## Spring 2019 Graduate Student Seminar

Meeting Tuesdays at 3:00pm in room WXLR A202.

Date |
Speaker |
Title & Abstract (click the title to expand) |

01/15/2019 | Joseph Wells |
*Room WXLR A546 |

01/22/2019 | Phillip Doi |
*Room WXLR A546 |

01/29/2019 | David Polletta |
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 |
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. |

03/12/2019 | Wendy Caldwell |
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. |

03/26/2019 | Sami Brooker |
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 $\mathbb{Z}^2$. |

04/02/2019 | Mary Cook |
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. |

04/16/2019 | Lauren Crider |
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. |

04/23/2019 | Chelsea Kennedy |
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. |