Research of Jarosław Buczyński
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...
More specifically, the problems I am (or I have been) working on are
related to the following notions and topics:
- 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.
Tools I am using in my research include:
- 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.