique hyperbolic affine sphere which is contained in $C$ and asymptotic to the boundary $\partial C$. The hyperbolic affine sphere is invariant under special linear automorphisms of $C$ , and carries an invariant complete Riemannian metric of negative Ricci curvature, the Blaschke metric. The Blaschke metric descends to a projective-invariantmetric on $\Omega $.
I will also address the relationship between the Blaschke metric and Hilbert metric, which is recent and is due to Benoist-Hulin. At the end, I will discuss applications to the geometry of real projective structures on surfaces.[-]

In the first part of this talk we will show how classical tools of Riemannian geometry can be used in the setting of stratfied spaces in order to obtain a lower bound for the spectrum of the Laplacian, under an appropriate assumption of positive curvature. Such assumption involves the Ricci tensor on the regular set and the angle along the stratum of codimension 2. We then show that a rigidity result holds when the lower bound for the spectrum is attained. These results, restricted to compact smooth manifolds, give a well-known theorem by M. Obata and A. Lichnerowicz.
Finally, we will explain some consequences of the previous theorems on the existence of a conformal metric with constant scalar curvature on a stratified space.[-]

A complete Riemannian manifold is called asymptotically hyperbolic if its ends are modeled on neighborhoods of infinity in hyperbolic space. There is a notion of mass for this class of manifolds defined as a coordinate invariant computed in a fixed asymptotically hyperbolic end and measuring the leading order deviation of the geometry from the background hyperbolic metric in the end. Asymptotically hyperbolic manifolds arize naturally in mathematical general relativity, in particular, as slices of asymptotically Minkowski spacetimes, in which case the mass of the slice coincides with the Bondi mass of the spacetime. Having reviewed these and related concepts, we will discuss our proof of the positive mass theorem in the asymptotically hyperbolic setting, which relies on the original ideas of Schoen and Yau and involves a blow-up analysis of the so-called Jang equation, a geometric PDE of mean curvature type.[-]