One-Day Workshop on Interactions between Mathematics and Physics

Time: 2018-9-18:

9:00-10:00 Urs Frauenfelder (I)

10:00-10:30 tea break

10:30-11:30 Thierry Levy

14:00-15:00 Robert Nicholls

15:00-15:30 tea break

15:30-16:30 Urs Frauenfelder (II)

Venue: Lecture Room 613, Math. Building at CNU

Urs Frauenfelder

Augsburg University

Title: Introduction to Scale smoothness and Polyfolds

Abstract: Reparametrization actions are continuous but not smooth. This led Hofer, Wysocki and Zehnder to the discovery of a new notion of smoothness in infinite dimensions referred to as scale smoothness for which reparametrization actions are smooth. As in the case of usual smoothness the chain rule holds for scale smoothness. On the other while due to Cartan's last theorem the fixed point set of a smooth retraction is a manifold this is not true for scale smoothness which leads to new spaces - M-polyfolds which are locally fixed point sets of scale smooth retractions. Unparametrized broken trajectories can be interpreted as M-polyfolds. This enables one to interpret some compactified moduli spaces of elliptic PDE's, appearing in Floer homology, Gromov-Witten theory and Symplectic Field Theory, as the zero set of a Fredholm section from a polyfold to a polyfold bundle. I plan to explain the basics of this theory and also indicate further directions concerned with Floer homology for delay equations whose gradient flow equations can be interpreted as a scale ODE on a scale manifold.

Thierry Levy

University Paris 6

Title: Stochastic quantization and stochastic partial differential equations

Abstract: One of the mathematical challenges raised by quantum field theory is to be able to construct certain measures on certain infinite dimensional spaces, and to be able to compute integrals with respect to these measures. This is a very difficult problem, to the point that there is probably not a single physically relevant situation in which the construction of the required measures is understood mathematically.

In 1981, Giorgio Parisi and Yong Shi Wu (吴咏时) proposed that some of these measures should be constructed and studied as the equilibrium distributions of random evolutions. Constructing these random evolutions amounts to solving a class of equations called stochastic partial differential equations, and this is unfortunately (but not surprisingly) very difficult as well.

The recent work of Martin Hairer on regularity structures provided a major breakthrough in the theory of stochastic partial differential equations by producing theorems of existence of solutions for a wide class of these equations - and earned him the Fields medal.

In this talk, I will describe in very elementary terms some of the ideas that underly Parisis and Wu's insight, and allude to the way in which Hairer's theory is now being applied, by him and people around him, to go a few steps further towards a mathematically rigourous theory of quantum fields.

Robert Nicholls

Augsburg University

Title: Syzygies for periodic orbits in the restricted three-body problem

Abstract: The existence of syzygies for all periodic orbits inside the bounded Hill’s region of the planer circular restricted three-body problem with energy below the second critical value. The proof will follow some ideas of Birkhoff to compute the roots of partial derivatives of the effective potential. Birkhoff’s methods are extended to higher energies and a new base case is created and shown to fulfil the requirements. Another step from Birkhoff is scrutinized to continue the statement to all mass ratios. The final step is achieved by integrating over periodic orbits. Applying the same methods to Hill’s lunar problem delivers similar results in that setting as well.