CAREER: Dehn Surgery and the Homology Cobordism Group

This project focuses on advancing a subfield of mathematics called low-dimensional topology. Low-dimensional topology is the study 3 and 4-dimensional shapes. Many familiar objects and phenomena can be described in terms of 3-dimensional and 4-dimensional shapes. For example, we naturally think of the world we live in as a 3-dimensional object. However, it is not static in time. Therefore, it is helpful to add an extra time dimension and view the universe as having 3 spatial dimensions and 1 time dimension. Topologists consider even more complicated (though frequently purely abstract) examples of 3 and 4-dimensional shapes. From a mathematical perspective, it is important to understand which 3-dimensional shapes can appear in the boundary of a 4-dimensional shape. For example, the universe at a fixed starting time is a 3-dimensional shape. Mathematically, we can think of this time slice as being the boundary of space time for all future times after our starting point. This is a simple example, but topologists have formulated several very famous and very challenging questions related to which 3-manifolds bound 4-dimensional shapes of a given complexity. The PI will investigate primarily using a tool called Floer homology, which is inspired by techniques from mathematical physics, such as Yang-Mill’s theory. The proposed project focuses on using Heegaard Floer homology to make progress on understanding the structure of the homology cobordism group. The structure of this group remains one of the most important questions in low-dimensional topology. The PI proposes developing a wide range of flexible tools that will be able to analyze the structure of this group. A major aim of this project is to develop tools to detect torsion in this group, which may lead to the solution of a long-standing open problem. The educational component of this grant will be focused on running a summer school at the University of Oregon for two summers during the grant period. This summer school will be for advanced undergraduates. Students will receive lectures on basic techniques in low dimensional topology and Floer theory and will be given approachable research problems to work on during the program. During the Summer research program, the PI will also run a small research conference in low-dimensional topology during one week of the summer research program, which the undergraduates will attend. The grant will fund further undergraduate and graduate research activities at the University of Oregon. 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. NSF Award ID: 2543889 | Program: 01003031DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Ian Zemke | Institution: University of Oregon Eugene, EUGENE, OR | Award Amount: $207,409 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2543889 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2543889.html