# Members Seminar

## Tracking trajectories in Hamiltonian systems using holomorphic curve tools.

## Geometry of 2-dimensional Riemannian disks and spheres.

I will discuss some geometric inequalities that hold on

Riemannian 2-disks and 2-spheres.

For example, I will prove that on any Riemannian 2-sphere there M exist

at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich).

This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.

## Positive geometries

Positive geometries are real semialgebraic sets inside complex varieties characterized by the existence of a meromorphic top-form called the canonical form. The defining property of positive geometries and their canonical forms is that the residue structure of the canonical form matches the boundary structure of the positive geometry. A key example of a positive geometry is a projective polytope.