Thesis advisor: Julia Elisenda Grigsby / Thesis advisor: David Treumann / In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In this thesis, we set up a framework to define knot invariants for each choice of a symmetric space. In order to address this task, we start by defining appropriate notions of singular bundles and singular connections for a given symmetric space. We can associate a moduli space to any singular bundle defined over a compact 4-manifold with possibly non-empty boundary. We study these moduli spaces and show that they enjoy nice properties. For example, in the case of the symmetric space SU(n)/SO(n) the moduli space can be perturbed to an orientable manifold. Although this manifold is not necessarily compact, we introduce a comapctification of it. We then use this moduli space for singular bundles defined over 4-manifolds of the form YxR to define knot invariants. In another direction we mimic the construction of Donaldson invariants to define polynomial invariants for closed 4-manifolds equipped with smooth action of Z/2Z. / Mathematics

In this thesis, topologically massive Yang-Mills theory is studied in the framework of geometric quantization. This theory has a mass gap that is proportional to the topological mass m. Thus, Yang-Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern-Simons theory with level number k. The focus of this research is the near Chern-Simons limit of the theory, where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of two copies of Chern-Simons with level number k/2, very similar to the Chern-Simons splitting of topologically massive AdS gravity model. As m approaches to infinity, the split parts add up to give the original Chern-Simons term with level k. Also, gauge invariance of the split CS theories is discussed for odd values of k. Furthermore, a relation between the observables of topologically massive Yang-Mills theory and Chern-Simons theory is obtained. It is shown that one of the two split Chern-Simons pieces is associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang-Mills theory observables in the near Chern-Simons limit. Finally, motivated with the topologically massive AdS gravity model, Chern-Simons splitting concept is extended to pure Yang-Mills theory at large distances. It is shown that pure Yang-Mills theory acts like two Chern-Simons theories with level numbers k/2 and -k/2 at large scales. At very large scales, these two terms cancel to make the theory trivial, as required by the existence of a mass gap.

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Cette thèse est divisée en quatre parties qui portent sur les centralisateurs des algèbres quantiques \(U_q(\mathfrak{sl}_N)\), les polynômes biorthogonaux avec propriétés bispectrales, les polynômes bivariés de Griffiths, et les schémas d'association avec structures polynomiales bivariées. Le fil conducteur principal entre ces parties est l'algèbre d'Askey–Wilson. Dans la première partie, l'idée principale est de combiner l'algèbre du groupe des tresses avec l'algèbre d'Askey–Wilson dans des situations qui impliquent les centralisateurs de \(U_q(\mathfrak{sl}_2)\). Ainsi, on obtient des représentations du groupe des tresses en termes de polynômes orthogonaux de \(q\)-Racah par le biais de matrices \(R\) de \(U_q(\mathfrak{sl}_2)\), on obtient une interprétation de l'algèbre d'Askey–Wilson dans le cadre de la théorie topologique des champs de Chern–Simons avec groupe de jauge \(SU(2)\) ainsi que dans le cadre des invariants d'entrelacs associés à \(U_q(\mathfrak{su}_2)\), et on offre une description algébrique complète du centralisateur de \(U_q(\mathfrak{sl}_2)\) dans un produit tensoriel de trois représentations irréductibles identiques de spin quelconque. Dans une optique différente, on offre aussi une présentation algébrique de certaines algèbres de Hecke fusionnées qui décrivent des centralisateurs de \(U_q(\mathfrak{sl}_N)\). Dans la deuxième partie, on étudie deux familles de polynômes biorthogonaux par des méthodes algébriques, offrant une extension du tableau qui existe pour les polynômes orthogonaux classiques de type Askey–Wilson. Les deux familles considérées sont les polynômes \(R_I\) de type Hahn et les polynômes de Pastro. Dans les deux cas, l'idée est d'introduire un triplet d'opérateurs ayant une action tridiagonale et d'obtenir les polynômes comme solutions à deux problèmes aux valeurs propres généralisés provenant de ce triplet. On trouve les propriétés de bispectralité et de biorthogonalité des polynômes en se servant des opérateurs du triplet, et on détermine l'algèbre réalisée par les opérateurs. Dans la troisième partie, on caractérise deux familles de