e-Geometry Day 2020

Andreas Savas-Halilaj

Curve shortening flow on singular Riemann surfaces

Abstract: We discuss curve shortening flow on Riemann surfaces with isolated singularities. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under natural geometric assumptions, we show short-time existence, uniqueness and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and prove some collapsing and convergence results. This work is joint with Nikolaos Roidos.

Konstantina Panagiotidou

An overview and recent results on real hypersurfaces in Hermitian symmetric spaces.

Abstract: In 1973 Takagi initiated the study of real hypersurfaces in complex projective spaces. He clas- sified real hypersurfaces in terms of their principal curvatures. The same problem in the case of the ambient space being the complex hyperbolic space was solved by Berndt. Many other geometers continue to study real hypersurfaces in non-flat complex space forms(complex projective and complex hyperbolic space) under other geometric conditions. Next real hypersurfaces have been studied in Hermitian symmetric spaces of rank 2, which are the complex two-plane Grassmannians, the complex hyperbolic two-plane Grassmannians, the complex quadric and the complex hyperbolic quadric. In this talk I will present results concerning real hypersurfaces in non-flat complex space forms and in complex quadric. The results are based on a joint work with G. Kaimakamis and J.de D. Perez.

Deepika

On the Fisher Information Matrix for Complex Gaussian Distributions

Abstract: The complex Gaussian distribution is a statistical model for complex random variables whose real and imaginary parts are normal distributed, which finds multiple applications in many fields of applied analysis, physics, and engineering. Complex Gaussian distributions are commonly parameterized by a complex mean vector, a Hermitian positive-definite covariance matrix and a symmetric complex relation matrix. In this paper we focus on the sub-family of complex Gaussian distributions characterized by zero relation matrix, which includes circularly symmetric complex random variables as special case, when the mean is set to zero. We provide explicit formulae for the entries of the Fisher matrix in three different parameterizations: the standard mean and covariance, and the natural and the expectation parameters of the exponential family, which are dual parameterizations of interest in information geometry. Also, we introduce conjugate parameters for complex Gaussians, and by using Wirtinger derivatives, we define a corresponding form for the Fisher information matrix based on conjugate scores. Finally we show that the well known formula by Miller for the Fisher information matrix for Gaussian distributions can be generalized to conjugate parameterizations. Compared to the real case, in the conjugate parameters the Fisher information matrix appears to have a more direct formulation.

Nikolaos Panagiotis Souris

An overview of geodesic orbit Riemannian manifolds

Abstract: A Riemannian manifold (M,g) is called geodesic orbit (g.o.) if all geodesics are orbits of one-parameter groups of isometries, or equivalently, integral curves of Killing vector fields. The simplest example of a g.o. manifold is the sphere, and more generally, the symmetric spaces. However, the full classification of g.o. manifolds is a longstanding open problem in Riemannian geometry. In this talk, we discuss several aspects of the classification problem as well as some partial classification results. The talk contains joint results with A. Arvanitoyeorgos and M. Statha.

Discussion