Professor Goncharov continues his study of the arithmetic aspects of polylogarithms and their generalizations, such as multiple polylogarithms and quantum polylogarithms, special values of L-functions of algebraic varieties, motivic fundamental groups, moduli spaces and higher quantum Teichmuller theory. Professor Goncharov investigates the structure of the motivic fundamental group of the projective line punctured at zero, infinity and all N-th roots of unity and its surprising relationship with the geometry and topology of modular varieties and mathematical physics. The arithmetic, side of the problem concerns the action of the absolute Galois group on the pro-l completion of the fundamental group of the projective line punctured as above. The analytic aspect of the story concerns the properties of multiple zeta values and their generalizations, multiple polylogarithms evaluated at N-th roots of unity. The relationship with the geometry of modular varieties as well as with mathematical physics are new tools to study this problem. Professor Goncharov investigates the higher quantum Teichmuller theory, which studies some new moduli spaces of G-local systems on a surface S, where G is a split reductive group, and its non-commutative deformations. This theory unites many different aspects of the representation theory which appear when S is simple but the group is general, and the classical Teichmuller theory corresponding to the case when G is the simplest possible, that is the group of two by two matrices, while S is general. The quantisation of these moduli spaces is governed by the motivic and quantum dilogarithms, and thus provides an example of fruitful relationship between mixed motives and mathematical physics.
This research is in the area of arithmetic algebraic geometry, which is the branch of mathematics that is concerned with questions about the integers, but approaches them using the ideas coming from investigation of geometric shapes. It is a modern version of number theory, which is a very old subject, but is full of difficult problems and significant conjectures. The theory of systems of polynomial equations with integer coefficients is important for many applications including questions in cryptography and coding theory. Just recently, ideas from physics started to influence the subject. The proposer will use the latest techniques in number theory, algebraic geometry and mathematical physics to study the L-functions and their special values at integer points.
