Algebraic geometry is a key area of mathematical research of international significance. It has strong connections with many other areas of mathematics (differential geometry, topology, number theory, representation theory, etc.) and also with other disciplines (in the present context, particularly theoretical physics). Moduli theory is the study of the way in which objects in algebraic geometry (or in other areas of mathematics) vary in families and is fundamental to an understanding of the objects themselves. The theory goes back at least to Riemann in the mid-nineteenth century, but moduli spaces were first rigorously constructed in the 1960s by Mumford and others. The theory has continued to develop since then, perhaps most notably with the infusion of ideas from physics after 1980.
The programme will focus on the following topics:
- Moduli of bundles and augmented bundles on algebraic curves: more specifically Higgs bundles, parabolic bundles, coherent systems, principal bundles.
- Relationship of moduli spaces to topology, Teichmüller theory and hyperbolic geometry - this relationship takes place in the study of representations of a surface group in a real Lie group and involves Higgs bundle theory, bounded cohomology, Anosov systems, cluster varieties, tropical algebraic geometry; there is a very rich geometric structure and the various points of view are complementary, the relationship between them yet to be understood.
- Moduli of algebraic varieties - recent work has made exciting connections with the subject of special metrics in Kähler geometry. This has led to the old unsolved question of giving an intrinsic criterion for the stability of algebraic varieties being revisited.
- Moduli in derived categories - the study of moduli spaces in derived categories is a new and highly promising area of research, involving in particular new notions of stability.
We should emphasise that the topics are not independent. There are obvious links between (i) and (ii), while (iv) has already had an impact on all the other topics and we believe that this impact is likely to get stronger.
The central aims of the programme are to bring together experts in various aspects of moduli theory and related areas, to advance these topics, and to introduce research students and post-docs to the welath of ideas and problems in them. As stated above, the interdependence of the topics we have identified is crucial to the development of the theory, and a major goal is to develop these ideas further. The programme will include an instructional course and three further workshops, one of which will be held outside Cambridge.