2019 Fall Colloquium and Teaching Seminar Schedule |
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September 18 | Teaching Seminar | |

September 25 | Colloquium | Dr. Xiyuan Liu (University of Central Florida) |

October 02 | Teaching Seminar | |

October 09 | Colloquium | Jake Fillman (Texas State University) |

October 16 | Colloquium | Naomi Krawzik (University of North Texas) |

October 23 | Colloquium | Felix Gotti (University of Florida) |

October 25 | Colloquium | Marly Gotti (University of Florida) |

October 30 | Colloquium | Doug Klein (Texas A&M - Galveston) |

November 06 | Colloquium | Jane Long (Stephen F. Austin State University) |

November 13 | Teaching Seminar | |

November 20 | Colloquium | Ram Kafle (Sam Houston State University) |

#### Colloquium November 20

##### Statistical Analysis of Trends using Joinpoint and Functional Regression Approaches

Ram Kafle

Statistical Analysis of Trends is designed to identify any existing patterns in data over time and apply it to predict upwards or downwards shifts. This talk aims to discuss two different statistical methods to identify and predict significant changes in the trends. Firstly, we apply a Bayesian Joinpoint statistical model to study the effect of smoking in the incidence of lung cancer trends. The resulting model estimates and predicts the rate of change of incidence in the time trends with the adjustment of smoking rate in the population along with other applicable covariates. Secondly, we apply the Functional Regression method to develop a differential equation to study the trends. In this approach, a differential operator is defined as a data smoother and we apply the penalized least square fitting criteria to smooth the data. The profile error sum of the square is optimized to estimate the differential operator using functional regression. The solution of the developed differential equation estimates and predicts the trends. We apply this model to estimate and predict the rate of change of carbon dioxide levels in the atmosphere. Finally, we compare these two methods and propose some possible future works.

#### Colloquium November 6

##### Valuations of Sequences Generated by Polynomials

Jane Long

Sequences of integers can be generated in many different ways, one of which is by evaluating a polynomial with non-negative integer coefficients at the natural numbers 1,2,3,... Interesting questions arise when we consider the prime decompositions of those outputs: is a particular prime *p* ever a factor? Always a factor? Is there a maximum power of *p* that divides some output? What impact do the characteristics of the polynomial have? These questions can all be framed in terms of the *p-adic valuation* of a natural number *n*, denoted *v _{p}(n)*, which is defined to be the exponent of the highest power of

*p*that divides

*n*. We will examine some surprisingly varied behavior that occurs for even simple (and similar) polynomial functions. Very little background is required or assumed, and this talk will be accessible to a general audience.

#### Colloquium October 30

##### Conjugated-Carbon Nano-Structures

Doug J. Klein

Three times in the last few decades Nobel prizes have been awarded for research on different novel conjugated-carbon nano-structures: fullerenes, linear polymers, & graphene. And conjugated-carbon nano-tubes have been studied, with no Nobel prize, yet. Intense interest in conjugated-carbon nano-structures has resulted. Natural questions arise: Why “Nano”? Why Carbon? Why “conjugated”? Why hexagon-rich? And what’s so special as viewed from fundamental science, molecular biology, eco-environmental science, & nano-engineering?

Here a main point is to indicate some relevant mathematical graph theory, particularly for subgraphs of “graphene” (the honeycomb net) or of “bucky-tubes” (honeycomb-based tubes). Besides a diversity of structures, attention is directed to the adjacency matrix eigen-spectrum as an indicator of the quantum-mechanical electronic energy levels. Within this eigen-spectral focus is more particularly directed to the “HOMO- LUMO gap”, which measures the paucity of levels for the least tightly bound electrons (occupying levels in the middle of the eigen-spectrum). Of special interest is this gap under the influence of decorations (or defects), such as we address with a theorematic approach.

#### Colloquium October 25

##### Applicability Domain in Data Science

Marly Gotti

There are times when a model's prediction should be taken with some skepticism. For example, if a new data point is substantially different from the training set, its predicted value may be suspect. In chemistry, it is not uncommon to create an "applicability domain" model that measures the amount of potential extrapolation new samples have from the training set. The methods used to define the applicability domain of a model can be applied to data sets not necessarily derived from chemistry. Here we describe various methods used to define the applicability domain of any model. In addition, we present the modeling R package applicable, which comprises different methods to measure how much a new data point is an extrapolation from the original data.

#### Colloquium October 23

##### On the elasticity of atomic monoids and domains

Dr. Felix Gotti

Let R be an integral domain. The elasticity of R is an algebraic statistic whose purpose is to measure how far is R from being a unique factorization domain (UFD). With the same purpose, the elasticity of certain classes of commutative monoids can be defined. Here we discuss the elasticity of some relevant classes of monoids and integral domains stemming from number theory, linear programming, and semigroup theory. In particular, we survey some old and recent advances on the elasticity of ring of integers, numerical monoids, and some natural generalizations of these two classes.

#### Colloquium October 16

##### Graded Hecke algebras for the symmetric group in positive characteristic

Naomi Krawzik

We discuss some noncommutative algebras and their deformations. When the characteristic of the underlying field is zero, deformations of skew group algebras have been well studied. For example, every Lusztig graded Hecke algebra is isomorphic to a Drinfeld Hecke algebra as filtered algebras. However, when the characteristic of the field divides the order of the group acting, new deformations arise, like when the symmetric group acts on a polynomial ring in positive characteristic. In this case, Poincaré-Birkhoff-Witt conditions can be used to classify the resulting graded Hecke algebras.

#### Colloquium October 9

##### Spectral properties of quasicrystals

Dr. Jake Fillman

Discovered by D. Shechtman in the early 1980s, quasicrystals are materials whose molecular structure is characterized by aperiodicity (the absence of translation symmetries) and long-range order. The study of these objects involves a beautiful synthesis of many areas of mathematics, including topology, dynamical systems, harmonic analysis, and spectral theory. We will introduce background and discuss operator-theoretic models of quasicrystals. Along the way, we will highlight some of the exotic features of these models.

#### Colloquium September 25

##### Conditional Random Fields with Lasso and its application

Dr. Xiyuan Liu

The classification problem of gene using gene expression level, more specifically, gene expression analysis, is a major research area in statistics. There are several classical methods to solve the classification problem. For example, Logistic Regression (LRM), Support Vector Machine (SVM), and hypothesis testing with Benjamini-Hochberg method. To apply these classical methods, the observations in the dataset should fit the assumption of independence. That is, the observations in the dataset are independent to each other, and the predictor (independent variable) should be independent. These assumptions are usually violated in gene expression analysis. Although the Classical Hidden Markov Chain Model (HMM) can solve the independence of observation problem, the classical HMM requires the independent variables in the dataset are discrete and independent. Unfortunately, the gene expression level is a continuous variable.

To solve the classification problem of Gene Expression Level data, the Conditional Random Field (CRF) is introduced. Finally, the Least Absolute Selection and Shrinkage Operator (LASSO) penalty, a dimensional reduction method, is implied to improve the CRF model.