Applications of parametarized gauge theory

Kansas State University, Manhattan KS

Investigators

Abstract

The study of higher-dimenional spaces has had applications to many different areas, ranging from computer vision to data science. These spaces have many unique and beautiful patterns. Studying these patterns is worthwhile in its own right and may lead to unexpected important applications in the future. Four dimensions is the setting for Einstein’s theory of General Relativity where it is referred to as space-time. Four dimensional spaces have many interesting and unexpected properties. For example, one may find pairs of surfaces so that one surface may be deformed into another, but any such deformation will fail to be smooth in an essential way. Many of the differences between smooth and continuous objects related to four-dimensional spaces disappear after one of several possible stabilization operations is performed enough times. Detecting the required number of stabilizations is a difficult problem because many of the tools that may be used to detect differences between smooth objects fail to work after one stabilization. Using invariants for families of objects is one approach, but many of the key problems are still difficult. The approach this project takes is to consider several related questions where more structure is present. Sometimes the extra structure will make the problem more difficult, but sometimes the extra structure will make the problem easier. The hope is that resolving some of these related questions will provide insight into the more difficult, and more fundamental questions. This is where this project derives its intellectual merit. In addition to the pure research, the PI shares mathematics with many different communities. Notably, the PI is the director of the Navajo Nation Math Circles. Several mathematical activities presented in this community were inspired by more technical problems from the PIs research. This outreach program is the most notable broader impact of the project. In more technical terms, the difficult problem is to find a pair of closed, simply connected, smooth 4 manifolds separated by more than one stabilization. The related approachable questions are analogues for embedded surfaces, diffeomorphisms, and families of such objects. The project will investigate a possible Arf-type invariant for pairs of surfaces, stabilization questions in the symplectic category as well as in families. It will also consider algebraic structures arising from these spaces. Some impacts arise because ideas developed and shared in this proposal may inspire work in adjacent fields. Other impacts arise due to the training and mentoring of future researchers and the mentoring of junior researchers that will take place in this research. The PI is an organizer for Gauge Theory Virtual and runs the MathCircles YouTube channel. The PI has close ties to the Navajo Nation and shares mathematics inspired by his research with this community. 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.