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.

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