Mathematics Theses and Dissertations
https://scholarsbank.uoregon.edu/xmlui/handle/1794/2044
2024-04-24T05:26:24ZA SPECIAL ENDOMORPHISM OF THE STANDARD GAITSGORY CENTRAL OBJECT OF THE AFFINE HECKE CATEGORY
https://scholarsbank.uoregon.edu/xmlui/handle/1794/29277
A SPECIAL ENDOMORPHISM OF THE STANDARD GAITSGORY CENTRAL OBJECT OF THE AFFINE HECKE CATEGORY
Hathaway, Jay
Using the combinatorial description of the standard Gaitsgory centralobject of the (extended, graded) affine type A Hecke category due to Elias, we
show the existence of and explicitly describe the unique endomorphism that lifts
right multiplication by the i-th fundamental weight on the i-th component of
the associated graded of its Wakimoto filtration. We give work in progress on
describing a conjectural program to categorify the Vershik-Okounkov approach
to the representation theory of the affine Hecke algebra. Here this endomorphism
will play a role. This is the affine version of the program described by Gorsky, Negut, and Rasmussen in finite type A.
2024-03-25T00:00:00ZScalar Curvature and Transfer Maps in Spin and Spin^c Bordism
https://scholarsbank.uoregon.edu/xmlui/handle/1794/29212
Scalar Curvature and Transfer Maps in Spin and Spin^c Bordism
Granath, Elliot
In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension5 or greater, the vanishing of a particular invariant α is necessary and sufficient
for the existence of a metric of positive scalar curvature. More precisely, there is
a map α: ΩSpin → ko (which may be realized as the index of a Dirac operator) ∗
which Hitchin established vanishes on bordism classes containing a manifold with
a metric of positive scalar curvature. Stolz showed kerα is the image of a transfer
map ΩSpinBPSp(3) → ΩSpin. In this paper we prove an analogous result for Spinc- ∗−8 ∗
manifolds and a related invariant αc : ΩSpinc → ku. We show that ker αc is the ∗
sum of the image of Stolz’s transfer ΩSpinBPSp(3) → ΩSpinc and an analogous map ∗−8 ∗
ΩSpinc BSU(3) → ΩSpinc . Finally, we expand on some details in Stolz’s original paper ∗−4 ∗
and provide alternate proofs for some parts.
2024-01-10T00:00:00ZC*-Correspondences, Hilbert Bimodules, and their L^p Versions
https://scholarsbank.uoregon.edu/xmlui/handle/1794/29208
C*-Correspondences, Hilbert Bimodules, and their L^p Versions
Delfin Ares de Parga, Alonso
This dissertation initiates the study of $L^p$-modules, which are modules over $L^p$-operator algebras inspired by Hilbert modules over C*-algebras. The primary motivation for studying $L^p$-modules is to explore the possibility of defining $L^p$ analogues of Cuntz-Pimsner algebras.
The first part of this thesis consists of investigating representations of C*-correspondences on pairs of Hilbert spaces. This generalizes the concept of representations of Hilbert bimodules
introduced by R. Exel in \cite{Exel1993}. We present applications of representing a correspondence on a pair of Hilbert spaces $(\Hi_0, \Hi_1)$,
such as obtaining induced representations of both $\Li_A(\X)$ and $\mathcal{K}_A(\X)$ on $\Hi_1$,
and giving necessary and sufficient conditions on
an $(A,B)$ C*-correspondences to admit a Hilbert $A$-$B$-bimodule structure.
The second part is concerned with the theory of $L^p$-modules. Here we present a thorough treatment of $L^p$-modules, including morphisms between them and techniques for constructing new $L^p$-modules. We then useour results on representations for C*-correspondences to motivate and develop the theory of $L^p$-correspondences, their representations, the $L^p$-operator algebras they generate, and present evidence that well-known $L^p$-operator algebras can be constructed from $L^p$-correspondences via $L^p$-Fock representations.
Due to the technicality that comes with dealing with direct sums of $L^p$-correspondences and interior
tensor products, we only focus on two particular examples for which a Fock space construction can be carried out. The first example deals with the $L^p$-module $(\ell_d^p, \ell_d^q)$, for which we exhibit a covariant $L^p$-Fock representation that yields an $L^p$-operator algebra isometrically isomorphic to $\mathcal{O}_d^p$, the $L^p$-analogue of the Cuntz-algebra $\mathcal{O}_d$ introduced by N.C. Phillips in \cite{ncp2012AC}. The second example involves a nondegenerate $L^p$-operator algebra $A$ with a bicontractive approximate identity together with an isometric automorphism $\varphi_A \in \op{Aut}(A)$. In this case, we also present an algebra associated to a covariant $L^p$-Fock representation, but due to the current lack of knowledge of universality of the $L^p$-Fock representation, we only show that there is a contractive map from the crossed product $F^p(\Z, A, \varphi_A)$ to this algebra.
This dissertation includes unpublished material.
2024-01-10T00:00:00ZStructures and Computations in Annular Khovanov Homology
https://scholarsbank.uoregon.edu/xmlui/handle/1794/29197
Structures and Computations in Annular Khovanov Homology
Davis, Champ
Let $L$ be a link in a thickened annulus. In [GLW17], Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $\exsltwo$, the exterior current algebra of the Lie algebra $\sltwo$. In this dissertation, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $\exsltwo$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $\exsltwo$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves.
In proving the above result, we include explicit formulas to compute the higher $L_\infty$-operations. Additionally, given a morphism $I: L' \to L$ of $L_\infty$-algebras, we define a restriction of scalars operation in the setting of $L_\infty$-modules and prove that it defines a functor $I^*: L-mod \to L'-mod$. A more abstract approach to this problem was recently given by Kraft-Schnitzer.
Finally, computer code was written to aid in the study of the above $L_\infty$-module structure. We discuss various patterns that emerged from these computations, most notably one relating the torsion in the annular Khovanov homology groups and the location of the inner boundary of the annulus.
2024-01-09T00:00:00Z