Low-Dimensional Topology, Floer Homology, and Categorification

该项目研究低维拓扑中的各种问题，研究3维和4维空间的整体形状以及其中包含的结和曲面。这门学科处于许多不同数学领域的交叉点，它有着广泛的应用，从宇宙学（宇宙的形状）到生物化学（DNA分子的打结）再到数学物理。令人惊讶的是，低维的许多问题通常比高维的类似问题更困难，需要使用超越传统代数拓扑的不变量。PI的特殊专业领域是Heegaard Floer同调，这是3维和4维流形的不变量集合，自21世纪初以来一直是低维拓扑研究中最富有成果的领域之一。这些工具汇集了几个不同的数学领域，包括表示论，微分几何和分析，PI希望阐明这些不同领域之间的联系，并扩大这些领域研究人员之间的话语。该项目的具体目标是（1）在4-流形拓扑中的各种具体问题上取得进展，包括结协调，奇异光滑结构，和嵌入的非定向表面;（2）理解之间的Heegaard弗洛尔同调的三维流形和拓扑性质，如存在的不可压缩的表面，拉紧叶理，和左序的基本组;（3）建立之间的关系结不变量产生的规范理论和辛几何和那些来自表示论和量子代数。

This project investigates a variety of questions in low-dimensional topology, the study of the global shapes of 3- and 4-dimensional spaces and of knots and surfaces contained within them. This subject lies at the crossroads of many disparate areas of mathematics, and it has a wide variety of applications ranging from cosmology (the shape of the universe) to biochemistry (the knotting of DNA molecules) to mathematical physics. Surprisingly, many problems in low dimensions are usually more difficult than their analogues in higher dimensions and require the use of invariants that go beyond traditional algebraic topology. The PI's particular area of expertise is in Heegaard Floer homology, a collection of invariants for 3- and 4-dimensional manifolds, which has been one of the most fruitful areas of research in low-dimensional topology since the early 2000s. These tools bring together several different fields of mathematics, including representation theory, differential geometry, and analysis, and the PI hopes to elucidate the connections between these different areas and expand the discourse among researchers in these fields.The specific goals of the project are (1) to make progress on a variety of concrete problems in 4-manifold topology, including knot concordance, exotic smooth structures, and embeddings of non-orientable surfaces; (2) to understand the relationship between the Heegaard Floer homology of a 3-manifold and topological properties such as the existence of incompressible surfaces, taut foliations, and left-orderings on the fundamental group; (3) to establish relationships between the knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory and quantum algebra.