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This research statement contains two parts, one on my interests in networks and complex systems and one on my education related research. The statement can also be downloaded at the bottom of the page.
Complex Systems
Complex systems, and especially networks, have received a growing amount of interest in many fields. Ideas from the field have influenced research in transportation, disease spread, social interaction, neural development and genome assembly just to name a few. Because the research focuses on how multiple components interact I anticipate it continuing to grow in importance as science looks at more complex and holistic systems of many parts. This interaction is where I am particularly fascinated and plan to continue developing new computational algorithms and analytics.
My dissertation research has focused on how best to utilize data describing how pieces of sequence DNA are connected in the assembly process. While techniques have been successfully developed holistically for the entire assembly process very little research has been done on efficient and effective algorithms for many of the individual steps. My dissertation takes the data that describes connections between partial sequencings of the DNA and views it as a network on which computations can be performed. Utilizing the research in networks from computer science and physics I have developed more efficient and accurate algorithms to solve parts of the assembly problem. Due to the network style of the algorithm it is also capable of identifying some of the bad data that exists, further improving assemblies. Additionally, auxiliary data generated in the process has suggested several promising future research avenues I hope to pursue such as applying community finding to the filtering and assembly process.
Previously I have examined several dynamical systems on networks, examining how the information passed (and how it was passed) between nodes affects the evolution of dynamics on, and communication between, nodes. This research focused on how chaos can develop in a network and how synchronization between chaotic systems could occur. Rather than developing algorithms as in the genomics research, this research involved optimizing differential equations solvers and simulating the evolution of connected experimental systems. An example bifurcation comparison between experimental and simulated work is shown in the figure, which was included in a publication in Philosophical Transactions of the Royal Society. The differences highlight why simulating, and not just solving the system was important. One need raised in this project, which I hope to address in the future, was the development of new numerical, integer-based integration schemes for delayed systems. Furthermore, no studies on changes in synchronization and dynamics caused by removing nodes from a network have been performed. With both of these research ventures I have found my diverse training in physics, computer science, mathematics and scientific computing to be invaluable. I’ve been able to bridge the gap between computer science algorithms, efficient programing, and physical and mathematical systems analysis to produce valuable and applicable results. In the future I plan to continue to mediate between these different worlds, fostering research that is applicable in the world and that utilizes knowledge passed across fields. I look forward to developing more collaboration with physicists, biologists, engineers, and those in my own department. Besides working with other faculty, the new projects I above each provide several opportunities to involve students. With synchronization, there are several standard, introductory dynamical systems which exhibit synchronization and could be simulated and investigated by students while providing meaningful results and insights. For the genomics work, many standard algorithms from sophomore through senior level courses are the starting points for adaptation and biological applications. Complex systems in general provide very appealing student projects as the mathematics are relevant to application problems but accessible to students within a few courses and computationally the implementations require careful data management and manipulation and tuning.
Figure 1: Experimental and Simulated Bifurcation of Mach-Zehnder system. Publication available online at: http://rsta.royalsocietypublishing.org/content/368/1911/343
University Education
My education related research has been more broadly focused, examining the expectations that students and faculty enter a classroom with. This work has largely been part of an exclusive fellowship program through University of Maryland’s Center for Teaching Excellence. At the core of the research though is the same underlying idea, that of interaction. In education however this takes the form of interaction between students and faculty or pre-knowledge and future knowledge.The initial results from our fellow’s group identified some major discrepancies between student and faculty expectations; it also identified some common knowledge changes in expectations not reported elsewhere. This work opens up a large range of follow-up investigations, several of which I would like to pursue over my next few years as an instructor. As an example, one of these is a closer examination of how closing the gap between student and faculty expectations can affect the classroom and answering questions like: Does it improve the work returned? Does it change how students are involved in the classroom? Can improving students understanding of the Meta-reasoning behind practices increase the impact of those practices? Or, in a different vein, does closing the gap of expectations improve student satisfaction with a course and retention or excitement in a subject, something I think has a significant relevance to requirements for general education mathematics.
A continuation of this work will provide significant opportunities for student involvement as well. Since the data consists of large survey results, statistical analysis is a key component of understanding and presenting results, as well as response coding. Both are components which the entrance knowledge required to provide relevant results is minimal and are ideal for undergraduate involvement, something I feel will be particularly valuable in developing meaningful and impactful results.
Figure 2: Comparison of Student and Faculty Perception of Student expectations of various learning assessments.
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