Moduli of Stable Log Maps and Applications

Boston College, Chestnut Hill MA

Investigators

Abstract

This project lies in the area of algebraic geometry, a branch of mathematics studying algebraic varieties, namely, the set of solutions of polynomial equations. The topic of this project is closely related to string theory from physics, where algebraic varieties are used to describe a piece of our universe. An amazing fact is that collections of varieties of certain type inside another bigger one are themselves varieties. The study of such collections of varieties provides not only powerfully tools to study problems in geometry in mathematics, but also useful methods to extract invariants interesting to physicists. The main topic of this project is aimed at developing a mathematical tool to calculate these invariants from string physics. Other topics of this project focus on using the above developed tool to solve problems from geometry and number theory. The major topic of this project is to study Gromov-Witten invariants via logarithmic smooth degenerations. The short term goal is to develop the theory of punctured maps which are, roughly speaking, stable logarithmic maps with "poles". The long term goal is to prove an effective degeneration formula which expresses Gromov-Witten invariants as a combination of invariants from punctured maps as the basic building blocks. Further topics of this project include (1) studying Teichmuller dynamics via the logarithmic compactification; and (2) studying density problem of integral points over function fields of algebraic curves. Both topics are further applications of the theory of stable logarithmic maps.