The NSF and the NSA provide generous funding and support for this
REU program.

**Where**: NC State University, Raleigh, NC .

**Stipend and support**: $4000 for ten weeks, all
housing provided, as well as a partial meal allowance. Travel
funds up to $300 per participant provided as needed.

**Topics for 2014 REU program**:

**Project 1: Mathematical Phylogenetics and the Space of Trees**

Faculty Mentor: Seth Sullivant

**Graduate Student Assistant: Colby Long**

Abstract:

Phylogenetics is the branch of biology concerned with reconstructing evolutionary trees of species from data. The set of all trees, with lengths on each edge, forms a geometric space called the "space of trees". The space of trees is an example of a negatively curved "CAT(0) space", and this implies that between each pair of points in the space of trees, any path that is locally shortest between a pair of trees is also a globally shortest path, and hence there is a unique geodesic between any pair of points. This fact means that it is possible to do a range of computations in tree space like compute averages of sets of trees.

In this REU group we will explore the geometry of tree space with the goal of developing algorithms for analysis of trees. Ideally, a student in this group should have a good background in multivariable calculus and discrete math, and be comfortable with computing software like Matlab.

**Project 2: Computational Modeling for OCT Imaging of the Human Eye**

Faculty Mentor: Mansoor Haider

Graduate Student Assistant: Micaela Mendlow

Abstract:

Corneal topography uses optical imaging to construct local curvature maps of the human cornea for diagnostic and treatment applications, e.g. contact lens design, LASIK, intraocular lens design. High resolution imaging of internal tissue microstructures can be achieved using optical coherence tomography (OCT), which is based on interferometry with near-infrared light. However, the associated curvature calculation is highly sensitive to small perturbations, and topographical mappings can only be as accurate as the algorithms used to process the OCT image data. This project will develop and refine 3-D mathematical models for curvature mapping of the human cornea and other internal surfaces. Models will also account for noise present in the raw data and are invaluable tools for validation of algorithms incorporated into diagnostic software.

Prerequisites: Linear algebra, vector calculus and an interest in biomedical research. Previous experience with Matlab or another programming language would be helpful, but not essential.

**Project 3: Multiple Beam, Permanent Magnet Focused Klystrons for the Next Generation of High Energy Accelerators**

Mentor: Dr. Lawrence Ives (Calabazas Creek Research, Inc.) and Hien Tran (NCSU)

Graduate Student Assistant: Amanda Coons

Abstract:

The Large Hadron Collider is the world’s highest energy accelerator and recently advanced our understanding of the universe by detecting the Higg’s Boson. The next major advance will require a still higher energy accelerator. To achieve these higher energies, more powerful and efficient RF sources will be required. The highest power RF sources available today are 10 MW, multiple beam klystron (MBKs) using single convergent electron guns and electrically powered solenoids.

The ultimate power limit for singly convergent MBKs is 20 MW; however, the next major accelerator system will require powers in excess of 40 MW. They also must operate at higher efficiency than existing klystrons. The only potential sources known today that might achieve these requirements are doubly convergent MBKs with permanent magnet focusing. Doubly convergent MBKs were invented at NCSU in a previous REU program.

This project will attempt to combine double convergent MBKs with periodic permanent magnet focusing. Permanent magnet focusing eliminates the power required for the confining magnetic field, significantly increasing the overall efficiency.

The initial task will design a double convergent electron gun consistent with a 40 MW klystron. The following task will interface the gun to a periodic permanent magnet field. This will be the world’s first design of a periodic permanent magnet focused MBK. Successful integration with the double convergent gun will represent an important step toward the world’s next major accelerator system.

**Project 4: Physiologically Based Pharmacokinetic Modeling for Acetone: How Much Do We Really Breathe In?**

Mentor: Dr. Marina Villafane Evans (US Environmental Protection Agency)

Graduate Student Mentor: Kathleen Schmidt

Abstract:

Physiologically based pharmacokinetic (PBPK) models are computational tools used to convert external exposure into internal organ doses. These internal dose estimates can then help determine potential risk for health effects for that particular chemical. PBPK models can account for differences in physiology, metabolism, and absorption across different species and routes of exposure. These models are based on the principle of mass balance, and make use of ordinary differential equations to describe transport in different organs needed for calculating the chemical's distribution. Environmental chemicals differ in water or fat solubility, making it necessary to estimate absorption in the upper respiratory tract when exposed to a water-soluble chemical via inhalation. The component that is not absorbed will continue into the inner lung and eventually be transferred into systemic blood. An accurate estimate of absorption should increase our confidence in calculations and result in better fits to experimental data. This group will learn basic physiology, biochemical, and mass transport concepts to derive the necessary equations. The simplest model able to explain the observed data will be selected. (This abstract does not necessarily reflect US EPA policy).

