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Diving Deeper Into Supercuspidal Representations, Prerna Agarwal Jul 2024

Diving Deeper Into Supercuspidal Representations, Prerna Agarwal

LSU Doctoral Dissertations

In 2013, Reeder and Yu introduced certain low positive depth supercuspidal representations of $p$-adic groups called \textit{epipelagic} representations. These representations generalize the simple supercuspidal representations of Gross and Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals appear in the automorphic data corresponding to the Kloosterman $l$-adic sheaf. In this thesis, we take a first step towards the construction of ``\textit{mesopelagic} representation (of Iwahori type)'' which are the higher depth analogues of simple supercuspidal representations. We see that these constructions can be done in a similar way …


Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain Jul 2024

Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain

LSU Doctoral Dissertations

The goal of this work is to develop an asymptotic formula for the behavior of a scattered electromagnetic field in the presence of a thin metamaterial known as a metasurface. By using a carefully chosen Green’s function and the single and double layer potentials we analyze the perturbed scattering problem in the presence of the metamaterial and a background scattering problem. By using Lippman-Schwinger type representation formulas for the two fields we develop the asymptotic formula for the perturbed field. From here we prove the asymptotic formula holds up to a specific error term based on the size of the …


The Lowest Discriminant Ideal Of Cayley-Hamilton Hopf Algebras, Zhongkai Mi Apr 2024

The Lowest Discriminant Ideal Of Cayley-Hamilton Hopf Algebras, Zhongkai Mi

LSU Doctoral Dissertations

Discriminant ideals are defined for an algebra R with central subalgebra C and trace tr : R → C. They are indexed by positive integers and more general than discriminants. Usually R is required to be a finite module over C. Unlike the abundace of work on discriminants, there is hardly any literature on discriminant ideals. The levels of discriminant ideals relate to the sums of squares of dimensions of irreducible modules over maximal ideals of C containing these discriminant ideals. We study the lowest level when R is a Cayley-Hamilton Hopf algebra, i.e. C is also a Hopf subalgebra, …


Reducibility Of Schrödinger Operators On Multilayer Graphs, Jorge Villalobos Alvarado Apr 2024

Reducibility Of Schrödinger Operators On Multilayer Graphs, Jorge Villalobos Alvarado

LSU Doctoral Dissertations

A local defect in an atomic structure can engender embedded eigenvalues when the associated Schrödinger operator is either block reducible or Fermi reducible, and having multilayer structures appears to be typically necessary for obtaining such types of reducibility. Discrete and quantum graph models are commonly used in this context as they often capture the relevant features of the physical system in consideration.

This dissertation lays out the framework for studying different types of multilayer discrete and quantum graphs that enjoy block or Fermi reducibility. Schrödinger operators with both electric and magnetic potentials are considered. We go on to construct a …


Finite Monodromy And Artin Representations, Emma Lien Apr 2024

Finite Monodromy And Artin Representations, Emma Lien

LSU Doctoral Dissertations

Artin representations, which are complex representations of finite Galois groups, appear in many contexts in number theory. The Langlands program predicts that Galois representations like these should arise from automorphic representations and many examples of this correspondence have been found such as in the proof of Fermat's Last Theorem. This dissertation aims to make an analysis of explicitly computable examples of Artin representations from both sides of this correspondence. On the automorphic side, certain weight 1 modular forms have been shown to be related to Artin representations and an explicit analysis of their Fourier coefficients allows us to identify the …


The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta Apr 2024

The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

LSU Doctoral Dissertations

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …


Subroups Of Coxeter Groups And Stallings Foldings, Jake A. Murphy Apr 2024

Subroups Of Coxeter Groups And Stallings Foldings, Jake A. Murphy

LSU Doctoral Dissertations

For each finitely generated subgroup of a Coxeter group, we define a cell complex called a completion. We show that these completions characterizes the index and normality of the subgroup. We construct a completion corresponding to the intersection of two subgroups and use this construction to characterize malnormality of subgroups of right-angled Coxeter groups. Finally, we show that if a completion of a subgroup is finite, then the subgroup is quasiconvex. Using this, we show that certain reflection subgroups of a Coxeter are quasiconvex.


Analytic Wavefront Sets Of Spherical Distributions On The De Sitter Space, Iswarya Sitiraju Apr 2024

Analytic Wavefront Sets Of Spherical Distributions On The De Sitter Space, Iswarya Sitiraju

LSU Doctoral Dissertations

In this work, we determine the wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space. Let G′ = O1,n(R) be the Lorentz group, and let H′ = O1,n−1(R) ⊂ G′ be its subset. The de Sitter space dSn is a one-sheeted hyperboloid in R1,n isomorphic to G′/H′. A spherical distribution is an H′-invariant eigendistribution of the Laplace-Beltrami operator on dSn. The space of spherical distributions with eigenvalue λ, denoted by DλH'(dSn), has dimension 2. We construct a basis for the space of …