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. For example, one categorifies sheaves and turns them into stacks obtaining a softer object which is far more amenable to calculations. I am especially interested in 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. For 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).