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    04-05 AR

2002-2003 Academic Year REU Projects

Kree Cole-McLaughlin
Faculty Mentor: Mladen Bestvina
Project Proposal:
The contour tree is an abstract characterization of the topology of a scalar field, w=F(x,y,x), obtained by contracting continuously each connected component of the level sets, F-1(w), to a single point. In practice, the nodes of the contour tree correspond to the critical points of F, and the arcs correspond to families of adjacent components of the level sets with equivalent topology. We have recently developed an algorithm that computes efficiently the contour tree of large scientific datasets and also characterizes the corresponding homology groups.

While the contour tree has been shown to be a very effective tool for the analysis of scientific data, its practical use remains limited to small test cases. This is due to the size that the tree can reach when used to analyze the output of large scientific simulations. We plan to analyze datasets generated by scientists at Lawrence Livermore National Laboratory.

Using Morse Theory, one can define a notion of topological persistence, where critical points are grouped in pairs and a ranking is assigned to each pair. Critical points with high rank "persist" at large scales, while low ranked critical points are only relevant in a fine scale analysis. This provides a natural way of defining a hierarchy of topological features of a scalar field and therefore a way of reducing the amount of data used at any given time by changing the level of detail or the locatility of our analysis.

We propose to study the notion of topological persisentce and its implication on the structure of the contour tree. Specifically, we plan to define a multi-resolution representation of the contour tree based on topological persistence. Moreover, we plan to develop an efficient algorithm that builds such multi-resolution contour trees without explicitly computing the correstponding sequence of topological contractions implied by the persistence algorithm.

Joshua Coon
South Summit High School, 1999
Hometown: Kamas, UT
Major: Mathematics & Physics
Year: Senior
Faculty Mentor: Andrej Cherkaev
(Project Beginning Spring 2003)

Project Proposal:
A major problem in many physical and theoretical systems is to find an efficient way to move large numbers of independent objects in an enclosed space of dimension d through a small opening in the enclosure. For my project, I would like to investigate the interactions of multiple objects moving through an opening in an enclosed space of dimension 2 and try to formulate algorithms that will make this process as efficient as possible. Specifically, the project will progress in three stages. First, I make a computer model of the system of particles, the enclosure, and the opening, giving the particles similar qualities to those found in particles in nature (such as angular momentum, conservation of energy, conservation of momentum). Next, I will study the percolation of the system and try to determine the speed and efficiency of the movement of the particles through the slit under various system conditions (such as varying aperture size, varying amounts of elasticity in the collisions with the walls, and varying amounts of particles). Finally, I will try to explore some real world implicatinos of my findings, particularly as they relate to applications in the movement of large numbers of computer characters ("sprites") in computer gaming.

Gary Crum
Major: Mathematics
Year: Second Bachelor's Degree
Faculty Mentor: Misha Kapovich

Project Proposal:
This research project involves the disciplines of geometry, topology and analysis. I propose research into relationships analogous to triangle inequalities, but in spherical domains.

The triangle inequalities in Euclidean space are a well-known part of analysis and calculus. If a, b and c are the vertices of a geodesic triangle in the plane R2 and d(x,y) denotes the distance from x to y, then d(a,b) + d(b,c) = d(a,c). This inequality is a necessary and sufficient condition for existence of a triangle in R2 with the given side lengths.

In this project we consider triangles in the 2-sphere S2 whose edges are geodesics in S2 and the angles alpha, beta, gamma at the corners. The goal of the project is to identify "angle-triangle inequalities", which would be conditions on three positive numbers alpha, beta, gamma which are necessary and sufficient for existence of a triangle T(alpha, beta, gamma) in S2 with the given angles alpha, beta, gamma.

We will begin by considering convex triangles T(alpha, beta, gamma) embedded in S2, where the concept of the angle is straightforward. For such triangles one can define a "dual" triangle T* embedded in S2 with the side-lengths Pi-alpha, Pi-beta, Pi-gamma. Thus the angle-triangle inequalities for T become side-length triangle inequalities for T*. Our first goal then is to identify side-length triangle inequalities for triangles T* in S2.

