Topology and Geometry at the Interface of Dimensions 3 and 4

理解称为光滑四维流形的四维形状是几何和拓扑学数学领域的一个基本挑战，并且通过规范理论的框架与理论物理密切相关。 世纪令人惊讶的结果表明，4-流形的分类在性质上与高维和低维的相应问题有很大的不同，而且更困难。 研究四维流形的一个富有成效的方法是将它们沿着沿着三维流形切割成更简单的片段，这样，从相对简单的三维流形研究中获得的有效工具和技术可以应用于四维领域。 该研究项目旨在使用这种方法加深对光滑四维流形的理解，使用称为“Floer同调”的工具，该工具源于规范理论和数学物理。 该项目将解决位于三维和四维边界的问题。 例如，它将探索称为同调配边群的代数结构，它允许通过考虑某些像电影帧一样在它们之间插值的4-流形来添加和减去3-流形。 该项目将解决这些结构的几个基本问题，并完善它们，以解释所涉及的空间的对称性。 本计画将邀请研究生参与研究，利用规范理论与辛几何的工具，研究低维的拓扑与几何物体。具体目标包括发展技术研究顺利等变同源配边组，这些技术的应用长期存在的问题结一致性，并表征结，约束复杂的曲线在斯坦域。 该项目还将为寻求学习实现这些目标所采用的工具的学生和研究人员创造资源。 在这方面的具体目标包括完成和传播一个非正式的介绍性文本Heegaard Floer同源性。 该项目借鉴了同调代数、规范理论、纽结理论、接触几何和辛流形中伪全纯曲线理论的方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估而被认为值得支持。和更广泛的影响审查标准。

Understanding 4-dimensional shapes called smooth 4-manifolds is a fundamental challenge in the mathematical fields of geometry and topology and is intimately related to theoretical physics through the framework of gauge theory. Surprising results from the 20th century indicate that classification of 4-manifolds is qualitatively much different, and more difficult, than corresponding questions in both higher and lower dimensions. A fruitful approach to studying 4-manifolds is to cut them into simpler pieces along 3-dimensional manifolds, so that effective tools and techniques available from the comparatively simpler study of 3-manifolds can be brought to bear on the 4-dimensional realm. This research project aims to deepen understanding of smooth 4-dimensional manifolds using this approach, employing tools called "Floer homologies," which stem from gauge theory and mathematical physics. The project will address questions that lie at the border of 3- and 4-dimensions. For instance, it will explore algebraic structures called homology cobordism groups, which allow 3-manifolds to be added and subtracted by considering certain 4-manifolds that interpolate between them like frames of a movie. The project will address several fundamental questions on these structures and refine them to account for symmetries of the spaces involved. The project will involve graduate students in the research.This project studies topological and geometric objects in low dimensions, with the aid of tools from gauge theory and symplectic geometry. Specific goals include the development of techniques for studying smooth equivariant homology cobordism groups, application of these techniques to long-standing questions on knot concordance, and characterization of knots that bound complex curves in Stein domains. The project will also create resources for students and researchers seeking to learn the tools employed in pursuit of these goals. Specific objectives in this direction include the completion and dissemination of an informal introductory text on Heegaard Floer homology. The project draws on methods from homological algebra, gauge theory, knot theory, contact geometry, and the theory of pseudo-holomorphic curves in symplectic manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.