Title: Surfaces in Triangulated 4-Manifolds
Faculty Supervisor: Lisa Piccirillo
Graduate student mentor: John Teague
Description: Embedded surfaces are natural objects of study in understanding the topology of 4-dimensional manifolds. Just as circles embedded in 3-manifolds can be knotted in nontrivial ways, so too can surfaces be knotted in 4-manifolds. One approach to understanding these complicated smooth objects is by discretizing them into easy-to-understand pieces, like triangles (or their 4-dimensional analogs).
The goal of this project is to explore how surfaces which are embedded subordinately to a given triangulation behave, especially in simple 4-manifolds like the 4-dimensional ball. It would be especially interesting to find embedded surfaces whose boundaries, which will be knots or links, are complicated. This can help us better understand classical questions about knots, like what surfaces knots in S³ can bound in B⁴ (e.g. if a knot bounds a disk in B⁴, it is called slice).
Recently, software has been developed which takes smooth descriptions of 4-manifolds and produces triangulations for them. Since we are interested in finding complicated surfaces, it would help to have triangulations of simple 4-manifolds which are themselves rather complicated. Ideally, this project will build upon the above software to accomplish this goal, as well as potentially begin the search for interesting embedded surfaces.