**Project 5: Generalized Symmetric Spaces and Their Applications**

Faculty Mentors: Aloysius Helminck (NCSU) and Ruth Haas (Smith College)

Graduate Student Assistants: Mark Hunnell and Amanda Sutherland

Abstract:

Generalized Symmetric Spaces occur in many areas of mathematics and physics. Symmetric Spaces were originally defined as the homogenous spaces G/H where G is a real matrix group and H is the fixed-point group of an automorphism of order 2 of G. The homogenous space G/H can be identified with nice subset Q of the group G. Generalizations of this from matrix groups to arbitrary groups have become of importance in many areas.

For the study of these generalized symmetric spaces and their applications one needs many properties. For example, one still needs a classification of these generalized symmetric spaces for many finite and matrix groups. Another example is the study of orbits in these symmetric spaces under the action of both G and H. How does a G-orbit split into H-orbits? Some of the questions can be studied with the help of computer algebra packages, like Maple, Mathematica or Sage.

This project will involve Abstract Algebra, Linear Algebra and Computer Algebra

**Project 6: Counting the Number of Real Roots of Random Polynomials**

Mentor: Dr. Dhagash Mehta

Abstract:

A classic mathematical question is: How many real roots does a random polynomials have? This simple looking question is still unsolved in a completely general setting. In this project, we will experiment with recently developed numerical methods to find 'certifiable solutions' of random univariate polynomials and will try obtaining some numerical answers to the above question.

**Project 7: Mean-Variance Portfolio Optimization and Black-Litterman Model**

Faculty Mentor: Tao Pang

Graduate Student Assistant: Cagatay Karan

Abstract:

In the modern portfolio optimization theory, the goal is to choose the optimal portfolio such that the expected mean portfolio return is maximized while the variance (risk) is under a certain level. The classical mean-variance optimization method uses the historical mean returns and co-variance matrix as input, but the optimal solution is often an extreme solution, which is often not feasible in practice. On the other hand, in Black-Litterman model, the subjective view of the asset returns can be integrated into the optimization process by virtue of the Bayesian estimation. The resulting portfolio usually has a better performance than the solution obtained in the classical mean-variance model. However, in the Black-Litterman model, the confidence level of the subjective view has some effects of the optimal portfolio. In this project, students will investigate the sensitivity of the optimal solution of the confidence level of the subjective view of expected returns. The solution performance will be evaluated by virtue of market data.

Qualified students should have strong background in linear algebra, probability and statistics. Background in financial mathematics is a plus but not required.

**Project 8: Geometric Flows of Plane Curves**

Faculty Mentor: Andrew Cooper

Graduate Student Assistant: George Lankford

Abstract:

We will study how a curve in the plane might ``evolve", based solely on its geometric properties, to a different shape. For example, what is the best way to shorten a curve? to lengthen it? to make it more symmetric? to straighten it? to make it mimic another curve to best of its ability?. ``Flows" of evolving curves model stretching and releasing an elastic band, as well as phenomena that occur when metal is annealed and honey is poured or stirred.

Our focus will be on the pure mathematics of curve evolution. To study the motion of a curve, we will regard it as the solution of a partial differential equation. The first such equation we will study is the curve shortening flow, which pulls the curve tight as fast as possible (and along the way solves one of the oldest problems in mathematics, the isoperimetric problem). Then we will explore some related flows that are interesting in their own right and might be used to solve some other longstanding geometric problems.

Students in this project will approach problems of curve evolution through both theoretical analysis and computer modeling. Some experience with mathematical programming (Matlab, Maple, Mathematica) will come in handy. All of our equations act in some ways like the ordinary heat equation, so an understanding of that equation should be considered a prerequisite.

**Project 9: Parameter Selection and Model Reduction Techniques for Uncertainty Quantification in Large-Scale Models**

Faculty Mentor: Ralph Smith

Graduate Student Assistant: Mami Wentworth

Abstract:

This project will focus on the development and implementation of parameter selection and model reduction techniques for physical and biological models. This is a critical step when quantifying the uncertainty of parameters and responses for large-scale models. Parameter selection is necessitated by the fact that models often have a large number of non-influential parameters, which must be identified and fixed before model calibration. Model reduction is required to obtain the efficiency needed to achieve the hundreds to millions of model evaluations required for Bayesian uncertainty analysis. Motivating applications include models arising in HIV treatment protocols, nuclear power plant design, and design of the flying robot "Robobee".

**Project 10: Data and Cluster Analytics**

Faculty Mentors: Carl Meyer and Shaina Race

Abstract:

Given unorganized data that may be derived from text or simply raw numerics, the objective is to learn and develop techniques for detecting, revealing, and analyzing hidden patterns and clusters of information that exhibit some sort of similarity or commonality. The size and diverse nature of the data sets of interest make this a formidable but extremely important problem. The first part of the project will be to learn and understand how to use some of the state-of-the art techniques by analyzing some selected practical applications. This will be followed by exploring the possibilities of developing some new methodologies and algorithms whose aim is to detect patterns and structure in unlabeled data where no value for error or accuracy can be placed on the final result.

Emphasis at the outset will be placed on text mining and community detection although the content eventually can be directed by the interests of the participants. Programming will be integral as students implement existing methods and develop their own improvements.

