Browsing by Author "Gillespie, Maria, committee member"
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Item Open Access Counting isogeny classes of Drinfeld modules over finite fields via Frobenius distributions(Colorado State University. Libraries, 2024) Bray, Amie M., author; Achter, Jeffrey, advisor; Gillespie, Maria, committee member; Hulpke, Alexander, committee member; Pallickara, Shrideep, committee member; Pries, Rachel, committee memberClassically, the size of an isogeny class of an elliptic curve -- or more generally, a principally polarized abelian variety -- over a finite field is given by a suitable class number. Gekeler expressed the size of an isogeny class of an elliptic curve over a prime field in terms of a product over all primes of local density functions. These local density functions are what one might expect given a random matrix heuristic. In his proof, Gekeler shows that the product of these factors gives the size of an isogeny class by appealing to class numbers of imaginary quadratic orders. Achter, Altug, Garcia, and Gordon generalized Gekeler's product formula to higher dimensional abelian varieties over prime power fields without the calculation of class numbers. Their proof uses the formula of Langlands and Kottwitz that expresses the size of an isogeny class in terms of adelic orbital integrals. This dissertation focuses on the function field analog of the same problem. Due to Laumon, one can express the size of an isogeny class of Drinfeld modules over finite fields via adelic orbital integrals. Meanwhile, Gekeler proved a product formula for rank two Drinfeld modules using a similar argument to that for elliptic curves. We generalize Gekeler's formula to higher rank Drinfeld modules by the direct comparison of Gekeler-style density functions with orbital integralsItem Open Access Determining synchronization of certain classes of primitive groups of affine type(Colorado State University. Libraries, 2022) Story, Dustin, author; Hulpke, Alexander, advisor; Adams, Henry, committee member; Buchanan, Norm, committee member; Gillespie, Maria, committee memberThe class of permutation groups includes 2-homogeneous groups, synchronizing groups, and primitive groups. Moreover, 2-homogeneous implies synchronizing, and synchronizing in turn implies primitivity. A complete classification of synchronizing groups remains an open problem. Our search takes place amongst the primitive groups, looking for examples of synchronizing and non-synchronizing. Using a case distinction from Aschbacher classes, our main results are constructive proofs showing that three classes of primitive affine groups are nonsynchronizing.Item Open Access Quantum Serre duality for quasimaps(Colorado State University. Libraries, 2022) Heath, Levi Nathaniel, author; Shoemaker, Mark, advisor; Cavalieri, Renzo, committee member; Gillespie, Maria, committee member; Gelfand, Martin, committee memberLet X be a smooth variety or orbifold and let Z ⊆ X be a complete intersection defined by a section of a vector bundle E → X. Originally proposed by Givental, quantum Serre duality refers to a precise relationship between the Gromov–Witten invariants of Z and those of the dual vector bundle E∨. In this paper we prove a quantum Serre duality statement for quasimap invariants. In shifting focus to quasimaps, we obtain a comparison which is simpler and which also holds for nonconvex complete intersections. By combining our results with the wall-crossing formula developed by Zhou, we recover a quantum Serre duality statement in Gromov-Witten theory without assuming convexity.