The investigator will study two different intersection theories in algebraic geometry: the arithmetic Chow rings of varieties defined over the integers and the quantum cohomology rings of complex manifolds. The varieties considered are homogeneous spaces of Lie groups, and the goal is to understand the arithmetic and quantum Schubert calculus. This allows one to effectively compute numerical invariants such as heights, on the one hand, and Gromov-Witten invariants, on the other. A further aim is to answer related questions about the combinatorics of modern Schubert calculus and the polynomials which represent degeneracy loci of vector bundles.
The study of the symmetries observed in geometric objects has been a meeting point of mathematics and physics for a long time. The many experiments and computations made in the 19th century eventually led to the cohomology and intersection theories of 20th century mathematics, which were applied to solve important open problems in both fields. At present, we face a similar situation in modern number theory and string theory, and computations of concrete examples are crucial to guide our intuition and to better understand the general theories. The investigator studies two new intersection theories, Arakelov theory and quantum cohomology, in many examples which are good testing grounds for both of them. The potential applications are a better understanding of Diophantine equations and approximation, used in coding theory and theoretical computer science, and the enumerative geometry of curves, related to classical algebraic geometry and quantum field theory in physics.
