TQFT, Links and Real Algebraic Curves

DMS-0203486Patrick Gilmer 该项目研究低维拓扑的拓扑量子场论 (TQFT) 下态射的完整性结果的应用。 强平移等价（SSE）是符号动力学中出现的等价关系。 吉尔默正在调查他最近发现的 TQFT 和 SSE 之间的联系。 使用 TQFT 可以定义结的各种 SSE 类不变量以及配备无限循环覆盖的其他空间。现在可以将其作为 SSE 类不变量的结果导出。Gilmer 正在尝试使用 TQFT 来查找经典结（即切片结）的障碍。 总的来说，Gilmer 使用 TQFT 作为低维拓扑的工具。在与 Stepan Orevkov 的合作中，吉尔默正在计算某些链接的进一步特征和无效性，他将这些链接与真实射影平面中的曲线集合相关联。 在之前的工作中，吉尔默发现，如果曲线集合与给定次数的实代数曲线同位素，则这些不变量会受到限制。 这些计算可能会对实代数曲线的拓扑产生新的一般限制。吉尔默还在图拉耶夫意义上探索实代数曲线和链接的阴影描述之间的关系。拓扑学是对内在形状的研究。它有时被称为“橡胶板几何”，因为所研究的物体可以扭曲和拉伸（但不能撕裂）而不会失去其特性。这是一门影响数学和科学许多领域的学科。 拓扑量子场论是拓扑学中最新、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 3 维流形的问题。 3 维流形是拓扑对象，其局部看起来像我们所生活的熟悉的空间。人们还可以考虑其他维度的流形。讽刺的是，第三维和第四维的流形却是最不为人所知的。人们可能会猜测，我们的直觉在这些维度上应该是最强的。结是 3 歧管中的闭合环。连杆是 3 流形中闭环的集合。 1900 年，希尔伯特列出了一份著名的数学家需要研究的问题清单。他的第十六个问题涉及实射影平面上的实代数曲线的拓扑，它仍然悬而未决，但它已经带来了许多漂亮的发展和部分解决方案。希尔伯特询问给定次数的非奇异实齐次多项式的零点集的分量（称为椭圆）如何在平面中排列，如果这些椭圆的数量对于给定的次数来说是最大的。该项目进一步研究了某些类型的链接，以在希尔伯特问题和相关问题。

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.