First steps into the world of systolic inequalities: From Riemannian to symplectic geometry

TL;DR

This paper introduces systolic inequalities, exploring their connections between Riemannian and symplectic geometry, and discusses recent developments like the Viterbo conjecture to provide a comprehensive overview for students.

Contribution

It offers an accessible introduction to systolic inequalities, highlighting their relationships across Riemannian and symplectic geometries, and discusses recent advances such as the Viterbo conjecture.

Findings

Explains classical systolic inequalities in Riemannian geometry.

Connects systolic inequalities to symplectic measurements.

Introduces the Viterbo conjecture and its significance.

Abstract

Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and the Viterbo conjecture. This will give us a perfect excuse to introduce the reader to some important ideas in Riemannian and symplectic geometry.