2013-2014 Seminar Series
Student presentations given during the 2013-2014 academic year. All talks are at noon in Hurst 101 (some exceptions apply) Each year senior mathematics majors participate in a seminar series. The students are responsible for learning about an advanced mathematics topic not generally covered in the curriculum. Each student has a faculty adviser, writes a paper, and gives a presentation. The presentations are open to all.
Modeling Sickle Cell Anemia: Genetic Distributions in a Population
I will be looking at mathematical models of allele frequencies and their resulting genotypes and phenotypes in a population. Beginning with the Hardy-Weinberg model, we will build upon it to show the complexity of real life models using difference equations and understanding stability of fixed points. These concepts will be applied to sickle cell anemia, which is one of few diseases to be positively selected for, to better understand how it will exist in a population.
Stabilizing an Inverted Pendulum
The inverted pendulum on a cart is a classic example of an optimal control theory problem. I will be deriving the equations of motion and investigating stabilization of the system using control theory techniques.
Nifty Matrix Decompositions
A matrix and a complex number are often thought of as two very different ideas. However, certain matrix factorizations demonstrate that matrices are quite similar to complex numbers. This uncanny connection produces some insightful results, which we will explore theoretically and visualize with illuminating pictures.
Designing a Recommender System
Using collaborative filtering techniques, I have designed a miniature recommender system, inspired by the Pandora system. There are two types of collaborative filtering strategies: user-based filtering and item-based filtering; depending on the responses from users, one is more advantageous than the other. The algorithm produced from these varying strategies computes how closely related the target song is to the previously rated song. By incorporating varying techniques and attributes of music, a hybrid similarity calculation can be made resulting in an even stronger recommender system!
Towers of Hanoi: a game of minimums
The Tower of Hanoi puzzle is a game that involves 3 pegs and 7 disks.
The goal is to get all 7 disks from one peg to another moving 1 disk at a time and not putting a large disk on a small one. A minimum number of moves can be achieved for 3 pegs. This game is a classic example of a simple game with certain constraints but when the constraints are loosened (adding a peg) becomes very hard. In this talk we discuss the graphs associated with the Tower of Hanoi and generalized Tower of Hanoi puzzles in hopes of getting insight in to why adding more pegs makes the minimization problem so hard to solve.
Detecting Centromere Boundaries with a Hidden Markov Model
Hidden Markov models are used in many machine learning problems to find the optimal explanation for given phenomenon. We assume that the phenomenon is produced by Markov model which is hidden from direct observation. The states of the Markov model emit characteristic patters which are observable. The problem is to find the set of states through which the model moves which best explains the observed emissions. In this talk we develop and implement a hidden Markov model for detecting centromere boundaries.
Information Flow in a Contact Network
The effort to employ mathematical methods to understand biological systems has undergone a transformation with combined recent advances in computational methods and biological datasets. We will explore the use of contact networks to model cellular interactions as information flow.
Following the work of mathematicians at the National Institutes of Health, we will examine current techniques for developing biologically relevant analyses of large networks of molecular interactions to illuminate the underlying structure of these systems.