Mathematical Controls and Sensitivity Analysis for Antibiotic-Resistant Infections (Yahdi):
Optimal control provides optimization methods for a dynamic system in order to achieve and/or optimize a certain result/output. It has major applications in science, engineering, economics and in finance. This project emphasis is on applications of optimal control theory into biology, more specifically on controlling the emergence of the antibiotic resistance VRE in intensive care units; a life-threatening concern. The goals include performing a global sensitivity analysis to determine most influential parameters, then constructing a bang-bang control, which is easier to implement that a continuous one. The ultimate goal is to determine the most efficient and economically favorable strategies to control VRE and prevent outbreaks.
After the modeling stage, and formulating problems appropriate to given scenarios and goals, the research work will focus on the merging of key methods to provide a global sensitivity analysis diagram, and the necessary conditions for the existence and characterization of a bang-bang optimal control, including the construction of appropriate switching functions, the computer simulations, and the role of critical parameters on the optimal controls.
This project would be accessible to students who are familiar with concepts from mathematical modeling, differential equations, multivariable calculus, and some mathematical software such as MatLab and Mathematica..
Discrete Morse theory and the Homology of Simplicial Complexes (Scoville):
Combinatorial topology is the study of combinatorial (counting) techniques in order to gain information about a topological space. One such technique is the use of discrete Morse theory invented by Robin Forman as an analogue of so-called smooth Morse theory. Many topological spaces are approximated by building them out of discrete “pieces” (called simplices). The resulting object is called a simplicial complex, and the techniques of discrete Morse theory allow one to count the number and kind of holes (among other things) in the simplicial complex, and hence in the original topological space.
In this project, we will be constructing discrete Morse functions on simplicial complexes which yield certain sequences of numbers based on the homology groups of the complex. The ultimate goal is to generalize several theorems relating a discrete Morse function on a simplicial complex to the homology groups of that complex. We will use linear algebra techniques and eventually a computer program to compute the simplicial homology groups.
Prerequisites: linear algebra and familiarity with proofs (either a formal course in proofs or a proof-based course like real analysis or abstract algebra). Knowledge of group theory is a plus.
We will go over the necessary definitions and background in graph theory and topology during the REU. This project will be accessible to those who are familiar with mathematical proofs and proof techniques.
Ontology driven health informatics approach to Translational Research on Inherited Metabolic Diseases (Dhawan):
The past few years have seen an increasing push to create, coordinate and develop Newborn Screening programs on a nationwide scale. In the US, this is being done under the administration Health Resources and Services Administration (HRSA). However, despite these efforts, Newborn Dried Bloodspot Screening (NDBS) programs need to evolve their scope beyond the collection and testing of screens.
As part of this project, we focus on creating a model health informatics system to collect and analyze LTFU data throughout the lifetime of children identified by newborn screening. The goal is to use semantic web techniques on this repository to improve identification, management and outcome of patients with a positive screen. As a pilot we pick Phenylketonuria (PKU) as an exemplar disease.
We plan on focusing on two aspects of the semantic web – ontology design and creation and end-user query interfaces. This project will be accessible to those who are familiar with Data Structures needed. Databases/Semantic Web courses will be a bonus.