Algebraic and Geometric Topology In Dimensions Three and Four

三维和四维的代数和几何拓扑

• 批准号: 2267124
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2267124
• 关键词: Algebraic; Geometric; Topology; Dimensions; Three

Knot theory in dimension 3 concerns the classifications of knots (continuous embeddings of some finite number of copies of the unit circle into the 3-dimensional euclidean space, up to isotopy, namely connectedness via a continuous path of embeddings). This means that the properties of interest in the field of knot theory are those which are stable under continuous deformation, and also concerns the classification of all knot types. A key method in this classification is the use of knot invariants: quantities assigned to knots which remain constant as the knot is continuously deformed. The discovery of knew knot invariants is a challenging and intriguing task involving the use of tools from areas such as representation theory, category theory, differential geometry and mathematical physics.The theory of knots in dimension 3 is well understood. Several tools have been developed for distinguishing inequivalent knots, such as the Jones and HOMFLY-PT polynomial, and the Kontsevich integral, the latter of which takes its values in an 'algebra of chord diagrams'. This is a diagrammatically defined algebra of 'chord diagrams', modulo some set of Lie theoretical relations called the 4-term relations., which ensure that the Kontsevich integral is a knot invariant.The main subject of my thesis will be knot theory in dimension 4, investigating the properties of surfaces embedded in 4-dimensional euclidean space, considered up to isotopy. We aim to study these properties by constructing invariants of knotted surfaces, which raises deep problems in geometry, topology and representation theory.Possible projects in this area include:A) Investigate diagrammatic algebras as targets for an analogue of the Kontsevich integral for knotted surfaces in dimension 4. This area was explored during a summer research project which I undertook with the University of Leeds during the summer of 2018. Key references would include: Cirio & Faria Martins: Infinitesimal 2-Braidings(arXiv:1309.4070v3, 16 Mar 2015) and Moutier: A Kontsevich integral of order 1 ( arXiv:1810.05747v1 , 12 Oct 2018)B) Investigate representations of 4-dimensional analogues of the braid group. One such analogue is the loop braid group, which can be presented by generators and relations in a similar manner to the braid group itself. As in Celeste Damiani: A journey through loop braid groups ( arXiv:1605.02323v3, 30 Sep 2016) (the author of which is now a research fellow at the university of Leeds). An potential aim for the project could be to study and improve the representations recently found by Bullivant, Martin and Faria Martins (All of whom are now at the university of Leeds) in Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. ( arXiv:1807.09551v2, 19 Dec 2018)C) Investigate invariants of knotted surfaces derived from higher gauge theory. This would have strong connections to algebraic topology.D) Investigate the higher categorical structure formed by knotted surfaces. The category of tangles can be represented by generators and relations within the language of monoidal categories. (This is essentially due to the fact that tangles form a monoidal category). Since knotted surfaces can be composed in two different directions, the 4-dimensional analogue of tangles(2-tangles) is no longer a category but a 2-category. This 2-category can also be presented by generators and relations, which renders the theory of knotted surfaces combinatorial. A project could focus on investigating the 2-category of knotted surfaces. This would have strong connections to my current master's project on the monoidal category of tangle

3维纽结理论关注的是纽结的分类（单位圆的有限个副本连续嵌入到3维欧几里得空间中，直到合痕，即通过连续嵌入路径的连通性）。这意味着在纽结理论领域感兴趣的性质是那些在连续变形下稳定的性质，并且还涉及所有纽结类型的分类。这种分类的一个关键方法是使用结不变量：分配给结的数量在结连续变形时保持不变。已知的结不变量的发现是一个具有挑战性和有趣的任务，涉及使用的工具，如表示论，范畴论，微分几何和数学物理领域。已经开发了几种工具来区分不等价的节点，如琼斯和HOMFLY-PT多项式，以及Kontsevich积分，后者在“弦代数”中取值。这是一个图解定义的“弦关系”代数，模一些李理论关系集称为4项关系。本文的主要研究内容是四维纽结理论，研究四维欧氏空间中曲面的性质，直到合痕。我们的目标是通过构造纽结曲面的不变量来研究这些性质，这在几何、拓扑和表示论中提出了深刻的问题。该领域可能的项目包括：A）研究图代数作为4维纽结曲面的Kontsevich积分模拟的目标。这一领域是在2018年夏天我与利兹大学进行的一个夏季研究项目中探索的。主要参考资料包括：Cirio & Faria Martins：无穷小2-辫子（arXiv：1309.4070v3，2015年3月16日）和Moutier：1阶Kontsevich积分（arXiv：1810.05747v1，2018年10月12日）B）研究辫子群的4维类似物的表示。一个这样的类似物是环辫群，它可以用生成元和关系以类似于辫群本身的方式表示。就像塞莱斯特达米亚尼：循环编织群之旅（arXiv：1605.02323v3，2016年9月30日）（作者现在是利兹大学的研究员）。该项目的一个潜在目标可能是研究和改进Bullivant，Martin和Faria Martins（他们现在都在利兹大学）最近发现的表示法，在环辫群的表示法和离散（3+1）维高规范理论中的Aharonov-Bohm效应。（arXiv：1807.09551v2，2018年12月19日）C）研究从高规范理论导出的打结曲面的不变量。这将与代数拓扑学有很强的联系。D）研究由纽结曲面形成的高级范畴结构。缠结范畴可以用么半群范畴语言中的生成元和关系来表示。(This本质上是由于缠结形成monoidal范畴的事实）。由于纽结曲面可以在两个不同的方向上组合，所以缠结的四维类似物（2-缠结）不再是一个范畴，而是一个2-范畴。这个2-范畴也可以用生成元和关系表示，这使得纽结曲面的理论是组合的。一个项目可以专注于研究2类打结表面。这将与我目前的硕士项目有很强的联系，