The area of my work is Algebraic Geometry, which is the study of geometric shapes which are defined by polynomial equations. These shapes are called algebraic varieties and occupy a prominent role in modern mathematics. The easiest example of algebraic varieties are lines and conics (circles, ellipses, hyperbolae, parabolae) in the plane, as they are defined by a single polynomial equation in two variables and degree up to 2. What happens if you increase the number of variables, the degree of the polynomial equation and the number of polynomial equations?
More specifically, within algebraic geometry I am mostly interested in the following areas of research.
- Fano varieties: these are the algebraic varieties with positive curvature (in some sense). I am particularly interested in studying how they deform and in studying geometric properties of moduli spaces of Fano varieties.
- Toric geometry: toric varietes are algebraic varieties which can be constructed in a combinatorial way from polytopes and fans of polyhedral cones; it is interesting and very fruitful to see how algebro-geometric properties of toric varietes are reflected in the combinatorial properties of their discrete-geometric avatars. This is a part of combinatorial algebraic geometry.
- Deformation theory: it is the study of how algebraic varieties deform when one perturbs their equations. It is a fundamental tool to grasp local properties of moduli spaces.
- Gromov-Witten theory: this enumerates/counts curves inside algebraic varieties.
- Mirror Symmetry: this is a very active area of research in geometry which takes inspiration from theoretical physics.