We then plan to study "abstract triangles" in S2, which are not necessarily embedded in the sphere and for which angles/side-lengths are allowed to be greater than 2*Pi. These abstract triangles appear in study of locally spherical structures with three conical singularities on the 2-sphere. It is currently unknown what are necessary and sufficient conditions for existence of such a structure on S2 with the prescribed cone angles 2*alpha, 2*beta, 2*gamma. The ultimate goal of this project is to find these conditions.

We also plan to use computer graphics to visualize results, i.e. how triangles on the 2-sphere deform as we vary their angles/side-lengths.

John Faust
Olympus High School, 1998
Hometown: Holladay, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Paul Bressloff
(Project Beginning Spring 2003)

Project Proposal:
In order to determine how the local circuitry of the primary visual cortex provides a substrate for the recurrent amplification and sharpening of the tuned response to local visual stimuli, it is necessary to specify how the isotropic organization of these connections is related to the columnar organization of coded features, such as stumulus orientation. Optical imaging of superficial cortical layers has shown that orientation preference changes slowly and continuously as a functin of cortical location except at singularities or pinwheel centers; away from the pinwheels, there exist linear zones within which iso-orientation regions form parallel slabs. Given that near the pinwheel centers, cells with orthogonal orientation preferences are close to each other, whereas they are far apart in the linear zones, it might be expected that the orientation tuning properties of cells would depend on their position relative to the pinwheel centers. In this project, we investigate the relationship between the pinwheel geometry of visual cortex and the response properties of cortical cells. We will approach the problem by treating each neuron as linear recurrent filter that transforms an input stimulus into an output firing rate. The recurrent filter will depend on both the feedforward receptive field properties of a cell and the recurrent interactions with other cortical neurons. The work will involve a mixture of linear algebra, Fourier analysis, and differential equations.

Troy Finlayson
Major: Physics
Year: Senior
Faculty Mentor: Ken Golden

Project Proposal:
As a continuation of the work we have been doing, I propose to finalize the comparison of thermal conductivity data with multi-component material bounds in order to theoretically and numerically estimate the thermal conductivity of sea ice. We have applied the mathematical theory of bounds on effective transport coefficients to the thermal conductivity of sea ice, viewed as a three component composite material (ice, air, and brine). In particular, we have applied the arithmetic and harmonic mean bounds, as well as the tighter three component Hashin-Shtrikman (HS) bounds to the thermal conductivity of sea ice. In our comparisons of the thermal conductivity data with the HS bounds, we have observed that, in both the two-component glacial ice case and the three-component sea ice case, the bounds capture the data well (for temperatures below and up to the critical temperature of -5 C for sea ice).

As a supplement to previous knowledge and observations of the existence of a critical point at a temperature of -5 C, another important aspect of the research would be the confirmation that the significant changes in thermal transport through sea ice are due to convective motion of brine through sea ice enabled by the percolation transition at -5 C. As termperature increases, the microstructural composition of sea ice changes. This affects the macro-thermal conduction of the sea ice as heat transfer between the ocean and atmosphere switches from primarily conduction to a regime where brine convection plays a significant role.

We now plan to obtain more thermal conductivity data on sea ice to compare with our bounds, particularly for brine volume fractions near and above the critical threshold. I am also beginning to work with Jingyi Zhu on obtaining thermal conductivity coefficients from noisy, raw temperature data. Another avenue that I will be pursuing is the improvement of the existing bounds, particularly in the "cold" regime below the critical temperature for percolation.

Rhett Hadley
Faculty Mentor: Ken Golden
(Project Beginning Spring 2003)

Project Proposal:
Sea ice is a complex and diverse structure that has many different components that contribute to its permeability. Air and fluid permeability depend on the composite structure that consists of pure ice and random brine inclusions. Algae make up a large part of the dynamic that live within the microstructure of sea ice. The algal community consists of many different species that interact based within the microstructure of the ice. Due to the complex composite nature, the algae must position themselves in the ice in order to maximize an ideal living environment. Algae, being photosynthetic, need to position themselves close to the surface in order to absorb the light. Another competing factor is the need for nutrients; if the algae are too high in the ice, they will not be able to maximize their ability to absorb the nutrients.

