I am currently working on my dissertation research with Assistant Professor Leo Lopes of the Systems & Industrial Engineering department on a class of stochastic shortest path network flow problems. Our interests involve networks where information is missing or unavailable. We are investigating a model in which random variables are used in place of unknowns. This allows us to search for paths that have a high probability of being optimal.
The Most Likely Path Problem (Under Review)In this paper, we present a stochastic shortest path problem that we refer to as the Most Likely Path Problem (MLPP). The MLPP is motivated by homeland security applications, but is also useful for reliability, design, and other applications. We prove that optimal solutions to the MLPP are composed of optimal subpaths, a property which is essential for solving the classical deterministic shortest path problem. On series-parallel networks, we produce analytical bounds for the probability of the Most Likely Path (MLP), which we compute efficiently via dynamic programming and ordinal optimization.
Below you will find descriptions of small research projects that I have worked on, dating back to my senior year of college. If you find yourself interested in reading more details, links to pdf versions of my work from graduate school and an html version of my work from college can be found below.
Spectra & PseudospectraAnalyzing spectra of nonnormal matrices often provides either useless or misleading information. To understand these nonnormal matrices, pseudospectral techniques have been developed and continue to be refined. This paper provides an introduction to the study of pseudospectra. Most of the content closely follows Spectra & Pseudospectra by Mark Embree and Nick Trefethen.
Young's InequalityThis paper follows the proof of Young's inequality in Analysis by Lieb & Loss. Virtually all steps are shown in great detail. Calculations may at times seem repetative, but this was done to make the proof readable to students not yet overly experienced in the realm of analysis (much like myself at the present time). For those of you already familiar with Young's inequality, you may wish to skip to the full proof with the sharp constant (section 8).
The Fluid DripOur research has been concerned with the phenomenon of fluid drip in three liquids - honey, latex, and a cornstarch-water mixture. Our explorations of the phenomenon of fluid drip have taken two directions. On the one hand, much of our work has been devoted to understanding the physical processes at play in fluid drip.The second focus has been to discern how the properties of the three fluids we are working with affect the processes of drop formation and snap-off. It is clear from observation that the fluid drip phenomenon is very different in latex, honey, and corn-starch. Some of this difference is explained by the distinction between Newtonian and non-Newtonian fluids.
Research in Cluster AnalysisWhat relationships, if any, are evident between chemical variability of volcanic rocks? What, if any, are evident between dynamics of plate tectonics and magma migration? What is the statistical description of global chemical variability? Are there correlations between elements, or between elements and isotopes? What can we discover about trace elements and in particular about their isotope ratios? Is clustering in chemical space possible? And will it lead to an accurate system of classifications?
