First order systems of PDEs on manifolds without boundary: understanding neutrinos and photons

Dmitri Vassiliev (University College London)

In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino or a photon. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) or the electromagnetic field (solution of the Maxwell system) propagate as a function of time? What are the eigenvalues (energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive eigenvalues) and the antineutrino (negative eigenvalues)? Why is the photon its own antiparticle? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons so different and, yet, so similar?

The speaker will approach the study of first order systems of partial differential equations from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.