Research team
Expertise
Algebraic geometry with connections to other fields. More specifically: algebraic curves, K3 surfaces, hyperkähler varieties, abelian varieties, sheaves/stable objects and their moduli spaces. I am also interested in algebraic cycles, Shimura varieties, modular forms, strata of differentials, and Teichmüller dynamics.
Non-Commutative and Homological methods in Geometry and Mirror Symmetry.
Abstract
Algebraic geometry is a field that underpins geometric intuition with the precision brought about by calculation from algebra. The algebraic information of an algebraic variety is packaged in sheaves and organized in categories, which makes it possible to import the tools of homological and (non-)commutative algebra. This categorical approach is natural for many leading questions in mirror symmetry, birational geometry, and theoretical physics. It is also fundamental, as many properties and even classification results are not possible without it. The project naturally splits into two parts. The first aims to investigate one of the most fundamental properties of curves, namely contractibility, using non-commutative deformation theory. It will generalize classical results using modern language, and produce many new explicit examples of rational curves in threefolds. Linking contractibility and non-commutative algebra will impact the birational and enumerative geometry of Calabi-Yau varieties. The second part involves Homological Mirror Symmetry for the unexplored setting of surfaces with mild positive curvature. The main objective is to prove a mirror theorem, packaging the physical information carried by the surface with tools of homological algebra. The novelty of the setting will require to formulate new definitions and heuristics, and open up an avenue to study other Fano varieties of higher dimension, where current techniques do not apply.Researcher(s)
- Promoter: Barros Ignacio
- Co-promoter: Lowen Wendy
Research team(s)
Project type(s)
- Research Project
New birational invariants and the geometry of moduli spaces.
Abstract
Algebraic geometry is the study of algebraic varieties. Moduli spaces are universal varieties, they parameterize all varieties of a certain kind. We aim to develop a better understanding of the new birational invariants measuring irrationality in the context of moduli spaces. Moduli theory is central in modern algebraic geometry with uses that go beyond mathematics. We investigate important moduli spaces from the point of view of birational geometry. We put special focus on recently constructed moduli spaces such as those of hyperkähler varieties with a Lagrangian fibration as well as strata of differentials, where almost nothing is known about their birational complexity. Determining if a variety is 'rational' (as simple as it can be) is a famously hard problem. On the opposite side, when a variety is of 'general type' (as complicated as it can be) not much is known about how to distinguish them from each other. Most moduli spaces fall into this category. In the last five years, a new set of invariants known as 'measures of irrationality' have attracted special interest. They will play a crucial role in understanding the geometry of moduli spaces and their complexity. We study these questions on concrete recently constructed collections of moduli spaces that have deep connections with modular forms and dynamics.Researcher(s)
- Promoter: Barros Ignacio
Research team(s)
Project type(s)
- Research Project
New birational invariants and the geometry of moduli spaces.
Abstract
Algebraic geometry is the study of algebraic varieties. Moduli spaces are universal varieties; they parameterise all varieties of a particular species. We aim at a better understanding of these new birational invariants in the context of moduli spaces. Moduli theory is central in modern algebraic geometry, with applications also outside mathematics. We investigate important moduli spaces from the perspective of birational geometry. We pay particular attention to recently constructed moduli spaces, such as those of hyperkähler varieties with a Lagrangian fibration and differential strata, for which almost nothing is known about their birational complexity. Determining whether a variety is 'rational' (as simple as possible) is a well-known difficult problem. On the other hand, for varieties of 'general type' (as complex as possible), not much is known about how to distinguish them from each other. Most moduli spaces fall into this category. Over the past five years, a new set of invariants, known as "measures of irrationality", attracted special interest. They will play a crucial role in understanding the geometry of moduli spaces and their complexity. We study these questions for concrete recently constructed collections of moduli spaces that have profound connections with modular forms and dynamics.Researcher(s)
- Promoter: Barros Ignacio
- Fellow: Barros Ignacio
Research team(s)
Project type(s)
- Research Project
Algebraic cycles on hyperkähler varieties.
Abstract
We study algebraic cycles on hyperkähler varieties. The goals are extending the knowledge on filtrations, tautological cycles, and tautological identities to the non-commutative setting as well as studying the Franchetta conjecture on small dimensional universal families of hyperkähler varieties with finite quotient singularities.Researcher(s)
- Promoter: Barros Ignacio
- Fellow: He Shi
Research team(s)
Project type(s)
- Research Project