Clearly, there is a relationship between these two competing factors: light availability and nutrient avilability. Sea ice plays a critical role in this in that sea ice permeability relates to the availability of nutrients. Sea ice also has optical properties that do not allow the light to pass cleanly through the ice due to the structure. What I will be trying to do is develop an equation model to describe the optimal algal position in sea ice that characterizes their preferred position as a maximum minimum problem. This also gives insight into the pattern formation of algae in sea ice.

Nathan Hancock
Weber High School, 1996
Hometown: Ogden, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Aaron Fogelson

Project Proposal:
When injury occurs to the walls of a blood vessel, platelets will aggregate at the site in order to start the healing process. If the injury includes rupture of the vessel, platelets must form a plug in order to stop blood loss and maintain vessel function until the wall can be repaired. Platelets stick together by forming fibrinogen cross-bridges to other nearby platelets. One end of the fibrinogen molecule bonds to specific receptors on a platelet called GPIIa/IIIb and the other end of the molecule binds to the same type of receptor on another platelet.

In the body where blood is constantly flowing, platelets attempting to aggregate have a force exerted on them by the fluid moving past. If this force is great enough, it can tear the forming aggregate apart. In order to develop an accurate model of the clotting process, it would be necessary to know the nature of these bonds and specifically how much force they can withstand before being torn apart.

A number of theories have been proposed as to the nature of these bonds and, recently, experiments have shed some light on the bond characteristics. For example, Weisel, et. al. used laser tweezers to determine the strength of these molecular bridge interactions. The force necessary to break the bonds is actually a stochastic distribution as thermal motion of the molecules is significant. One interesting thing about their experiement is that the amount of force needed to break the bond did not depend on the loading rate of the force as predicted by Evans and Ritchie. Loading rate dependence of the rupture force has been seen experimentally in studies of bonds between simpler (and smaller) molecules, but Weisel's experiments do not show this for fibrinogen-GPIIb/IIIa bonds.

We propose to develop a model from first principles which explores the experiment performed by Weisel. In particular, we want to see whether loading rate should be important under his experimental conditions and, if not, to understand why the theory of Evans and Ritchie fails to explain these experiments. It may be that the model comfirms the theory of Evans and Ritchie, but this would be surprising as they make a number of ad hoc assumptions. This model would include stochastic ordinary differential equations (Langevin equations). These systems will be studied through simulation and statistical analysis.

Amy Heaton
Vermillion High School, 2001
Hometown: Vermillion, SD
Majors: Mathematics, Chemistry
Year: Sophomore
Faculty Mentor: Ken Golden
Project Proposal:
The problem of finding viable bounds on the fluid permeability of sea ice is of substantial significance in the sea-ice world. No work has been done in this area, and such progress would be beneficial to better understanding important physical effects, both local and worldly, of which sea ice plays a major role. I am interested in nailing down some good bounds and comparing them to experimental measurements of fluid flow through sea ice. Theoretical investigation in this area has led me to several fluid permeability bounds for porous media of different given physical properties, and the relevance of these bounds is to be investigated as experimental data is obtained. A model of parallel pipes could also lead to practical bounds of this type, as it characterizes a microstructure similar to that of sea ice. Optimal geometries could be simulated given a specific volume fraction of brine to ice, and the optimal geometry would lead to optimal flow, thereby the upper bound to the problem.

The fluid flow through sea ice is central to the problem of thermal conductivity through the same medium. Thermal conductivity measures the flow of heat through a given medium, and the movement of fluid through that medium is a major transporter of that heat. These two properties are closely related and, once the flow of fluid is understood, it can be tied together with current work being done on thermal conductivity.

