The well-known Yau’s uniformization conjecture states that any
complete noncompact Kaehler manifold with positive bisectional curvature is
bi-holomorphic to the complex Euclidean space. The conjecture for the case
of maximal volume growth has been recently confirmed by G. Liu. In this
talk, we will consider the conjecture for manifolds with non-maximal volume
growth. We will show that the finiteness of the first Chern number is an
essential condition to solve Yau’s conjecture by using algebraic embedding
method. Furthermore, we can verify the finiteness in the case of minimal
volume growth. In particular, we obtain a partial answer to Yau’s
uniformization conjecture on complex two-dimensional Kaehler manifolds with
minimal volume growth. This is a joint work with Bing-Long Chen.

Entropy has been an important topic in the study of Ricci flow
ever since it was invented by Perelman. We consider Perelman's entropy
defined on an ancient solution, and prove a gap theorem for its backward
limit: If Perelman's entropy limits to a number too close to zero as time
approaches negative infinity, then the ancient solution must be the trivial
Euclidean space.

A deep result of X.-J. Wang states that a convex ancient solution to mean curvature flow either sweeps out all of space or lies in a stationary slab (the region between two fixed parallel hyperplanes). We will describe recent results on the construction and classification of convex ancient solutions and convex translating solutions to mean curvature flow which lie in slab regions, highlighting the connection between the two. Work is joint with Theodora Bourni and Giuseppe Tinaglia.

 I will discuss some recent work with David
Wiygul to determine the index and nullity of the
Lawson surfaces desingularizing two orthogonal
great two-spheres in the round three-sphere.
 

Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Instead of using metric for measuring distances on such manifolds, these applications often use “divergence functions” for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance Kullback-Leibler divergence, Bregman divergence, f-divergence, etc. Divergence functions are tied to generalized entropy (for instance, Tsallis entropy, Renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. It turns out that divergence functions enjoy pleasant geometric properties – they induce what is called “statistical structure” on a manifold M: a Riemannian metric g together with a pair of torsion-free affine connections D, D*, such that D and D* are both Codazzi coupled to g while being conjugate to each other. Divergence functions also induce a natural symplectic structure on the product manifold MxM for which M with statistical structure is a Lagrange submanifold.  We recently characterize holomorphicity of D, D* in the (para-)Hermitian setting, and show that statistical structures (with torsion-free D, D*) can be enhanced to Kahler or para-Kahler manifolds. The surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages.