Abstracts

Graduate Student Talks

The presenters will be:

Undergraduate Research Talks

Carling Sugarman, Let's Learn Topology: Using Topology to Explore Mathematics Education Reform

Mathematics education is a constant topic of conversation in the United States. Many attempts have been made historically to reform teaching methods and improve student results. Particularly, past ideas have emphasized problem-solving to make math feel more applicable and enjoyable. Many have additionally tackled the widespread problem of ''math anxiety'' by creating lessons that are more discussion-based than drill-based to shift focus from speed and accuracy. In my project, I explored past reform goals and some added goals concerning students' perceptions of mathematics. To do so, I created and tested a pilot workshop in topology, a creative and intuitive field, for use in 4th-6th grade classrooms. Preliminary results suggest some success in altering student views on mathematics.

Jackie Mok, Ramsey Theory

The main philosophical idea behind Ramsey Theory is finding ''structure'' in ''randomness.'' We may mathematically formalize these ideas using combinatorial methods. In particular, we will study the natural numbers, and finding "structure" with k-term monocromatic arithmetic progressions.

Brent Allman, DNA and Knot Theory: Topological Applications to Biology

Knot theory is traditionally a field of pure mathematics, stigmatized as''recreational doodling of bored topologists''. However, knot theory can be used to solve biological problems! Enzymes and DNA strands are structures that are topologically equivalent to ''3-ball'' and knots respectively. Tangle addition can be used to determine where on a DNA molecule an enzyme will bind to perform a conformational change to the binding site, and what the end product will be. DeWitt Sumners outlines proofs of these biologically relevant questions using knot theory. The talk will be interactive and informative of the applications of knot theory to biology.

2014 Keynote Speaker

Rachel Levy, Harvey Mudd College

Abstract: Surfactants are chemicals that lower the surface tension of a fluid. This talk will describe experiments that observe the motion of thin fluid films caused by the presence of surfactant as well as differential equations that model this scenario. The talk will be accessible to anyone who has had Calculus. For those who specialize in PDEs, the talk will include a coupled system of fourth- order nonlinear hyperbolic-parabolic partial differential equations. But don't let that scare you. This math could save premature babies!

Want to know what kinds of talks we've had before? See below for the titles of the talks from last years conference.

2013 Keynote Speaker

Crazy quaternions: Why won't they commute? , Amy Irwin Stout, USD

Abstract: Addition, subtraction, multiplication, and division of real numbers motivates most of the math that we learn up through high school. But why not consider these operations for points in 2-space, 3-space, or more? While answering this question, we will define complex numbers and Hamilton quaternions, discuss the motivation behind their creation, and see why they are useful. By the end, we will cover all the normed division algebras over the reals.

2012 Keynote Speaker

The Shape of Space, Diane Hoffoss, USD

2011 Keynote Speaker

The Millenium Problems, Alissa Crans, Loyola Marymount

2010 Keynote Speaker

When Topology Meets Chemistry, Erica Flapan, Pomona College

Abstract: Mirror image symmetry plays an important role in predicting the behavior of molecules. Recently, knots and links and other non-planar molecules have been synthesized which are large enough that they do not have the rigidity that is characteristic of small molecules. In order to understand the symmetries of such molecules we need to understand how they can be deformed. Topology is the area of mathematics that analyzes how geometric objects can be deformed and which properties of such an object will be preserved by deformations. In this talk we will discuss how topology can be used to help us analyze the symmetries of flexible molecules.

2010-2013 Students Presentations

A Comparison and Catalog of Intrinsic Tumor Growth Models, Elizabeth Sarapata

Determining the dynamics and parameter values that drive tumor growth is of great interest to mathematical modelers, experimentalists and practitioners alike. We provide a basis on which to estimate the growth dynamics of ten different tumors by fitting growth parameters to at least five sets of published experimental data per type of tumor. These timescale tumor growth data are also used to determine which of the most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor. In order to compute the best-fit parameters, we implemented a hybrid local-global least squares minimization algorithm based on a combination of Nelder-Mead simplex direct search and Monte Carlo Markov Chain methods.

A Dynamic Programming Approach to Edit Distance Alignment, Qianxue (Amy) Lu

Sequence alignment is important in biology so we can compare similarities in DNA strings and learn more about their evolutionary relationship. Through mutations, DNA strings can deviate from each other. Given two strings, we wish to find the minimal number of edit operations needed to transform one string into the other. To find the optimal alignment, we will look at several algorithms for pairwise alignment which will utilize dynamic programming.

Quadrotor Obstacle Avoidance, Peter Fedak

Microflyers, a class of small, agile unmanned aerial vehicle (UAV), offer remarkable capabilities for mapping and search tasks. Navigating through three-dimensional space and manipulating rotational dynamics demands sophisticated mathematical models and high computational capacity. Autonomous navigation of ground-based robots, by contrast, is much better understood, with mature methods available. The Microsoft KinectŠ and similar devices (as well as extensive hobbyist interest that has encouraged development of small computers that can handle point cloud data) remove a substantial hurdle from the UAV navigation problem. We extend an existing algorithm for ground obstacle avoidance to quadrotors in particular, incorporating the state of the art in several subsidiary areas.

Keeler's Theorem and Products of Distinct Transpositions, Lihua Huang

Abstract: An episode Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community indulges in a mind-switching spree, the question is asked, "Can the switching be undone so as to restore all minds to their original bodies?" Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two ``outsiders". We refine Keeler's result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group $S_n$. Finally, we give necessary and sufficient conditions on m and n for the identity permutation to be expressible as a product of m distinct transpositions in S_n.

The Links Between Smale's Mean Value Conjecture and Convergence. - Hayley Miles-Leighton

Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes - Jacob Miller

Quantum Computation, Marguerite Manela, Scripps College

Pappus' Hexagon Theorem: from Ancient to Projective Geometry, Katherine Ford, Scripps College

A Surprise, Leilani Gilpin, UC San Diego

Coloring of Some Distance Graphs, Aileen Sutedja, CSULA

Poster: Statistical Analysis of Nutritional Food Choices by Nugget(c) Patrons, Tania Miller, Cal State Long Beach