A concept of central importance in mathematics is that of symmetry. One used to think of symmetry as a property of geometric shapes, but in the 19th century Evariste Galois extended the concept of symmetry to algebraic objects, and today his insights are completely fundamental to pure mathematics. The underlying goal of this proposal, which is situated between Algebra, Number Theory, and Topology, relying also on techniques from Probability Theory and Additive Combinatorics, is to study symmetries of arithmetic and geometric objects.
Number Theory is an ancient mathematical discipline with a rich history of over 2000 years, but also with spectacular developments in recent years. Some of the most impressive recent advances have happened in the area of Number Theory called Arithmetic Statistics: the groundbreaking contributions of Manjul Bhargava have been rewarded with a Fields Medal in 2014. The aim of Arithmetic Statistics is to understand the behaviour of arithmetic objects, such as (ray) class groups, in families. The birth of this area goes back to Gauss, who formulated some concrete conjectures concerning the behaviour of class groups of quadratic fields. It was given a huge boost in the 1980s, when Cohen and Lenstra proposed a general model that implied all the conjectures of Gauss, and more. Roughly speaking, they postulated that class groups of imaginary quadratic fields obey a probability distribution that assigns to a finite abelian group X a probability that is inverse proportional to the number of symmetries of X. This is, in fact, a very natural model for random algebraic objects. This was later generalised to other number fields by Cohen and Martinet, but in more general cases the probability distributions looked more mysterious. The CohenLenstraMartinet Heuristics have been used as a guiding principle in Arithmetic Statistics since then, and have found applications in many other areas, such as the theory of Elliptic Curves, in Combinatorics, and in Differential Geometry. This project will consist of a blend of theorems, conjectures, and computations. I will:
 show that the original conjectures are false, as stated,
 find the correct formulations,
 put them on a more conceptual footing, by explaining the mysterious looking probability weights of CohenMartinet using a theory of commensurability of algebraic objects that I have been developing together with Hendrik Lenstra,
 extend the scope of the heuristics, e.g. to ray class groups.
Two other kinds of very basic objects whose symmetries one studies are finite sets and finite dimensional vector spaces. An old problem in Representation Theory, with applications to Number Theory and Differential Geometry, is to compare symmetries of sets with symmetries of vector spaces, and in particular to determine which symmetries of sets become isomorphic (essentially the same) when the sets are turned into vector spaces. There are two incarnations of this problem: one where the vector spaces are over a field of characteristic 0, e.g. over the real numbers, and one where they are vector spaces over a field of positive characteristic. In previous joint work with Tim Dokchitser we have solved the case of characteristic 0, thereby settling an over 60 year old problem. Using the techniques that we developed, and new ones, this project will settle the case of positive characteristic.
Finally, I will also investigate symmetries of lowdimensional manifolds. These are the basic objects studied by modern geometry and topology, and it is an old and fruitful line of investigation to determine what one can say about the topology of the manifold from knowing its symmetries. In recent joint work with Aurel Page, I have introduced a new representation theoretic tool into the area, which I had worked on in number theoretic contexts. Using these new techniques, I am planning to shed more light on the connection between symmetries and the topology of the manifold.
