Event Details

Two short talks about open research problems, given by staff and postgraduate students, accessible to undergraduate students.

Speaker 1: Paul Levy (senior lecturer)

Title: Classifying symplectic singularities, or getting the Lie of the land

Abstract: Mathematics abounds with classification problems. Perhaps the most ancient is the classification of the regular polyhedra, otherwise called the Platonic solids: there are five of them, with 4, 6, 8, 12 and 20 sides. The first known proof of this classification dates to 360 BC.

In more modern times, a famous classification problem was that of finite simple groups. Almost all of the finite simple groups are of Lie (pronounced "lee") type - very roughly speaking, this means that they are simple groups of matrices (over finite fields, whatever that means). They are organised into families: labels A_n, B_n, C_n, D_n for the classical ones, and E_6, E_7, E_8, F_4, G_2 for the exceptional groups. These labels are encapsulated in certain graphs called Dynkin diagrams. It turns out that the Platonic solids can be related to the Dynkin diagrams E_6, E_7 and E_8!

A current area of research concerns certain geometric structures called symplectic singularities. The word "singularity" means that the structure is not smooth, i.e. it has some jagged edges or points; "symplectic" indicates that there is a uniform measure of area. Symplectic singularities have not yet been classified, but some important subfamilies are known to correspond to the Dynkin diagrams. In particular, this is true for symplectic singularities of dimension 2 (the smallest possible). The next dimension up is 4, and the latest research (including some results by my PhD student Callum Berry) show most, but not all, symplectic singularities are "of Lie type".

Speaker 2: Elliot Gathercole (PhD student)

Title: Symplectic Geometry, Planetary Orbits and Floer Theory

Abstract:

Symplectic geometry can be thought of as the geometry of phase spaces, which describe the possible states of a system in classical physics, and how they can evolve over time. A (T-)periodic orbit is a state which returns to itself at time T. The problem of counting periodic orbits has inspired the development of Floer theory, which can be used to show that certain counts of periodic orbits are invariant under perturbations of the system.

This talk will introduce these ideas via the example of the 2 (and 3)-body problems, which arise, for instance, from the motion of a satellite under the gravitational influence of a planet (and its moon). We will see how these problems can be described in the mathematical language of symplectic geometry, and discuss what Floer theory can tell us about these problems, including an application to space mission design.