de Rham theory in derived differential geometry (arXiv:2505.03978)
The main question this text addresses is: What is the correct notion of de Rham cohomology for derived manifolds? We identify two natural answers to this question: one in terms of the cotangent complex and one in terms of the so-called C-infinity de Rham stack. It turns out that while the first version is very algebraically nice, it might not satisfy the Poincaré lemma for general derived manifolds. On the other hand, the second version is very hard to compute. Still, it gives a kind of de Rham isomorphism with the cohomology of the constant sheaf on the underlying topological space of a derived manifold. We also provide sufficient conditions for when the de Rham cohomology calculated using the cotangent complex is isomorphic to the constant sheaf cohomology.
Equivalent models of derived stacks (arXiv:2303.12699)
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.
Equivariant automorphisms of the Cox construction and applications (arXiv:2403.02465)
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.
Infinite transitivity for automorphism groups of the affine plane (arXiv:2202.02214)
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.