Donaldson-Thomas theory and tropical geometry

该项目是代数几何，广泛地说，位于热带几何和计数几何的交叉点上。计数几何是一个经典的和当代感兴趣的学科，它的核心目标是通过提取与几何对象集(例如丛或子流形)相关的数值信息来探索某些空间(代数流形)的几何。学生将探索对数Donaldson-Thomas(DT)理论，这是一门年轻的学科，并探索它与热带几何以及Gromov-Witten理论的关系。本论文可能的方向是研究新构造的对数Hilbert点格式及其基本拓扑，进而在曲面的情况下对log DT不变量进行一般研究。

The project is in algebraic geometry and broadly speaking, lies at the intersection of tropical geometry and enumerative geometry. Enumerative geometry is a subject of classical and contemporary interest, and its core goals are to probe the geometry of certain spaces (algebraic manifolds) by extracting numerical information associated to the set of geometric objects, e.g. bundles or submanifolds. The student will explore logarithmic Donaldson-Thomas (DT) theory, which is a young subject, and explore its relationship to tropical geometry, and to Gromov-Witten theory. The likely direction of the thesis will be a study of the newly constructed logarithmic Hilbert scheme of points and its basic topology, progressing towards a general study of log DT invariants in the case of surfaces.