Research
I am mainly interested in derived homotopical geometry. The key idea here is to apply methods of homotopy theory to categorify various classical geometric structures.
Example: a higher categorical version of a sheaf is a stack. Stacks are softer objects more amenable to calculations, that's why we are interested in them. I am especially focused on the analytic context for homotopical geometry, i.e. geometry over Fermat theories. This is the subject of my paper (arXiv:2303.12699).
Another area of interest for me is toric geometry and the beautiful combinatorics that it entails. As it turns out, torus actions allow describing very complicated objects of algebraic geometry in elegant combinatorial terms.
Example: Methods of toric geometry can be used to study automorphism groups of highly non-algebraic complex manifolds that nevertheless admit a kind of "transverse" algebraic structure. This approach is initiated in my paper (arXiv:2403.02465)
Example: the infinite-dimensional geometry of the automorphism group of the affine plane can be studied by means of simple combinatorics of the two-dimensional lattice. This was done in Alisa Chistopolskaya's and mine paper (arXiv:2202.02214).