There are two themes to the proposed project. The first is the study of A-infinity algebras that arise naturally in algebraic geometry. While originally introduced in topological contexts, A-infinity algebras have proven essential in the understanding of structures arising in both algebra and geometry, and the PI proposes to study them from an algebraic-geometric point of view. Of particular interest is the study of their Hochschild theory, which governs the way A-infinity algebras deform. The second theme is the study of explicit examples of numerical invariants, known as Pandharipande-Thomas invariants, which have been recently introduced to understand counting-of-curves problems in geometric and string-theoretic situation. The PI proposes to try to explain certain unexpected coincidences observed by physicists between invariants of apparently unrelated spaces.
Algebraic geometry, which is the geometric study of solutions of polynomial equations, has seen in the last few years major developments, especially in terms of its applications in other fields of science. Of particular importance are applications to modern theoretical physics, in particular in string theory. The present project will increase our general understanding of algebraic structures that arisein algebraic geometry and physics.
