Elementary
  differential
  geometry
  Zhengchao
    Wan
 Introduction
 Overview            Elementary differential geometry
 Differentiable
 manifolds
 Tangent
 vectors and                     Zhengchao Wan
 tangent spaces
 Vector fields
 and tensor                       Peking University
 fields
 Connections
 Flatness
 Riemannian                       May 9, 2016
 connection
 Submanifolds
   Elementary
    differential
    geometry                                                        Information geometry
   Zhengchao
       Wan
  Introduction        Information geometry is a branch of mathematics that applies
  Overview            the techniques of differential geometry to the field of
  Differentiable       probability theory. This is done by taking probability
  manifolds
  Tangent             distributions for a statistical model as the points of a
  vectors and         Riemannian manifold, forming a statistical manifold. The
  tangent spaces
  Vector fields        Fisher information metric provides the Riemannian metric.
  and tensor
  fields
  Connections         Information geometry reached maturity through the work of
  Flatness            Shun’ichi Amari and other Japanese mathematicians in the
  Riemannian          1980s. Amari and Nagaoka’s book, Methods of Information
  connection
  Submanifolds        Geometry, is cited by most works of the relatively young field
                      due to its broad coverage of significant developments attained
                      using the methods of information geometry up to the year 2000.
   Elementary
    differential
    geometry                                                                            Applications
   Zhengchao
       Wan
  Introduction        Information geometry can be applied where parametrized
  Overview            distributions play a role. Here an incomplete list:
  Differentiable
  manifolds                • statistical inference
  Tangent                  • time series and linear systems
  vectors and
  tangent spaces           • quantum systems
  Vector fields
  and tensor               • neural networks
  fields
  Connections              • machine learning
  Flatness                 • statistical mechanics
  Riemannian
  connection               • biology
  Submanifolds             • statistics
                           • mathematical finance
   Elementary
    differential
    geometry                                                                                  Overview
   Zhengchao
       Wan
  Introduction
  Overview
  Differentiable           • Differentiable manifolds
  manifolds
  Tangent                 • Tangent vectors and tangent spaces
  vectors and
  tangent spaces          • Vector fields and tensor fields
  Vector fields
  and tensor              • Connections
  fields
  Connections             • Flatness
  Flatness                • Submanifolds
  Riemannian              •
  connection                  Riemannian connection
  Submanifolds