Georgia Institute of Technology College of Sciences

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.

A CAT(0) space is a geodesic metric space where triangles are thinner than comparison triangles in a Euclidean plane. Prime examples of CAT(0) spaces are Cartan-Hadamard manifolds: complete simply connected Riemannian spaces with nonpositive curvature, which include Euclidean and Hyperbolic space as special cases. The triangle condition ensures that every pair of points in a CAT(0) space can be connected by a unique geodesic. A subset of a CAT(0) space is convex if it contains the geodesic connecting every pair of its points. We will give a quick survey of classical results in differential geometry on characterization of convex sets, such the theorems of Hadamard and  of Chern-Lashof, and also cover other background from the theory of CAT(0) spaces and Alexandrov geometry, including the rigidity theorem of Greene-Wu-Gromov, which will lead to the new results in the second talk.
 

Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex with a latent feature vector sampled from a mixture of Gaussians, and we add edge if and only if the feature vectors are sufficiently similar. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features---for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component).

In this talk, we focus on clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors goes to infinity as the size of the network goes to infinity. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures and begin to sketch out the information-computation landscape for clustering and embedding in these models.

This is based on joint work with Tselil Schramm.

It is known for many years that various inequalities in convex geometry have information-theoretic analogues. The most well known example is the Entropy power inequality which corresponds to the Brunn-Minkowski inequality, but the theory of optimal transport allows to prove even better analogues. 

At the same time, in recent years there is a lot of interest in the role of symmetry in Brunn-Minkowski type inequalities. There are many open conjectures in this direction, but also a few proven theorems such as the Gaussian Dimensional Brunn-Minkowski inequality. In this talk we will discuss the natural question — do the known information-theoretic inequalities similarly improve in the presence of symmetry?  I will present some cases where the answer is positive together with some open problems. 

Based on joint work with Gautam Aishwarya.