Geometry and Physics of Higgs Bundles

This workshop focused on interactions between the various perspectives on the moduli space of Higgs bundles over a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and mathematical physics, and the goal was to promote interactions between these various branches of the subject. The main current directions of research were well represented by the participants, and the talks included many from both senior and junior participants

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on $\mathbb{P}^1.$ In particular, we derive both formulas from their common generalization.Comment: 49 pages, 7 figure

Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem

We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.Comment: 62 page

Teichmüller Space (Classical and Quantum)

This is a short report on the conference “Teichmüller Space (Classical and Quantum) ” held in Oberwolfach from May 28th to June 3rd, 2006

Stability and Arithmetic

Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using semi-stable parabolic bundles & for p-adic number fields using what we call semi-stable filtered (phi,N;omega)-modules; and non-abelian zeta functions for function fields over finite fields using semi-stable bundles & for number fields using semi-stable lattices.Comment: 121 page