Prerequisites: The mathematics employed involves linear algebra, probability and statistics, networks and graphs, and computer programming.

**Project 11: Modeling the Interaction Between Autonomic Neural Regulation and Inflammation Using Pre- and Postoperative Data from Hip and Knee Replacement Surgery**

Faculty Mentor: Mette Olufsen

Graduate Student Assistant: Andrew Wright

Abstract:

Early postoperative mobilization is essential for rapid functional recovery after surgery and is considered a cornerstone in the treatment. This strategy has improved patient outcomes after surgery and reduced the length of hospital stays. However, some patients experience postoperative symptoms of dizziness and fainting due to failed orthostatic cardiovascular regulation. Tests to determine if a given patient is prone to fainting include head-up-tilt and sit-to stand, during which blood pressure and heart rate are measured. During this project, we will develop and analyze mathematical models that can predict dynamics observed during these tests and try to understand how inflammation impacts autonomic cardiovascular control system. Models will be based on ordinary differential equations, that we will solve using Matlab, and potentially compare to data measured before and after surgery.

**Project 12: Computational Modeling of the Thyroid Hormones Hemostasis and its Manipulation by Environmental Chemicals**

Mentor: Dr. Hisham El-Masri (US Environmental Protections Agency)

**Graduate Student Assistant: Ariel Nikas **

Abstract:

In the 1990's, some scientists proposed that certain chemicals might be disrupting the endocrine systems of humans and wildlife. A variety of chemicals have been found to disrupt the endocrine systems of animals in laboratory studies, and compelling evidence shows that endocrine systems of certain fish and wildlife have been affected by chemical contaminants, resulting in developmental and reproductive problems. A number of chemicals can interfere with endocrine function of the thyroid causing disruption of the highly regulated hypothalamic–pituitary–thyroid (PTH) axis that has been shown to cause neurodevelopmental toxic effect on children born to affected mothers. The purpose of this project is to develop computational models describing the biological events of the PTH axis and its possible manipulations by chemicals.

**Project 13: Differential Games and Level Set Methods**

Mentors: Adam Attarian, Reed Jensen, and Dan Finkel (MIT Lincoln Laboratory); Zhilin Li (NCSU)

Graduate Student Assistant: Guanyu Chen

Abstract:

Differential games are the continuous time analogue to classical multi-player game theory. In differential games, the modeling and analysis of conflict can be formed by systems of continuous ordinary and partial differential equations (PDEs) where each equation models a player his or her strategy. Games can range from areas as diverse as economics, pursuit, and combat. Some of the most well known problems include the “homicidal chauffeur problem” and the “princess and monster game”. Results from game theory can have wide-ranging impacts on many other areas of mathematics and applied science.

A current challenge in the analysis of differential games is the lack of methods for obtaining reliable solutions. Computing numerical solutions to the resulting system of equations is challenging, and can be ill posed. One possible solution technique is level set methods, which can be used to approximate the solution to time-varying PDEs common to both differential games and other areas. Level set methods are powerful numerical techniques for analyzing boundaries of the complex topologies that can arise in differential game theory.

This project will be a combination of game theory, optimal control, the calculus of variations, and the associated numerical methods. Over the course of the summer, students will formulate several classic differential games and implement their numerical solutions using an existing level set toolbox. In some cases, closed form solutions can be derived and compared to the numerical output. Students completing this project will become versed in optimal control theory, game theory, and the numerical solutions to PDEs.

**Project 14: Model Refinement through Verification and Validation Methods and Anomaly Detection**

Mentors: Lori Layne, Brian Lewis, and David Padgett (MIT Lincoln Laboratory); Hien Tran (NCSU)

Graduate Student Assistant: Phuong Hoang

Abstract:

A mathematical model of an observable process is the basic tool needed to study and develop a thorough understanding of the process. Models can be assessed and refined as more observations are collected, but can be skewed by corrupt or anomalous data observations, inconsistencies in the process observation method, or the inability of the model to sufficiently capture physical phenomenology. Model assessment based on the incorporation of continued observations and the identification of corrupted observations suggests two separate model refinement pathways: verification and validation (V&V) and anomaly detection. Verification is the assurance that a model conforms to a given description of a process, and validation is the assurance that the underlying description of the process is accurate. Verification and validation techniques together determine the extent to which a given model is representative of the process of interest. Furthermore, techniques exist to determine whether or not a given measurement is representative of the process of interest or is an outlier or anomaly.

This REU project is centered on developing a model using data gathered from observations of specific processes. Over the course of the summer, students will not only design a model using real-world data, but also use statistical methods to develop V&V techniques for model assessment and refinement. Additionally, students will apply several anomaly detection techniques to identify measurements that may not be representative of the process being modeled. The utility of these anomaly detection techniques will be determined for different types of measurement data and models. Students completing this project will not only learn what constitutes successful models, but also understand the complications that can arise when working with real-world data and how to improve their models given new information.

This work is sponsored by the Department of the Air Force, under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. Government.