**October 19, 2020 at 10:00 - October 19, 2020**
- BCAM

**Jean-Pierre Françoise (Laboratoire Jacques-Louis Lions,
,Sorbonne-Université Paris, France)
,**

**DATES:** 19-23 October (5 sessions)

**TIME:** 10:00 - 12:00 (a total of 10 hours)

**LOCATION:** BCAM Seminar room

**ABSTRACT:**
This mini-course proposes an introduction to information Geometry and a special focus on its use introduced by Nakamura to study the Hierarchy of integrable gradient systems of Jacobi-Toda. This Hierarchy is equivalent to an averaged learning equation.

**PROGRAMME:**
1. Information Geometry:

Connections, their curvature and torsion, flat connections, Levi-Civita connection associated to a Riemannian metric, Dualistic structure and Hessian manifolds, Gradi- ent Flows of dual connections in a Hessian manifold, Examples: The set of positive- definite symmetric matrices and the manifolds of probability distributions;

2. The Toda system:

Toda Lattice in Flaschka’s form, Moser’s system, equivalence with an averaged learn- ing equation, equivalence with a linear flow on the set of rational functions, Hankel matrices;

3. Moment problem and associated Stieljes measure:

The Hamburger moment problem, the Jacobi-Toda Hierarchy, associated Lax repre- sentation of double bracket form, completely integrable gradient flows;

4. The tau-function of the Hierarchy:

Nakamura’s construction of the tau-function, positivity of the tau-function and ex- istence of the Stieljes measure, N-solitons of the Hierarchy;

5. The tau-function and the Information Geometry:

Hankel matrices and ∇-affine and ∇∗-affine coordinates, dualistic potentials and the tau-function.

**REFERENCES: **
[1] S. I. Amari, Information Geometry and its Applications Applied Mathematical Sciences, vol. 194, Springer, 2016.

[2] N. Ay, J. Jost, H. V. Le, Hong Van and L. Schwachhofer, Information geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. A Series of Modern Surveys in Mathematics 64, Springer, 2017.

[3] A. M. Bloch, R. W. Brocket and T. S. Ratiu, Completely Integrable Gradient Flows, Commun. Math. Phys. 147: 57–74, 1992.

[4] Nakamura, Yoshimasa, A tau-function for the finite Toda molecule, and information spaces, Symplectic geometry and quantization (Sanda and Yokohama, 1993), 205–211, Contemp. Math. 179, Amer. Math. Soc., Providence, RI, 1994.

[5] H. Shima The Geometry of Hessian Structures, World Scientific Publishing, Singapore, 2007.

*Registration is free, but mandatory before 16th October. To sign-up go to

https://forms.gle/MkXCmKMWo5gSqnEK8 and fill the registration form. Student grants are available. If you need support for travel and accommodation expenses, please, let us know in the form before September 14th 2020.