## Spring 2019 Graduate Student Seminar

Meeting Tuesdays at 12:00pm in room WXLR A202 unless otherwise specified.

 Date Speaker Title & Abstract (click the title to expand) 01/15/2019 Joseph Wells *Hybrid subgroups of complex hyperbolic lattices *Room WXLR A546In the 1980’s Gromov and Piatetski-Shapiro presented a technique called ”hybridization” wherein one starts with two arithmetic hyperbolic lattices and uses them to produce new hyperbolic lattices (and notably, nonarithmetic lattices). It has been asked whether there exists an analogous hybridization technique for complex hyperbolic lattices. In this short talk I’ll present a potential candidate hybridization technique and some recent results for both arithmetic and nonarithmetic lattices in $\operatorname{PU}(2,1)$. Some of this is joint work with Julien Paupert. 01/22/2019 Phillip Doi *Stratified Formulas and NF Set Theory *Room WXLR A546Proposed in 1937 by Willard Van Orman Quine, New Foundations Set Theory (NF) is a theory of sets that allows for the formulation of a universal set as well as unconventional orderings via the membership relation. Because of these features, it deviates from traditional set theory orthodoxy where the universe of sets forms a cumulative hierarchy. Unlike the more ubiquitous Zermelo-Fraenkel-Choice theory (ZFC), the NF is a succinct system, similar to the systems of Cantor and Frege, which were shown to be inconsistent via Russell’s Paradox. In this talk, I will discuss how mathematics might be done within NF and demonstrate how it presumably avoids Russell’s Paradox via formula stratification. Finally, a few mathematical arguments for and against NF will be mentioned. 01/29/2019 David Polletta A Presentation for $\operatorname{PU}(2,1;\mathcal{O}_2)$ There has been a great deal of study devoted to discrete subgroups and lattices in semisimple Lie groups. In particular, the study of arithmetic lattices, which can be roughly described as lattices obtained by taking matrices with entries lying in the integer ring of some number field. Julien Paupert and Alice Mark devised a general method for computing presentations for any cusped hyperbolic lattice, $\Gamma$, by applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. In this talk, I will discuss my application of their method to obtain a presentation for the Picard modular group, $\operatorname{PU}(2,1;\mathcal{O}_2)$ 02/05/2019 Mary Cook An introduction to Riemannian holonomy In this talk, I will define the Riemannian holonomy group and describe some of the major results surrounding it. Time permitting, I will also discuss how this group behaves under the Ricci flow. 02/12/2019 02/19/2019 02/26/2019 03/12/2019 Wendy Caldwell Differential equation models for understanding phenomena beyond experimental capabilities Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ODE and PDE models for two problems: Vicodin abuse and impact cratering. The prescription opioid Vicodin is the nation’s most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I analyze two mathematical models of Vicodin use and abuse to demonstrate their biological relevance. I prove that solutions to each model are non-negative and bounded, and I show that each model has a unique solution. I also derive conditions for which these solutions are asymptotically stable. Verification and Validation are necessary processes to ensure accuracy of computational methods used to solve problems key to vast numbers of applications and industries. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the FLAG hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes. Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also indicate that Psyche is likely metallic and porous. 03/19/2019 03/26/2019 Sami Brooker Title TBD Abstract TBD 04/02/2019 Mary Cook Title TBD Abstract TBD 04/09/2019 Chelsea Kennedy Title TBD Abstract TBD 04/16/2019 Lauren Crider Title TBD Abstract TBD 04/23/2019 Nikolay Krantsevich Title TBD Abstract TBD