*Room WXLR A546

In 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.

*Room WXLR A546

Proposed 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.

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, $\operatorname{PU}(2,1;\mathcal{O}_2)$

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.

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.

In recent years, there has been some interest in $C^*$-algebras arising from certain oriented combinatorial data, such as directed graphs, subsemigroups of discrete groups, and various generalizations of these objects. The idea of using a directed graph in order to define a $C^*$-algebra began with a construction of Cuntz and Krieger in the 80's, and was itself based on shifts of finite type in symbolic dynamics. The use of a semigroup goes back to a theorem of Coburn characterizing the Toeplitz algebra as the $C^*$-algebra of the subsemigroup $\mathbb{N} \subseteq \mathbb{Z}$. The present talk is concerned with this construction in the case of naturally defined semigroups in $\mathbb{Z}^n$, namely the {\it simplicial cones}. There is a rich proliferation of such semigroups even in the simplest case of $\mathbb{Z}^2$.

Ricci flow is a technique introduced by Hamilton in the 1980's, and it allows one to deform various smooth manifolds while still retaining certain important geometric properties. Famously, Perelman utilized Ricci flow in his proof of Thurston's geometrization conjecture (from which the Poincare conjecture followed as a corollary). If that isn't enough to convince you that Ricci flow is basically the coolest thing ever, you should come to this talk and find out more about it.

Everything I know isn’t much. I promise. This will be an introduction to Morse theory (note: different from Morse code!). In a sentence, Morse theory allows us to study the topology of manifolds through the behavior of critical points of a given function (you remember critical points from good ol' days of Calc I!). Morse theory is a sweet algebraic-topological tool used most recently as a foundation for so-called Topological Data Analysis problems, i.e., what is the shape of my data? The plan of this talk is to start at ground zero with some basic definitions, include a few theorems, and touch a bit on Morse homology, with heavy focus on examples.

A computerized adaptive test (CAT) is a test with questions chosen real-time, based on the examinee's prior responses; the goal is to measure the relevant trait of interest more precisely, using significantly fewer questions. Computerized adaptive tests have traditionally been designed using item response theory (IRT), where the questions are selected to maximize information gain. However, real time computation of information gain based on Fisher information can be costly. An alternative approach for CAT design involves fitting a single tree to training data (question responses combined with known exam scores), where each node of the tree consists of an exam question and response threshold; taking the CAT amounts to traversing the tree, with the score stored in the leaf nodes. The single-tree approach has an intuitive appeal and is computed only once, thus drastically reducing computational burden while administering the test. However, a single tree fit to data can suffer from lack of stability with respect to small variations in the training data.

Our present research explores a three step approach for fitting the single tree that represents the CAT. First, we use an ensemble of trees (several methods are compared) to fit the exam score as a function of the question responses to the training data described above. Second, a large set of synthetic data is generated, consisting of fictitious question responses and an exam score predicted using the function in the first step. Finally, a single tree is fit to this synthetic data using Brieman et al's Classification and Regression Trees algorithm. This three-step approach provides a more accurate CAT than a single tree fit directly to the data, while maintaining the ease of interpretability and reduced real-time computational cost.