Michael Hofmann
Highland High School, 1999
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Gordan Savin
(Project Beginning Spring 2003)

Project Proposal:
My research project is to investigate the properties of elliptic functions. These functions take the form y2 = ax3 + bx2 + cx + d. These functions are interesting to me because so many branches of mathematics are involved, including number theory, group theory, algebraic topology, and complex analysis.

I plan to study properties like the group structure of elliptic curves, uniformization of elliptic curves, and the Weierstrass P-function, which has the form P(z) = 1/Z2 + Sigmaw in L(1/(z-w)2 - 1/w2) where L = mw1 + nw2 is a lattice defined in the complex plane by two vectors w1 and w2, and has the interesting property that it is the solution to the differential equation [P'(z)]2 = 4P3(z) + aP(z) + b.

My goal is to learn more about elliptic curves and to learn more about the different branches of mathematics that are connected with elliptic curves as well as how they interrelate. I am also looking forward to experiencing mathematical research.

Ronald McKay
Penncrest High School, 1988
Hometown: Media, PA
Major: Mathematics
Year: Senior
Faculty Mentor: Davar Khoshnevisan
(Project Beginning Spring 2003)

Project Proposal:
The topic I would like to do research on is Ito diffusions. An Ito diffusion is a model of a random walk-like process. A random walk is a mathematical model for molecular motion. A particle starts at an initial position, and a random process (such as flipping a fair coin) dictates which direction the particle moves at each time step. Each random process, and thus each step, is independent of the preceding step. In other words, let S0=0 be the position of the particle at time n=0. The values X1, X2, ... are independent random variables that take values +1 or -1 with probability 1/2 each, and represent the displacement values at times 1, 2, ... The position at time n is given by Sn=X1 + ... + Xn. The study of random walks has certain inherent interest, but one important development is summed up in Donsker's Theorem (Monroe Donsker, 1951). Donsker's Theorem states that, if run for a long time, a simulated random walk with mean zero and variance one looks like Brownian motion. Brownian motion is the name of the phenomenon of random collisions between particles. Just like random walks, the study of Brownian motion has intrinsic interest, as well as applications in various fields such as polymer theory, quantum physics, financial mathematics, and population genetics.

To simulate one-dimensional Brownian motion from the discrete steps of the random walk, Sn = X1 + ... + Xn, we let X1, X2, ... be independent identically distributed random variables with mean zero and variance one. For large n, the random graph of S1/square root of n, S2/square root of n, ... Sn/square root of n is very close to the graph of Brownian motion until time one. The 1/square root of n term is a scaling term that acts to compress the axes of the graph so that the Brownian motion remains to scale for large values of n (n to infinity gives the approximation of Brownian motion).

Certain applications lead to the question of how random walks and Brownian motion behave in an inhomogeneous space. This is the situation known as Ito diffusions. The discrete time model is constructed by motivating independent molecular fluctuations, that are independent random variables, X1, X2, ... that are equal to +1 or -1 with probability 1/2 each. Let Y0 be the origin, that is Y0:=0. With Y0...Yk, define Yk+1:=Yk + a(YkXk+1, where the function a influences the fluctuations of the random variables, depending on where the diffusion Y is at time k. A drift term, some function b, can also be added that acts to model the effects of a push. The equation becomes,

Yk+1 = a(Yk)Xk+1 + b(Yk).

A continusous-time diffusion can be approximated in the same manner that Brownian motion was from the discrete random walk. Let Y(0) := 0, and with the functions a and b as above,

Y(k+1/n) := Y(k/n) + a(Y(k/n))(Xk+1/square root of n) + b(Y(k/n))(1/n).

The goal of my research project will be to examine the behavior of Ito diffusion simulations, particularly, how do variations in the functions a and b influcence the diffusion. Also, I would like to develop a firm understanding of the theory that motivates the above equations and subsequently, stochastic differential equations and Ito's formula. Finally, I would like to gain insight into how Ito diffusions can be applied to various fandom phenomenons such as stock prices and gene frequency models.

