Students and Faculty Dive into Summer Research
John Shamshoian
Math major John Shamshoian investigated techniques to predict cardiac alternans at the National Institute for Mathematical and Biological Synthesis (NIMBioS) in Knoxville, Tenn. Alternans are an unstable, beat-to-beat alternation in action potential duration that lead to ventricular fibrillation. Ventricular fibrillation is a fatal arrhythmia and is a precursor to sudden cardiac arrest.
Unfortunately, specifying a model to fully describe cardiac dynamics may be impossible. Furthermore, the full state space may not be physically measured. However, action potential duration can be measured easily.
To take advantage of this, researchers have developed a state space reconstruction technique by solely analyzing action potential duration. Shamshoian, along with two mentors and three fellow researchers, expanded the technique by introducing random disturbances to the pacing rate. The new technique is more robust, rendering it more suitable for experimental analyses.
Eric Brussel with Anthony Kling and Zach Straus
Anthony Kling and Zach Straus completed summer research with Professor Eric Brussel on the subject of the arithmetic of projective conics. They found right away that even the definition of a simple object such as a conic was problematic over a field that was not algebraically closed, but that this was easily fixed using the concept of the functor of points. They then easily computed the fine moduli space of projective conics and identified the set of degenerate conics as a closed subset.
They subsequently took up the classification problem, first proving that all conics with a rational point are projectively equivalent. They noted that a fine moduli space could not exist for the projective equivalence classes of conics, even over C, primarily because of the existence of continuous families of conics that included both degenerate and nondegenerate specimens. The research experience culminated in the classification of conics up to projective equivalence over the fields C, R, the p-adic field Q_p, and finally the rational field Q (using the famous Hasse-Minkowski theorem). Kling presented the results at the fall 2014 Nevada-Southern California Mathematics Association of America meeting at Pomona College.
Emily Hamilton with Michael Campbell
Knot theory, which is over a century old, is an active area of modern mathematics. The study of knots has led to important applications in DNA research and the synthesis of new molecules. Moreover, knot theory is having a significant impact on statistical mechanics and quantum field theory.
In a series of papers, mathematician Colin Adams studied multi-crossing numbers of knots and links. He produced results and examples for triple and quadruple crossing numbers. However, there are no examples in the literature of knots or links with known 5 - crossing numbers. In this project, we created four tangle moves, which allowed us to move from double crossings to a 5 - crossing. We then used these moves to compute the 5 - crossing numbers for a collection of knots and links.
Danielle Champney with David Kato, Jordan Spies and Kelsea Weber
Professor Danielle Champney worked with undergraduate students David Kato, Jordan Spies and Kelsea Weber on a summer research project studying students' perceptions of their different approximation strategies and their ideas about how those strategies are or are not appropriate in physics, mathematics and chemistry contexts. They analyzed existing data with a developed coding scheme to highlight the ways that students' epistemological framing tied to their ideas about the appropriateness of their strategies. Additionally, the research team designed a second phase of the study and collected data for it during fall 2014.
They presented their summer research results, titled "Students' perceptions of the disciplinary appropriateness of their approximation strategies," at the Research in Undergraduate Mathematics Education Conference in February 2015. Spies and Weber intend to continue working with the second phase data for their senior projects.
Erin Pearse with Chad Eckman, Jon Lindgren and Zach Zhang
Professor Erin Pearse worked with students Chad Eckman, Jon Lindgren and Zach Zhang on machine learning problems related to function interpolation. Using techniques from applied linear algebra and numerical optimization, the team was able to implement an algorithm that can reconstruct damaged data sets (or recover lost data) with remarkable accuracy, even when up to 70% of the data has been destroyed.
The basic strategy is to transform the data set into a network by assigning similarity values to pairs of data points, and then study the geometry of the resulting network. If the data is amenable (technically: if the data has the form of a random sample taken from a low-dimensional submanifold of a high-dimensional Euclidean space), then the network provides a greatly simplified model of the data that retains the essential geometric structure of the original data set.
Their technique has applications to statistics and scientific data analysis, as well as image and video reconstruction and security/military applications. A patent has been filed, and the report is in review with a scientific journal.
Matthew White with Warren (Michael) Shultz
Warren (Michael) Shultz and Professor Matthew White worked on one of the approaches to deciding if the Burau representation of the 4-strand braid group is faithful. Burau's original paper appeared in 1936, so this problem is about 80 years old. It’s been known for a long time that the 3-strand representation is faithful, and in the late 1990s it was shown that for n > 4, the representation is not faithful. This leaves the 4-strand question open.
One approach to the 4-strand Burau representation uses topological methods by viewing the representation as a map on the first homology of a certain covering space of the 4-punctured disc. Michael used graph-theoretic methods to try to understand the behavior of particular arcs in the 4-punctured disc, the existence of which would provide an element in the Burau representation's kernel.
Charles Camp with Andrew Gallatin, Ryan Smith, Katelyn Veyna, Darren Marotta, Camille Vernon and Benjamin Brown
Cal Poly undergraduates Andrew Gallatin, Ryan Smith, Katelyn Veyna, Darren Marotta, Camille Vernon and Benjamin Brown worked with Professor Charles Camp on the analysis and validation of conceptual climate models for the Pleistocene climate. There are a great variety of such models, with different underlying physical assumptions, which attempt to explain the observed climate variability. These projects focused on using modern time-series analysis techniques to compare the model outputs to the analysis of paleoclimate data records in order to better determine the consistency of the models with the empirical records. Research on this project is continuing during the 2014-15 academic year with support from the National Science Foundation via the Mathematics and Climate Research Network.
