Smooth Isotopy, Gauge Theory and Contact Geometry

One of the key areas of mathematics research involves understanding the shape of an object. In mathematical terms, exploring different types of mappings on and from the object aids in visualizing the space. Consequently, the space comprising all such mappings is pivotal in comprehending an object from a mathematical standpoint. The PI will delve into such a space, specifically focusing on the space of diffeomorphisms on a manifold, with a primary emphasis on dimension four. Historically, dimension four has been one of the most enigmatic dimensions, as various mathematical theories falter here. Therefore, the PI aims to develop new theories that can enhance our comprehension of this space. As a component of this initiative, the PI intends to establish connections and partnerships with various branches of mathematics. Moreover, the PI is actively engaged in mentoring at all levels. The projects offer a welcoming avenue for undergraduate and graduate students to engage in research. One aspect of this project involves advancing theoretical frameworks for the smooth mapping class group of 4-manifolds, followed by utilizing diverse gauge theories to identify unconventional phenomena. The PI will pioneer new methodologies in this realm and endeavor to address longstanding conjectures concerning diffeomorphisms on 4-manifolds. Additionally, alongside investigating diffeomorphisms, the PI will explore the behavior of various embeddings of lower-dimensional manifolds within 4-manifolds. In the second phase, the PI will also allocate attention to three dimensions, seeking to discern relationships between different geometric structures. An illustrative example involves investigating the correlation between hyperbolic geometry and contact geometry. 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.