Low-Dimensional and Contact Topology of Links of Surface Singularities

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

This project focuses on several interrelated topics in geometry of surfaces in dimensions three and four. On a complex surface defined by algebraic equations, a cusp-like sharp point in an otherwise smooth region is called an isolated singularity. In nature, singularities occur when sudden changes or catastrophic events happen, thus the study of singularities from various viewpoints is important to all branches of science. In geometry, an object known as the link of singularity encircles the sharp point on the surface; this object resembles a sphere in the absence of a sharp point but typically has a complicated shape encoding properties of the singularity. The PI will study the properties of such shapes and the additional structures they support. A particular focus will be on collections of planes called contact structures that arise in physics. Plamenevskaya's research will contribute to several areas of mathematics: low-dimensional topology, symplectic and contact topology, algebraic geometry, and combinatorics. Her other activities related to the project will make significant contributions to undergraduate education and graduate and postdoctoral training. Plamenevskaya is working to increase awareness of current research in topology among undergraduate students, as well as contribute to participation of women in mathematics. She co-organizes research seminars, conferences, and workshops for a diverse audience, from graduate students to distinguished researchers. Specifically, the PI will work with the canonical contact structure induced by the complex tangencies on the link of singularity. A particular goal of the project is to understand the relation between Stein fillings/cobordisms of such contact structures and smoothings/deformations of the surface singularity, and to address a variety of questions concerning the interplay of the singularity theory and symplectic and contact topology. The PI's previous work (with Ghiggini and Golla) shows that rational surface singularities with reduced fundametal cycle, whose deformation theory is well-understood, correspond precisely to the class of planar contact structures, whose fillings can be studied via surface diffeomorphisms. This opens the door to extending certain results from algebraic geometry to symplectic topology as well as to applications of symplectic topology to singularity theory. Plamenevskaya will pursue these questions using a range of tools, including low-dimensional constructions such as open book decompositions and symplectic caps, complex and symplectic line arrangements, and certain Floer-homological and combinatorial invariants. She intends to discover further connections between the aforementioned constructions and ideas, which may additionally lead to new results on mapping class groups and combinatorics of line arrangements. Plamenevskaya will also study a number of related questions, such as certain constructions in lattice cohomology (a combinatorial theory introduced by Nemethi for links of singularities) and their relation to Heegaard Floer homology. Several collaborators will be involved in the PI's work. 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.