Jaclyn Lang
Explicit Class Field Theory
Class field theory was a major achievement in number theory
about a century ago that presaged many deep connections in mathematics
that today are known as the Langlands Program. Class field theory
associates to each number field an special extension field, called the
Hilbert class field, whose ring of integers satisfies unique
factorization, mimicking the arithmetic in the usual integers. While
the existence of this field is always guaranteed, it is a difficult
problem to find explicit generators for the Hilbert class field in
general. The theory of complex multiplication of elliptic curves is
essentially the only setting where there is an explicit version of class
field theory. We will briefly introduce class field theory, highlight
what is known in the theory of complex multiplication, and end with an
example for the field given by a fifth root of 19. There will be many
examples!
Jan Sbierski
The strength of singularities in general relativity
One of the many curious features of Einstein’s theory of general relativity is that the theory predicts its own breakdown at so-called gravitational singularities. The gravitational field in general relativity is modelled by a Lorentzian manifold — and thus a gravitational singularity is signalled by the geometry of the Lorentzian manifold becoming singular. In this talk I will first review the classical definition of a gravitational singularity along with a classification of their strengths. I will conclude with outlining newly developed techniques which capture the singularity at the level of the connection of Lorentzian manifolds.