polynômes bivariés de Griffiths. La première famille est une généralisation des polynômes de Griffiths de type Krawtchouk qui dépend d'un paramètre \(\lambda\). On trouve leurs relations de bispectralité et leur biorthogonalité en utilisant les propriétés des polynômes de Krawtchouk à une variable. Les relations de contiguïté des polynômes univariés jouent un rôle essentiel dans les calculs. On utilise des méthodes semblables pour caractériser la deuxième famille, qui est formée de polynômes de Griffiths de type Racah. Ceux-ci sont orthogonaux. Dans la quatrième partie, on propose une généralisation bivariée des propriétés \(P\)- et \(Q\)-polynomiales pour les schémas d'association et de concepts reliés. Plusieurs exemples de schémas vérifiant la propriété \(P\)-polynomiale bivariée sont obtenus. On montre que les schémas de Johnson non-binaires ainsi que leurs analogues \(q\)-déformés, les schémas définis à partir d'espaces atténués, sont \(P\)- et \(Q\)-polynomiaux bivariés en étudiant les propriétés bispectrales des polynômes bivariés associés. Les structures algébriques reliées à ces schémas sont explorées. On propose aussi une généralisation multivariée des graphes distance-réguliers, et on montre que ceux-ci sont en correspondance avec des schémas \(P\)-polynomiaux multivariés. Finalement, on étudie une sous-classe de paires de Leonard de rang 2 qui font intervenir des polynômes bivariés factorisés. / This thesis is divided in four parts concerning centralizers of quantum algebras \(U_q(\mathfrak{sl}_N)\), biorthogonal polynomials with bispectral properties, bivariate Griffiths polynomials, and association schemes with bivariate polynomial structures. The main topic relating all these parts is the Askey–Wilson algebra. In the first part, the main idea is to combine the braid group algebra with the Askey–Wilson algebra in situations involving the centralizers of the quantum algebra \(U_q(\mathfrak{sl}_2)\). Hence, we obtain representations of the braid group in terms of \(q\)-Racah orthogonal polynomials using \(R\)-matrices of \(U_q(\mathfrak{sl}_2)\), we obtain an interpretation of the Askey–Wilson algebra in the framework of Chern–Simons topological quantum field theory with gauge field \(SU(2)\) as well as in the framework of link invariants associated to \(U_q(\mathfrak{su}_2)\), and we provide a complete algebraic description of the centralizer of \(U_q(\mathfrak{sl}_2)\) in the tensor product of three identical irreducible representations of any spin. In a different perspective, we also provide an algebraic presentation of some fused Hecke algebras, which describe some centralizers of \(U_q(\mathfrak{sl}_N)\). In the second part, we study two families of biorthogonal polynomials using algebraic methods, hence extending the picture that exists for the classical orthogonal polynomials of the Askey–Wilson type. The two families that we consider are the \(R_I\) polynomials of Hahn type and the Pastro polynomials. In both cases, the idea is to introduce a triplet of operators with tridiagonal actions and obtain the polynomials as solutions of two generalized eigenvalue problems involving this triplet. We find the bispectrality and biorthogonality properties of the polynomials using the operators of the triplet, and we determine the algebra realized by the operators. In the third part, we characterize two families of bivariate Griffiths polynomials. The first family is a generalization of the Griffiths polynomials of Krawtchouk type which depends on a parameter \(\lambda\). We find their bispectrality relations and their biorthogonality by using the properties of univariate Krawtchouk polynomials. The contiguity relations of the univariate polynomials play a key role in the computations. We use similar methods to characterize the second family, which is formed by Griffiths polynomials of Racah type. These are orthogonal. In the fourth part, we propose a bivariate generalization of the \(P\)- and \(Q\)-polynomial properties of association schemes and related concepts. Several examples of schemes satisfying the bivariate \(P\)-polynomial property are obtained. We show that the non-binary Johnson schemes and their \(q\)-deformed analogs, the schemes based on attenuated spaces, are bivariate \(P\)- and \(Q\)-polynomial by studying the bispectral properties of the associated bivariate polynomials. The algebraic structures related to these schemes are explored. We also propose a multivariate generalization of distance-regular graphs, and we show that these are in correspondence with multivariate \(P\)-polynomial schemes. Finally, we study a subclass of rank 2 Leonard pairs involving factorized bivariate polynomials.