My main research interests are in the area of **Algebraic Geometry.**

It has relations and overlaps with many other areas of Mathematics and Science, including:

- Combinatorics,
- Representation Theory,
- Differential Geometry,
- Computational Complexity,
- Commutative Algebra,
- Algebraic Topology...

- secant varieties,
- Hilbert schemes, multigraded Hilbert schemes
- ranks of tensors and symmetric tensors (and partially symmetric ones, etc),
- toric varieties,
- Mori Dream spaces (MDS) and Cox rings,
- toric degenerations of Calabi-Yau varieties and K3 surfaces,
- contact Fano manifolds and LeBrun-Salamon conjecture,
- complex Legendrian varieties,
- Hilbert basis of a lattice cone,
- Tensor network states.

- homogeneous spaces and varieties with an open orbit (quasi-homogeneous varieties),
- toric geometry (and more generally, torus actions), and related combinatorics (lattices, cones, fans, polytopes, etc), and related algebra (Cox rings, so multigraded rings, etc),
- geometry and deformations of minimal rational curves on projective manifolds,
- computational algebraic geometry tools, such as Magma,
- birational geometry,
- apolarity,
- cactus varieties and its analogues.