Bertrand's Theorem tells us that the only classical central forces with the property that all bounded orbits are periodic are the Kepler and the elastic force. What happens for relativistic forces? The orbits of the Kepler force are no longer all periodic, a fact that was used to confirm Einstein's general relativity by measuring Mercury's perihelion precession.
In this project we study relativistic examples with the property that all orbits are periodic.
Let K be the unit ball of a norm in R^n and let K^* be the ball of the dual norm in R^n. Mahler's conjecture gives a sharp lower bound for the product vol(K)vol(K^*) of the volume of the ball and of its dual. The conjecture was proved by Mahler for n=2 and, recently, by Iriyeh and Shibata for n=3. In general, the conjecture is open and has attracted a lot of attention among symplectic geometers since it is implied by a conjecture of Viterbo about symplectic capacities.
In this project we study the relationship between Mahler's conjecture for n=2 and the symplectic ball capacity of the product KxK^*. In other words, we ask: what is the size of the largest symplectic ball that can be embedded in KxK^*?
Samanyu Sanjay, The local systolic inequality for odd-symplectic forms. Formal supervisor: Umberto Hryniewicz, RWTH Aachen. October 2021 - present.
Valerio Assenza, Magnetic curvature and existence of closed magnetic geodesics on low energy levels. Formal supervisor: Peter Albers, Heidelberg University. September 2019 - December 2023.
Johanna Bimmermann, On the Hofer-Zehnder capacity of twisted tangent bundles. Formal supervisor: Peter Albers, Heidelberg University. January 2020 - April 2023.
Coming soon...
Coming soon...