TQFT, Links and Real Algebraic Curves

DMS-0203486 Patrick Gilmer本项目研究拓扑量子场论（TQFT）下的完整性结果形式同构在低维拓扑中的应用。 强移位等价（SSE）是在符号动力学中产生的一种等价关系. Gilmer正在调查他最近发现的TQFT和SSE之间的联系。 使用TQFT一个定义各种SSE类不变量ofknots和其他空间配备了一个无限循环cover.这现在可以推导出的后果SSE类invariants.Gilmer试图使用TQFT找到障碍经典knotsbeing切片knots. 一般来说，Gilmer使用TQFT作为低维拓扑的工具。在与Stepan Orevkov的联合工作中，Gilmer正在计算与真实的投影平面中的曲线集合相关的某些链接的进一步签名和零值。 在以前的工作中，Gilmer发现了这些不变量的限制条件，如果曲线的集合与给定次数的真实的代数曲线是同位素的。 这些计算可能导致对真实的代数曲线拓扑的新的一般限制。吉尔默还探索关系之间的真实的代数曲线和阴影描述的联系，在Turaev的意义。拓扑学是研究内在的形状。它有时被称为“橡皮片几何”，因为被研究的物体可以被扭曲和拉伸（但不会被撕裂）而不会失去它们的特性。这是一个涉及数学和科学许多领域的课题。 拓扑量子场论是拓扑学中最新和最令人兴奋的领域之一，与高能物理以及其他数学领域（例如数论和符号动力系统）有着密切的联系。吉尔默正在应用这个主题来回答有关结、链接和三维流形的问题。 一个三维流形是一个拓扑对象，它在局部看起来像我们所熟悉的空间。人们也可以考虑其他维度的流形。具有讽刺意味的是，三维和四维的流形最不容易理解。有人会猜测，我们的直觉应该在这些维度上最强。纽结是三维流形中的一个闭环。链环是三维流形上的闭回路的集合。 在1900年，希尔伯特给了一个著名的问题清单，供数学家研究。他的第十六个问题涉及拓扑的真实的代数曲线在真实的射影平面，它仍然是未解决的，但它导致了许多美丽的发展和部分解决方案。希尔伯特问如何组成一个给定次数的非奇异真实的齐次多项式的零点集的椭圆（称为椭圆）可以在平面上排列，本项目进一步研究了某些类型的链环，使希尔伯特问题及相关问题得到进一步的进展。

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.