Low-dimensional topology and links of singularities

In nature, singularities are associated with sudden changes or catastrophic events. In case of a complex surface the link of a singularity is a three-dimensional object encircling the singularity, or the sharp point, within the surface. This object often has a complicated shape reflecting properties of the singularity. In this context, the PI will study a number of questions relating algebraic and topological properties of the link and its fillings that are four-dimensional shapes with given boundary. The project includes applications of these shapes to other much-studied questions in geometry and the role they may play in certain new constructions. The research has applications in several other areas of mathematics and science. The PI's activities related to the project will make significant contribution to undergraduate education, curriculum development, and graduate as well as postdoctoral training. The PI will co-organize research seminars, conferences, workshops, and continue her editorial work at Quantum Topology, a research journal. The PI’s collaborative research will make significant contributions to important areas in topology of three- and four-manifolds, with connections to algebraic geometry and combinatorics. This project will further develop and expand recent results of the PI and her collaborators, where a new perspective and novel tools were introduced to the study of symplectic and contact topology of links of singularities. One of the important questions to be addressed is the comparison between symplectic fillings that arise in the algebraic context and those that have more general nature. In addition to fillings, symplectic cobordisms will be studied. The PI plans to find applications of the newly-developed tools to constructions of exotic smooth four-manifolds and exotically knotted surfaces. Another goal is to understand further connections between combinatorial and symplectic phenomena, and in particular, to use invariants such as Khovanov homology to detect "unexpected" monodromy factorizations and symplectic fillings. The project goals include work on a long-standing conjecture about finiteness of certain types of fillings. 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.