TQFT, Links and Real Algebraic Curves

patrick gilmer本项目研究拓扑量子场论（TQFTs）低维拓扑下完整性结果形式的应用。强移位等价（SSE）是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与上交所之间的联系。使用TQFT可以定义结点和其他具有无限循环覆盖的空间的各种SSE类不变量。这现在可以作为SSE类不变量的结果派生出来。Gilmer正尝试使用TQFT来寻找经典结（切片结）的障碍。一般来说，Gilmer使用TQFT作为低维拓扑的工具。在与Stepan Orevkov的合作中，Gilmer正在计算与真实投影平面上曲线集合相关的某些链路的进一步特征和零值。在以前的工作中，Gilmer发现了这些不变量的限制，如果曲线的集合是给定度的实代数曲线的同位素。这些计算可能导致对实际代数曲线拓扑结构的新的一般限制。吉尔默也在探索真实的代数曲线和图拉耶夫意义上的链接的影子描述之间的关系。拓扑学是对内在形状的研究。它有时被称为“橡胶板几何”，因为被研究的物体可以被扭曲和拉伸（但不能被撕裂）而不会失去其特性。这是一门涉及数学和科学许多领域的学科。拓扑学量子场论是当前最令人兴奋的拓扑学领域之一，与高能物理以及数学的其他领域密切相关，例如数论和符号动力学系统。Gilmer正在应用这门学科来回答关于结、链接和三维流形的问题。三维流形是一种拓扑物体，它在局部看起来就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，人们对三维和四维的流形了解得最少。有人会猜测，我们的直觉在这些维度上应该是最强的。一个结是3-歧管中的一个闭环。连杆是三流形中闭环的集合。1900年，希尔伯特给了数学家们一个著名的问题清单来研究。他的第16个问题是关于实投影平面上的实代数曲线的拓扑结构，这个问题至今仍未得到解决，但它已经导致了许多美丽的发展和部分解。希尔伯特提出了一个问题，如果给定阶次的非奇异实齐次多项式的零集合的分量（称为椭圆）在给定阶次下是最大的，那么这些分量（称为椭圆）是如何排列在平面上的。本课题进一步研究某一类环节，在希尔伯特问题及相关问题上取得进展。

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.