In 1934, Wilhelm Blaschke’s attention focused on a recent construction in metric geometry proposed by Dan Barbilian as a generalization of various models of hyperbolic geometry. It was the year when S.-S. Chern started his doctoral program under Blaschke’s supervision in Hamburg and when in several academic centers in Europe scholars were interested in generalizations of Riemannian geometry. Introduced originally in 1934, Barbilian’s metrization procedure induces a distance on a planar domain through a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to more general domains. In our discussion we plan to show that these spaces are naturally related to Gromov hyperbolic spaces. In several works written with W.G. Boskoff, we explore this connection. We conclude our talk by stating several open problems related to this content.

The Nash embedding theorem states that every Riemannian
manifold can be isometrically embedded into some Euclidean space with
dimension bound. Isometric means preserving the length of every
path. Nash's proof involves sophisticated perturbations of the
initial embedding, so not much is known about the geometry of the
resulted embedding.
     In this talk, using the eigenfunctions of the Laplacian
operator, we construct canonical isometric embeddings of compact
Riemannian manifolds into Euclidean spaces, and study the geometry of
embedded images. They turn out to have large mean curvature
(intuitively, very bumpy), but the extent of oscillation is about the
same at every point. More can be said about global quantities like
the center of mass. This is a joint work with Xiaowei Wang.

We will review some recent work on free boundary minimal
hypersurfaces. In particular, we will explain a geometric classification
of the critical catenoid (joint with Ivaldo Nunes) and discuss what
information about such hypersurfaces in a general ambient manifold one can
extract from the knowledge of their Morse index (joint with Alessandro
Carlotto and Ben Sharp).

In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α- energy. More specifically, we show that for the SU(2) Hopf fibration over S4, for sufficiently small α values the SO(5, 1)-invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold. 

In 2012, Tseng and Yau introduced several Laplacians on symplectic manifolds that are related to a system of supersymmetric equations from physics.  In this talk, we will discuss these "symplectic Laplacians" and their relations with cohomologies on compact symplectic manifolds with boundary.  For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds.  Their properties and importance will be discussed.