Current Trends in Mathematical Cosmology

TL;DR

This paper provides an overview of current challenges and unsolved problems in mathematical cosmology, emphasizing singularities, solution structures, and the validity of cosmological models.

Contribution

It introduces the fundamental concept of 'a cosmology' and highlights key unresolved issues, especially concerning singularities and solution definitions.

Findings

Identification of key unsolved problems in cosmology

Discussion on the nature and role of singularities

Analysis of the limitations of current cosmological models

Abstract

We present an elementary account of mathematical cosmology through a series of important unsolved problems. We introduce the fundamental notion of `a cosmology' and focus on the issue of singularities as a theme unifying many current, seemingly unrelated trends of this subject. We discuss problems associated with the definition and asymptotic structure of the notion of cosmological solution and also problems related to the qualification of approximations and to the ranges of validity of given cosmologies.