TQFT, Links and Real Algebraic Curves

Louisiana State University, Baton Rouge LA

Investigators

Abstract

DMS-0203486 Patrick Gilmer This project investigates applications of integrality results for morphisms under Topological Quantum Field Theories (TQFTs) for low-dimensional topology. Strong Shift Equivalence (SSE) is an equivalence relation which arose in symbolic dynamics. Gilmer is investigating a connection between TQFT and SSE which he has recently discovered. Using a TQFT one defines various SSE class invariants of knots and other spaces which are equipped with an infinite cyclic cover. This can now be derived as a consequence of a SSE class invariant. Gilmer is attempting to use TQFT to find obstructions to classical knots being slice knots. In general, Gilmer is using TQFT as a tool in low-dimensional topology. In joint work with Stepan Orevkov, Gilmer is calculating further signatures and nullities of certain links which he associated to collections of curves in the real projective plane. In previous work, Gilmer found restrictions on these invariants if the collection of curves is isotopic to a real algebraic curve of given degree. These calculations may lead to new general restrictions on the topology of real algebraic curves. Gilmer is also exploring relations between real algebraic curves and shadow descriptions of links, in the sense of Turaev. Topology is the study of intrinsic shape. It is sometimes called "rubber sheet geometry" because the objects under investigation can be twisted and stretched (but not torn) without losing their identity. It is a subject which impinges on many areas of mathematics and science. Topological Quantum Field Theory is one of the most current and exciting areas of topology with intimate connections to high energy physics as well as other areas of mathematics, for instance number theory and symbolic dynamical systems. Gilmer is applying this subject to answer questions about knots, links and 3-dimensional manifolds. A 3-dimensional manifold is a topological object which looks locally like the familiar space we live in. One may also consider manifolds of other dimensions. It is ironic that manifolds of dimension three and four are least well understood. One would guess that our intuition should be strongest in these dimensions. A knot is a closed loop in a 3-manifold. A link is a collection of closed loops in a 3-manifold. In 1900, Hilbert gave a famous list of problems for mathematicians to study. His sixteenth problem concerns the topology of real algebraic curves in the real projective plane, It is still unsolved but it has lead to many beautiful developments and partial solutions. Hilbert asked how the components (called ovals) of the set of zeros of a nonsingular real homogenous polynomial of given degree can be arranged in the plane, if the number of these ovals is maximal for the given degree. This project further studies certain type of links to make progress on Hilbert's problem and related questions.