Research Area DE: ‘Analytic, algebraic, and combinatorial aspects of moduli theory’

The Research Area DE started in November, 2012.

The former Research Areas D and E have joined forces. Certain topics like Shimura varieties have been central to both areas, but were studied from different angles, and so we decided to bring automorphic forms and moduli theory closer together. Topological and differential-geometric aspects of moduli spaces are less in the focus of the new research area.

The theory of automorphic forms is a central area of mathematics with a rich history and deep connections to number theory, harmonic analysis and algebraic geometry. Recently p-adic versions have gained much attention in connection with the deformation theory of Galois representations. The interplay of arithmetic and analytic information of automorphic forms is one of the most important unifying ideas of modern mathematics. Moduli spaces of varieties, sheaves, representations, etc., form themselves varieties or stacks, rigid analytic varieties, Berkovich spaces, depending on the situation. Their study requires various techniques, including modern combinatorial methods and homological algebra like cluster algebras and triangulated categories. Currently, one of the important trends in this broad area is the study of numerical and motivic invariants using moduli spaces, vast generalizations of Donaldson and Gromov-Witten invariants. The numerical information is often encoded by modular forms and the combinatorics is described in terms of wall crossing formulas and cluster algebras.

To learn more, read a detailed description of the Research Area's goals and its predecessors' achievements.

Leaders of the Research Area