Mathematical Foundations for Origins

Currently there are two major mathematical challenges in theoretical
physics: a) making sense of elementary particles, and b) making
sense of cosmology. We outline below the mathematical issues that we
plan to address in these two subject areas.

In modern theoretical physics the mathematical description of
elementary particles is provided by two theories: Quantum
Electrodynamics which describes the most basic elementary particles
(photons, electrons, neutrinos and corresponding antiparticles) and
the Standard Model which describes the legion of other particles.
Both of these theories are unsatisfactory from the mathematical
point of view. The issue with Quantum Electrodynamics is that it is
a perturbative theory, i.e. it gives a formal asymptotic expansion
with respect to a small parameter (fine structure constant) but does
not reveal the partial differential equation (?) containing a small
parameter which generates this expansion. With the Standard Model
the additional feature is its shear complexity (ugliness?): a
mathematician can be forgiven for thinking that the Standard Model
is an empirical construction in which agreement with experiment is
achieved by the appropriate choice of sufficiently many physical
parameters (the number currently stands at 29). There does not
appear to be a coherent underlying mathematical structure. Moreover,
the Large Hadron Collider will in 2008 reach an energy regime above
the electroweak symmetry-breaking scale. This means the Standard
Model can no longer be treated as a low energy effective theory
without the Higgs mechanism, and thus its mathematical form will be
challenged in a way unprecedented in the last two decades.

With Cosmology the situation is, in a sense, better in that it has a
solid mathematical basis in the form of Einstein't equation.
However, modern experimental data indicates that the expansion of
the universe is accelerating and explaining this acceleration turns
out to be a serious mathematical problem. It has been suggested that
agreement between experiment and theory could be achieved by the
introduction of "dark energy" and "dark matter". These are hot
topics in modern theoretical physics and the analysis of the
corresponding mathematical models has hardly begun.

The subjects of Particle Physics and Cosmology deal with physical
phenomena at opposite scales - microscopic scale and scale of the
whole universe - but they have one major issue in common which
provides a natural link: for some reason neutrinos play an important
role in both. Neutrinos have traditionally been of great interest to
mathematicians and pursuing their mathematical study would be a good
starting point for the mathematical foundations for Origins.