Benjamin Murphy
Bountiful High School, 1994
Hometown: Bountiful, UT
Major: Mathematics, Physics
Year: Senior
Faculty Mentor: Kenneth Golden

Project Proposal:
Last summer, I did a lot of reading to familiarize myself with the modeling of electromagnetic interaction with multi-component composite media. In particular, I studied the properties of electrorheological fluids and sea ice when perturbed by electromagnetic waves. Many different approaches have been used to model these complicated interactions but none have been entirely successful.

I have done a considerable amount of reading on the Stieltjes integral representation of the complex effective permittivity developed by Bergman, Milton, Golden, and Papanicolaou. This integral representation allows bounds on the effective parameters to be obtained. It also suggests many similarities to a statistical mechanics approach to the problem - methods that would be greeted with open arms if available.

Using the current representation, parameter bounds are obtained for the quasi-static case where the wavelength is very long compared to the micro structural scale. The goal of this next semester is to generalize this technique to the scattering regime, where the wavelength is comparable to the micro structural scale. For many important phenomena, such as the interaction of microwaves with sea ice or composite photonic crystals, the scattering regime model is imperative for much of the research.

I am presently formulating a one-dimensional model of interactions of electromagnetic waves with composite media by using a Green's function integral representation of the solution to the inhomogeneous Hemholtz equation. In one-dimension, the represenation formulas are relatively simple and I am able to capitalize on computer packages that allow me to graph the approximate solutions. This approach allows me to visualize the geometry of the solution while giving me more of an intuitive understanding of the problem. The next step after completing the analysis on this one-dimensional problem is to apply function analysis techniques to resolvent operators associated with the Hemholtz equation in higher dimensions. The goal is to derive an integral representation for the solution of the Hemholtz equation in homogeneous media, analogous to what has been previously found for the quasistatic case.

There are many areas of mathematics that are necessary in the approach I plan to take. Spectral theory, the theory of resolvents, and analytical function theory will all be used in the theoretical formulation of the problem. Extensive numerical analysis will also be used to get a feel for the problem before attacking head on with theoretical tools.

There are many useful applications to other areas of research as well as industrial applications that would benefit from this information. A few of which are: wave properties and the scattering effects off of random media, the electrorheolgical fluid problem, sea ice research, photonic crystals (the analogue of semiconductors), and many more.

It is a unique experience working on problems that have not yet been solved. There are many techniques of modeling and many fields of mathematics that I have been introduced to that have vastly expanded my view on mathematics as a whole. It is extremely exciting to be able to use my knowledge of mathematics and to apply the physics I've learned throughout the years. I am looking forward to continuing my research this year and am thrilled to be working toward my PhD thesis while I am still an undergraduate.

Nancy Newren
Woods Cross High School, 2004
Hometown: Bountiful, UT
Major: Mathematics
Year: Freshman
Faculty Mentor: Aaron Fogelson
(Project Beginning Spring 2003)

Project Proposal:
The research project I am intending to work on has to do with the elastic properties of fibrin meshes formed when blood clots. Blood clots form to fill holes in blood vessels. They also form in the arteries of the heart and brain and cause many heart attacks and strokes. It turns out that fibrin can form "fine" or "coarse" meshes, as explained below.

These meshes differ in their elastic properties; and also differ in that fine fibrin meshes seem to be related to the clots that cause heart attacks. It might be that these fine clots are too hard for the body to break down or for the blood flow to push them close to the wall. So it's important to understand what gives a fibrin mesh its elastic properties. I want to use computer modeling to help with this.

Let me tell you a little more about fibrin: Thrombin is an enzyme produced by a complicated set of chemical reactions when a blood vessel is damaged. Fibrinogen is a soluble protein present in the blood at high concentration. When thrombin cuts off two pairs of short peptide fragments from the fibrinogen molecule, what is left is a fibrin monomer. Fibrin monomers polymerize by non-covalently bonding to one another. This produces thin strnds called protofibrils, which can then bond side to side, sort of like wires in a cable, to form thicker fibrin fibers. When protofibrils bond along one portion of their length, branches in the fibers can develop. The overall result is a fiber mesh that forms around aggregated blood cells called platelets.

