Titles and Abstracts

All talks and sessions take place in Ken Edwards Lecture Theatre 3. A timetable is here. Where exist, slides of the talks can be downloaded by clicking on the titles.

Minhyong Kim (University of Oxford, UK) Diophantine geometry and principal bundles

Abstract: In the 17th century, Pierre de Fermat initiated the modern theory of Diophantine equations. He also stated the first least-action principle to explain the trajectory of light. Even though Fermat was quite isolated in his arithmetic interest at the time, over the next few hundred years, Diophantine equations became firmly established as a source of vanguard problems in the development of algebra, analysis, and geometry. As number theory itself became mainstream, Diophantine problems called forth the tools of algebraic number theory, class field theory, the Langlands programme, and arithmetic geometry. The principle of least action, in the meanwhile, became the key theoretical foundation of gauge theory, which in turn underlies quantum field theory and its applications to geometry and topology. In this lecture, we will describe some recent attempts to unify Fermat's two directions of investigation, whereby classical solutions to arithmetic gauge theory leads to solutions of Diophantine equations.

Leila Schneps (Institut de Mathématiques de Jussieu Paris, France) Pro-unipotent Grothendieck-Teichmüller theory: surprising connections with number theory

Abstract: Grothendieck-Teichmüller theory was originated by Alexander Grothendieck as a way to study theabsolute Galois group of the rationals by considering its action on fundamental groups of varieties,in particular of moduli spaces of curves with marked points: the special properties of the Galois action with respect to inertia generators and the fact of respecting the relations in the fundamental group gave rise to the definition of the group GT which contains G_Q. The group GT is profinite, but its defining relations can also be used to give a pro-unipotent avatar, and an associated graded Lie algebra grt. The study of the Lie algebra grt reveals many unexpected relations with number theory that are completely invisible in the profinite situation. We will show how Bernoulli numbers, cusp forms on SL_2(Z) and multiple zeta values arise in the Lie algebra context.

Fabrizio Catanese (University of Bayreuth, Germany) Mathematical Mysteries behind the Interplay of Algebra and Topology in Moduli Theory

Abstract: Complex algebraic curves are Riemann surfaces: the interplay of algebra and topology goes back to this well known truth. The famous Riemann existence theorem shows in an indirect way the existence of certain polynomials, or algebraic functions satisfying certain very special properties. The topological approach, via Riemann’s existence theorem, is particularly useful when analyzing the moduli spaces of curves with symmetries. In turn, the study of curves with certain groups G of automorphisms, is crucial for the construction of higher dimensional varieties, and for the description of their moduli spaces. I shall very briefly illustrate the two different approaches to the construction of moduli spaces, one via topology, called Teichmüller theory, and the other through algebra, called GIT (Geometric Invariant Theory). Each approach has its own advantages, and disadvantages. GIT allows to define an action of the Absolute Galois Group (group of Automorphisms of the algebraic numbers) on moduli spaces, which is particularly interesting in the case of projective classifying spaces (projective varieties whose universal cover is contractible). An interesting question, pioneered by a famous theorem of Kazhdan, is whether a Galois conjugate of a projective classifying space is again one such. I shall show how the question is open, even if the fundamental group or the complex structure of the universal cover may change.

In the talk, after illustrating basic examples of projective classifying spaces, and by now classical constructions (Hirzebruch-Kummer covers associated to line configurations), I shall review several results, some obtained jointly with Ingrid Bauer, Michael Lönne and Fabio Perron, some due to other authors. Finally, I shall concentrate on quite recent results and work in progress concerning rigid complex manifolds and projective classifying spaces, especially in the crucial complex dimension 2.