Bridgeland stability structure/condition on a triangulated category is a vast generalization of the notion of an ample line bunlde (or polarization) in algebraic geometry. The origin of the notion lies in string theory, and is applicable to derived categories of coherent sheaves, quiver representations and Fukaya categories. In a category with Bridgeland stability every objects carries a canonical filtration with semi-stable pieces, an analog of Harder-Narasimhan filtration.

It is expected that for categories over complex numbers Bridgeland stability structures often admit analytic enhancements, similar to the relation between ample bundles and usual Kaehler metrics. In a sense, this should be a generalization Donaldson-Uhlenbeck-Yau theorem which syas that a holomorphic vector bundle over compact Kaehler manifold is polystable if and only if it admits a metrization satisfying hermitean Yang-Mills equation.

In my course I will talk about a non-archimedean analog of analytic Bridgeland stability. I will show several examples, results and conjectures. In particular, I’ll introduce non-archimedean moment map equations, generalized honeycomb diagrams, and hypothetical stability on derived categories of coherent sheaves on maximally degenerating varieties over non-archimedean fields.

Paraphrasing Alexander Polyakov, « Conformal Field Theory is a way to learn about elementary particles by studying boiling water ».
There is a technical statement behind this joke: Euclidean Conformal Field Theory, under certain conditions, can be rotated to the Lorentzian signature, and vice versa. This means that even statistical physicists studying finite-temperature phase transitions on a lattice should learn about the Minkowski space! The goal of this course will be to explain various classical and recent results pertaining to this somewhat surprising conclusion.

Plan of the course:
– elementary introduction to Euclidean CFT in d>2 dimensions
– the Osterwalder-Schrader theorem about the Wick rotation of general reflection-positive Euclidean Quantum Field Theories, and its limitations
– the Luescher-Mack theorem about continuation of CFT correlation functions to the Lorentzian cylinder, and its limitations
– recent results about the analytic structure of Lorentzian CFT correlators