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.

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.