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University of Texas – Austin

1. Moss, Gilbert Samuel. Interpolating gamma factors in families.

Degree: PhD, Mathematics, 2015, University of Texas – Austin

URL: http://hdl.handle.net/2152/31509

In this thesis, we extend the results of Jacquet, Piatetski-Shapiro, and Shalika [JPSS83] to construct interpolated local zeta integrals and gamma factors attached to families of admissible generic representations of GL[subscript n](F) where F is a p-adic field. Our families are parametrized by the spectrum of an ℓ-adic coefficient ring where ℓǂp. To show the importance of gamma factors, we prove a converse theorem in families, which says that suitable collections of interpolated gamma factors of pairs uniquely determine a family of representations, up to supercuspidal support. To prove the converse theorem we re-prove a classical vanishing Lemma, originally due to Jacquet and Shalika, in the setting of families. This is done by extending the geometric methods of Bushnell and Henniart to families, via Helm's theory of the integral Bernstein center.
*Advisors/Committee Members: Helm, David, doctor of mathematics (advisor), Voloch, José Felipe (advisor), Ben-Zvi, David (committee member), Ciperiani, Mirela (committee member), Schedler, Travis (committee member).*

Subjects/Keywords: Local Langlands; Families; Whittaker; Gamma factor; P-adic

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Moss, G. S. (2015). Interpolating gamma factors in families. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/31509

Chicago Manual of Style (16^{th} Edition):

Moss, Gilbert Samuel. “Interpolating gamma factors in families.” 2015. Doctoral Dissertation, University of Texas – Austin. Accessed August 04, 2020. http://hdl.handle.net/2152/31509.

MLA Handbook (7^{th} Edition):

Moss, Gilbert Samuel. “Interpolating gamma factors in families.” 2015. Web. 04 Aug 2020.

Vancouver:

Moss GS. Interpolating gamma factors in families. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2015. [cited 2020 Aug 04]. Available from: http://hdl.handle.net/2152/31509.

Council of Science Editors:

Moss GS. Interpolating gamma factors in families. [Doctoral Dissertation]. University of Texas – Austin; 2015. Available from: http://hdl.handle.net/2152/31509

University of Texas – Austin

2. Orem, Hendrik Nikolas. Coordinate systems and associative algebras.

Degree: PhD, Mathematics, 2015, University of Texas – Austin

URL: http://hdl.handle.net/2152/31505

This dissertation applies and extends the techniques of formal algebraic geometry in the setting of certain "smooth" associative algebras and their globalizations, noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a principal bundle for an appropriate group of local coordinate changes. These bundles are shown to carry a natural flat connection with properties analogous to the classical Gelfand-Kazhdan structure. Every noncommutative manifold has an underlying smooth variety given by abelianization. A basic question is existence and uniqueness of noncommu- tative thickenings of a smooth variety, i.e., finding noncommutative manifolds abelianizing to a given smooth variety. We obtain new results in this direction by showing that noncommutative coordinate systems always arise as reductions of structure group of the commutative bundle of coordinate systems on the underlying smooth variety; this also explains a relationship between D-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf. The lower central series invariants M[subscript k] of an associative algebra A are the two-sided ideals generated by k-fold nested commutators; the M[subscript k] give a decreasing filtration of A. We study the relationship between the geometry of X = Spec A[subscript ab] and the associated graded components N[subscript k] of this filtration. We show that the N[subscript k] form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. We then construct the N[subscript k] in terms of the bundle of coordinate systems on X and the N[subscript k] invariants for the free associative algebra; in particular, since this is independent of A, we exhibit the N[subscript k] as natural vector bundles on the category of smooth schemes.
*Advisors/Committee Members: Ben-Zvi, David, 1974- (advisor), Freed, Daniel (committee member), Hadani, Ronny (committee member), Nadler, David (committee member), Nevins, Thomas (committee member), Schedler, Travis (committee member).*

Subjects/Keywords: Noncommutative algebra; Algebraic geometry

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Orem, H. N. (2015). Coordinate systems and associative algebras. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/31505

Chicago Manual of Style (16^{th} Edition):

Orem, Hendrik Nikolas. “Coordinate systems and associative algebras.” 2015. Doctoral Dissertation, University of Texas – Austin. Accessed August 04, 2020. http://hdl.handle.net/2152/31505.

MLA Handbook (7^{th} Edition):

Orem, Hendrik Nikolas. “Coordinate systems and associative algebras.” 2015. Web. 04 Aug 2020.

Vancouver:

Orem HN. Coordinate systems and associative algebras. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2015. [cited 2020 Aug 04]. Available from: http://hdl.handle.net/2152/31505.

Council of Science Editors:

Orem HN. Coordinate systems and associative algebras. [Doctoral Dissertation]. University of Texas – Austin; 2015. Available from: http://hdl.handle.net/2152/31505

3. Zhu, Yuecheng. Compactification of moduli spaces and mirror symmetry.

Degree: PhD, Mathematics, 2015, University of Texas – Austin

URL: http://hdl.handle.net/2152/31508

Olsson gives modular compactifications of the moduli of toric pairs and the moduli of polarized abelian varieties A [subscript g,δ] in (Ols08). We give alternative constructions of these compactifications by using mirror symmetry. Our constructions are toroidal compactifications. The data needed for a toroidal compactification is a collection of fans. We obtain the collection of fans from the Mori fans of the minimal models of the mirror families. Moreover, we reinterpretate the compactification of A [subscript g,δ] in terms of KSBA stable pairs. We find that there is a canonical set of divisors S(K₂) associated with each cusp. Near the cusp, a polarized semiabelic scheme (X, G, L) is the canonical degeneration given by the compactification if and only if (X, G, Θ) is an object in A P [subscript g,d] for any Θ ∈ S(K₂). The two compactifications presented here are a part of a general program of applying mirror symmetry to the compactification problem of the moduli of Calabi–Yau manifolds. This thesis contains the results in (Zhu14b) and (Zhu14a).
*Advisors/Committee Members: Keel, Seán (advisor), Hacking, Paul (committee member), Schedler, Travis (committee member), Perutz, Timothy (committee member), Neitzke, Andrew (committee member).*

Subjects/Keywords: Compactificaiton; Moduli space; Abelian varieties; Toric pairs; KSBA stable pairs; Mirror symmetry; Toroidal compactification

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Zhu, Y. (2015). Compactification of moduli spaces and mirror symmetry. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/31508

Chicago Manual of Style (16^{th} Edition):

Zhu, Yuecheng. “Compactification of moduli spaces and mirror symmetry.” 2015. Doctoral Dissertation, University of Texas – Austin. Accessed August 04, 2020. http://hdl.handle.net/2152/31508.

MLA Handbook (7^{th} Edition):

Zhu, Yuecheng. “Compactification of moduli spaces and mirror symmetry.” 2015. Web. 04 Aug 2020.

Vancouver:

Zhu Y. Compactification of moduli spaces and mirror symmetry. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2015. [cited 2020 Aug 04]. Available from: http://hdl.handle.net/2152/31508.

Council of Science Editors:

Zhu Y. Compactification of moduli spaces and mirror symmetry. [Doctoral Dissertation]. University of Texas – Austin; 2015. Available from: http://hdl.handle.net/2152/31508