Depending on how much thrombin is around when the clot is formed, on the pH and the ionic strngth of the blood, and on which of several forms of fibrinogen a person has, the mesh formed can be "fine" - many short thin fibers with lots of branches and small pores between the fibers - or it can be "coarse" - fewer, longer, and thicker fibers iwth fewer branches and larger pores.

The structure of the mesh is called its architecture. I am interested in exploring how the different architectures along with the elastic properties of individual fibrin fibers, together give a clot its overall (macroscopic) elastic properties. I plan to construct computer models of fibrin networks; giving each strand properties to calculate the elastic properities of the whole network. By doing different computer models of fine and coarse meshes, I hope to help understand how the architecture of the mesh influences its elasticity.

Reza Sarijlou
Brighton High School, 1996
Hometown: Tehran, Iran
Major: Mathematics, Computer Engineering
Year: Senior
Faculty Mentor: Elena Cherkaev

Project Proposal:
My research project under Professor Elena Cherkaev will be the study of random networks. Networks provide efficient descriptions of complex phenomena starting from internet web to finite-dimensional approximation of processes in porous medium. Networks are used to model nonlinear behavior of biological materials and rapidly growing infrastructure of large industrial companies. We will start with modeling of electric transport in porous medium. We will use a random network as a tool to try to computationally model the response of differently structured networks. When studying the structure of the network, we will be using the spectral function, this function contains all the information about the structure of this netowrk. I will come up with this function by using computer simulation. This network is a random network so I will write a program that will use the rand() function to generate the points in the network and compute the response of this random network to applied current. I will be using C++ for this siumlation. Once I have produced the data from the simulation, the next step will be to construct hte spectral function from these data. I will study how different structures of the network are related to the specral function. I can then use the computed spectral function to find other properties of the netowrk and to characterize its structure.

Michael Woodbury
East High School, 1997
Hometown: Salt Lake City, UT
Majors: Mathematics
Year: Senior Faculty Mentor: Aaron Fogelson

Project Proposal:
Over the course of the summer and the first few weeks of the school year, I have been involved in research of blood clot lysis and mathematical applications of this process. This research has increased my interest in mathematics dramatically. I am excited to continue in the particular research with which I have been involved.

Specifically, I have worked on a model that imitates the experiments performed by J. W. Weisel et. al., as outlined in their paper, "Influence of fibrin network conformation and fibrin fiber diameter on fibrinolysis speed". The work I have done has not been concluded. I believe that I can refine and add to the current model, thereby gaining an increased understanding of the lysis process.

As I work more on the clot lysis model, I also will be working with my mentor, Professor Aaron Fogelson, and another undergraduate student, Nate Hancock, on a related topic. While this other topic will be primarily the work of Nate, his focus will be on another aspect of the clot lysis. Also, I will participate in his work (as he participates in mine) through a weekly meeting with Professor Foegelson in which we will discuss what we are doing as a group. I feel that this will add another dimension of collaboration beyond that of just the Senior Seminar and, as a result, I hope to learn and experience more.

I am looking forward to many more discoveries in mathematics research and expect my interest to grow exponentially as I work on this project.

Benjamin Young
Granite High School, 1994
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Lajos Horvath
(Project Beginning Spring 2003)

Project Proposal:
I am thinking about getting a Master's degree in biostatistics and I would like to have some experience with research. Dr. Horvath suggested the following project. Mr. Vanuiter in the Computer Science Department measured brain activities under different circumstances. There are ten locations and two types of matter (gray and white). At each location, ten measurements were taken using different levels of stimulu. Mr. Vanuiter has asked the following questions: Is there a difference between locations? Does the material matter for the response? Dr. Horvath suggested using polynomial fitting to the data and analyse. We should check if the model fits the data and, after that, answer the questions biologists would like to know.
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