Combinatorial Methods in Low-Dimensional Topology

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Topology refers broadly to the study of shapes, and low-dimensional topology refers specifically to their study in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world, and from a mathematical perspective, since the phenomena they exhibit and the collection of techniques used to study them are rather different from those in higher dimensions. Combinatorics refers to the study of discrete structures, such as networks and flows, and it is influential in the development of codes and algorithms. Combinatorial methods have long influenced topology. Amongst the various techniques that come to bear on low-dimensional topology are graphs of surface intersections; lattice-theoretic methods; and extremal combinatorics. Each technique has led to sensational progress on the main problems in low-dimensional topology, and they lend very different perspectives on the subject. The unifying goal of this research project is to advance combinatorial methods towards problems in low-dimensional topology. Alongside the research component, the PI will conduct activities that integrate his research interests with education and training initiatives that reach audiences from the high school level to postdoctoral researchers. For instance, the PI is actively involved with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and Mathematical Staircase, Inc. In the context of these programs and in other mentoring activities, he seeks to inspire the discovery process and aid in the exposition of beautiful mathematics. He ran a graduate summer school focused on a central theme in low-dimensional topology, Dehn surgery, and is in the process of editing a book based on it. He has also written a survey article on Heegaard Floer homology for the Notices of the AMS, the most widely-read general interest periodical targeted at professional mathematicians, and he is committed to more expository work aimed at a wide mathematical audience. The award provides funds for supporting graduate students.The PI will develop combinatorial methods in low-dimensional topology, in continuation of an established program. The main projects focus on studying graphs of surface intersections in application to exceptional Dehn surgery and embeddings of surfaces in the four-sphere; lattice embeddings and their refinements in application to the study of rational homology balls, slice links, and surface isotopy; and extremal combinatorics in application to properties of curves on surfaces. The surface intersection techniques are more direct and rely on the development of graph theoretic tools in order to draw topological conclusions. Floer homology methods are less direct but apply heavy machinery to a vast collection of problems. This project will more closely bind combinatorics and low-dimensional topology. The PI currently advises three PhD students and has just graduated one student. The PI runs original graduate courses on various themes, including advanced combinatorics and the knot complement problem, and will continue to do so. This award will support graduate students during summer months and assist them with travel funds. At the postdoctoral level, the PI mentors and collaborates with post-doctoral fellows. This award will support them with travel funds. At the scholarly outreach level, the PI intends to undertake more expository projects with the aim of reaching and educating a wide audience.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.