TQFT, Links and Real Algebraic Curves

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DMS-0203486Patrick GilmerThis project investigates applications of integrality results formorphisms under Topological Quantum Field Theories (TQFTs) forlow-dimensional topology. Strong Shift Equivalence (SSE) is anequivalence relation which arose in symbolic dynamics. Gilmer isinvestigating a connection between TQFT and SSE which he has recentlydiscovered. Using a TQFT one defines various SSE class invariants ofknots 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 knotsbeing slice knots. In general, Gilmer is using TQFT as a tool inlow-dimensional topology. In joint work with Stepan Orevkov, Gilmer iscalculating further signatures and nullities of certain links which heassociated to collections of curves in the real projective plane. Inprevious work, Gilmer found restrictions on these invariants if thecollection of curves is isotopic to a real algebraic curve of givendegree. These calculations may lead to new general restrictions on thetopology of real algebraic curves. Gilmer is also exploring relationsbetween real algebraic curves and shadow descriptions of links, in thesense of Turaev.Topology is the study of intrinsic shape. It is sometimes called "rubbersheet geometry" because the objects under investigation can be twisted andstretched (but not torn) without losing their identity. It is a subjectwhich impinges on many areas of mathematics and science. TopologicalQuantum Field Theory is one of the most current and exciting areas oftopology with intimate connections to high energy physics as well as otherareas of mathematics, for instance number theory and symbolic dynamicalsystems. Gilmer is applying this subject to answer questions about knots,links and 3-dimensional manifolds. A 3-dimensional manifold is atopological object which looks locally like the familiar space we live in.One may also consider manifolds of other dimensions. It is ironic thatmanifolds of dimension three and four are least well understood. One wouldguess that our intuition should be strongest in these dimensions. A knotis a closed loop in a 3-manifold. A link is a collection of closed loopsin a 3-manifold. In 1900, Hilbert gave a famous list of problems formathematicians to study. His sixteenth problem concerns the topology ofreal algebraic curves in the real projective plane, It is still unsolvedbut it has lead to many beautiful developments and partial solutions.Hilbert asked how the components (called ovals) of the set of zeros of anonsingular real homogenous polynomial of given degree can be arranged inthe plane, if the number of these ovals is maximal for the given degree.This project further studies certain type of links to make progress onHilbert's problem and related questions.