Selected Research Accomplishments

As indicated by the nearly two thousand citations of W. Fulton’s treatise on this topic, Excess Intersection Theory plays a very important role in modern algebraic geometry. It is often used to determine the contributions to the Euler class of a vector bundle from degenerate components of the vanishing locus of a specific, typically geometrically meaningful, section. This approach is rigid and global in nature and involves blowing up the degenerate loci to smooth them out. My PhD thesis introduces an approach to this problem which is soft and local in nature. It applies in finite-dimensional singular and infinite-dimensional Fredholm settings. In contrast to the standard approach, it determines the contributions from (not necessarily compact) strata of the vanishing locus and involves no blowups (which may not be even defined in some of the applicable settings). The local excess intersection approach fits ideally with the virtual fundamental class construction in symplectic topology, as the latter is fundamentally about incorporating degenerate contributions. The rudiments of this approach, in both finite-dimensional and infinite-dimensional settings, appear in [18]. It is developed further and applied to a wide range of problems in classical enumerative geometry in [19]. Along with (2) below, this approach in a Fredholm setting underpins the proof of the long-standing BCOV mirror symmetry conjecture described in (3).