Algebraic and Geometric Topology In Dimensions Three and Four

三维的纽结理论涉及纽结的分类(单位圆的某些有限个副本的连续嵌入到三维欧几里得空间中，直到等距，即通过连续嵌入路径的连通性)。这意味着纽结理论领域感兴趣的性质是那些在连续变形下是稳定的性质，也涉及到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：分配给在结连续变形时保持不变的结的量。已知纽结不变量的发现是一项具有挑战性和耐人寻味的任务，涉及到表示论、范式论、微分几何和数学物理等领域的工具的使用。已经开发了几种工具来区分不等价的纽结，如Jones和HOMFLY-PT多项式，以及Kontsevich积分，后者的值在“弦图代数”中。这是一个图解定义的‘弦图’代数，模上一组称为四项关系的李理论关系，从而确保Kontsevich积分是一个纽结不变的。我的论文的主要主题是4维的纽结理论，研究嵌入到4维欧氏空间中的曲面的性质，被认为是等比的。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、拓扑和表示理论中提出了深刻的问题。这一领域可能的项目包括：a)研究图代数作为模拟4维纽结曲面的Kontsevich积分的目标。这一领域是我在2018年夏天与利兹大学合作的夏季研究项目中探索的。主要参考文献包括：Cirio&Faria Martins：infinitesimal 2-Braidings(arxiv：1309.4070v3,2015年3月16日)和Moutier：a Kontsevich 1阶积分(arxiv：1810.05747v1,2018年10月12日)B)研究辫子群的4维类似物的表示。一个这样的类比是环辫子群，它可以用与辫子群本身类似的方式由生成元和关系表示。《塞莱斯特·达米亚尼：环状辫子群之旅》(arxiv：1605.023v3,2016年9月30日)(作者现为利兹大学研究员)。该项目的一个潜在目标可能是研究和改进最近由Bullivant、Martin和Faria Martins(他们现在都在利兹大学)在离散(3+1)维高规范理论中的Loop Braid群和Aharonov-Bohm类效应的表示中发现的表示。(arxiv：1807.09551v2,2018年12月19日)C)研究源自高规范理论的纽结曲面的不变量。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更高范畴结构。纠缠范畴可以用生成元和么一范畴语言中的关系来表示。(这主要是因为缠结形成了一个单一的范畴)。由于纽结曲面可以在两个不同的方向上组合，因此缠结的4维模拟(2缠结)不再是一个范畴，而是一个2-范畴。这种2-范畴也可以用生成元和关系来表示，这使得纽结曲面理论具有组合性。一个项目可以专注于研究2-类打结曲面。这将与我目前硕士的关于纠缠的单一化类别的项目有很强的联系

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