Derived differential geometry

The main result of the paper is an equivalence of models for analytic geometry over a Fermat theory in characteristic zero. We also prove a conjecture of Behrend-Liao-Xu, which characterizes weak equivalences between derived manifolds. It also serves as our master's thesis at HSE. The advisor for this project is prof. Dmitri Pavlov of the Texas Tech University.

Toric geometry and related topics

The paper describes automorphisms of a Cox construction associated with a generalized fan equivariant with respect to the action of a "large enough" subgroup of the large torus acting on the Cox construction. The idea is that generalized fans (unlike ordinary fans of toric geometry) admit projections along arbitrary linear maps. This allows one to give a purely combinatorial description of the equivariant automorphism group in terms of the Demazure roots associated with the generalized fan obtained via a projection corresponding to the given subgroup of the torus. This paper is the first in a two-part series of papers dedicated to the study of automorphism groups of moment-angle manifolds and other complex manifolds admitting large abelian symmetry groups.

This is a joint work with Alisa Chistopolskaya. We prove a combinatorial criterion for infinite transitivity for automorphism groups of the affine plane generated by collections of root subgroups. As it turns out the action is infinitely transitive iff the collection of integer vectors associated to these root subgroups spans the lattice.