PhD Project in Mathematics Modular Invariants and Galois Algebras

The discovery of symmetry in nature is one of the most fundamental and universal intellectual achievements. The mathematical language for analyzing symmetries is the theory of groups and their invariants: objects or phenomena of interest and their properties are described mathematically in terms of solutions of systems of equations, involving numerical functions that depend on chosen coordinates. Changes of coordinates are then described by a suitable transformation group, acting on the system and its ingredients. Those functions which are unchanged by that group action, the `invariants', reveal the objective nature and the underlying symmetries of the studied phenomenon. It is therefore a major goal of invariant theory to provide general principles how to find all such invariants for a given group and how to perform efficient computations with them. Traditionally one looked at functions with real or complex coefficients, but more recent developments ask for invariants over more general coefficients, including modular fields. In that situation many of the “classical” results theory are unknown or known to be false, in which case one is looking for appropriate replacements. Key open questions include: How efficiently can such a ring be constructed? When is a modular ring of invariants a polynomial ring, or a Cohen-Macaulay ring? How do invariant rings behave under standard ring theoretical operations (e.g. localisation, completion etc.?)