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04-05 AR

# 2001-2002 Academic Year REU Projects

Brian Budge
Alta High School 1997
Hometown: Sandy, UT
Majors: Mathematics, Computer Science
Year: Senior
Faculty Mentor: Klaus Schmitt

Project Proposal:
People have been generating fractals and making use of properties of fractals ­- in image compression for example -- for many years. The most common way to generate these entities is through what is called an Iterated Function System (IFS). An IFS works on the basis of Banach's Contraction Mapping Principle. Basically what happens is the image is taken though any number of affine linear transformations (these include scaling, shearing, rotation, translation, and mirroring) to produce a new image. This new image is then taken through the transformations, and so on. More formally:

Let f(x): x element of S → y element of Snew | f is an affine transformation.
Let F(S): union over all x elements of S of f(x) → Snew
Then we can say that if set S is the original set of points, then
F(S) = S0
F(S0) = S1...
F(SN-1) = SN...
Until F(B) = B
This limit (where F is a map from a set to itself) is called the fractal.
I would like to work with Dr. Klaus Schmitt to create fractal images, and possibly image compression techniques based on this principle. We hope to introduce some new ideas to this including creating images from higher dimension fractals (4 or 5D), and also we would like to look into using non-linear transformations to generate fractals and fractal image compression techniques.

Rex Butler
Viewmont High School, 1998
Hometown: Tooele, Utah
Major: Mathematics
Year: Junior
Faculty Mentor: Henryk Hecht

2001-2002 AY Project Proposal:

My research project, under Professor Henryk Hecht, will be the study of the representation theory of finite groups. Finite groups arise in many areas of mathematics. One may study them through purely algebraic methods of group theory, or through the perspective of representation theory, which studies groups through their representations as linear transformations of various spaces.
As it is well known, every irreducible representation of a finite group is fully determined by its character, or in other words, the trace of the resulting linear transformation. Thus, in order to study representations, we may, equivalently, study their characters, which are functions on the group in question. I propose to compute characters of a variety of concrete representations of specific groups and study their features beginning fall semester 2001 through spring semester 2002.
This subject is of interest to me for multiple reasons. Representation theory involves groups ­ structures that lie at the heart of abstract algebra and mathematics as a whole. This subject will also allow me to work with tangible examples while also developing the ability to deal with abstract structures.

Aaron Cohen
East High School, 2000
Hometown: Salt Lake City, Utah
Major:
Year:
Faculty Mentor: Paul Bressloff

Project Proposal:
My project will be concerned with the study of spontaneous pattern formation in activation-inhibition systems. I will begin by studying the classical Turing instability in diffusion-driven systems, in order to gain insight into some of the modeling and mathematical issues concerned. I will then focus on the role of pattern formation in cortical dynamics, including a theory of geometric visual hallucinations that has recently been developed by Dr. Paul Bressloff (my faculty mentor) and Dr. Jack Cowan (University of Chicago).
One of the major goals of the project will be to gain an understanding of the role of symmetry and group theory in pattern formation, especially within the context of geometric hallucinations and the emergent patterns of certain cellular automata.

Matthew Dalton
Mountain View High School, 1998
Hometown: Orem, Utah
Major: Mathematics (Physics and Spanish minor)
Year: Senior
Faculty Mentor: Mladen Bestvina
Project Proposal:
There is a special type of automorphism for closed surfaces with hyperbolic structure, known as PseudoAnosov automorphisms. Under iterations of such an automorphism, it has been shown that the length L of a geodesic grows as L(n) = C(lambda)fn, where C is a constant and lambda is determined by the automorphism f. It has also been shown that lambda -> 1 as the genus of the surface goes to infinity. There are still, however, many things not known about lambda. For instance, fora given surface, what is the smallest possible lamda?
My proposal for the REU program is to (1) bring up my level of understanding of hyperbolic surfaces, PseudoAnosov automorphisms, and the origin of this l factor, and (2) attempt to answer some of these unanswered questions about l. If the problem is satisfactorily completed by the end of this semester, I will seek a new project for next semester.

Song Du
Hometown: Wuhan, China
Majors: Mathematics, Electrical Engineering
Year: Freshman
Faculty Mentor: Paul Fife
Project Proposal:
This is a project in "experimental mathematical materials science." Its purpose is to test the validity of the well-known "motion by curvature law" for the motion of grain boundaries of many varieties, when the motion is modeled by a system discrete in space and time. The approach will be to approximate a two-phase (or two-grain) material by a lattice structure with the state of the material being specified in some discrete manner at each lattice point. Then various supposed dynamical laws regarding how the system changes from one discrete time to the next are tested by numerical simulation. There will be an approximate boundary between two grains, and its motion will be simulated. Its curvature will also be calculated, and the validity of the afore-mentioned law tested within the confines of these various approximate models. The simulations will be performed through use of Matlab or C++.

Troy Finlayson
HS
Hometown:
Major: Physics
Year: Junior
Faculty Mentor: Kenneth Golden
Project Proposal:
I'd like to work with Professor Ken Golden on theoretically and numerically estimating the thermal conductivity of sea ice, and its role in mediating heat transfer between the ocean and the atmosphere in the polar regions. Seawater freezes into a composite material, containing pockets of air and inclusions of brine with high concentrations of salt throughout the ice. As the micro-structural composition of the sea ice changes, through variations in temperature and growth processes, the thermal conductivity of the sea ice changes. The amount of trapped air and brine in the sea ice is directly correlated with the rate at which the seawater freezes. Large variations in temperature, due to meteorological or perhaps longer term global warming effects, can significantly affect the thermal conductivity properties of the ice, which can in turn affect growth processes, leading to ice of a different composition. Subsequently, the heat transfer that occurs between the vast ocean and the air is also affected.
Sea ice is made up of three components; ice, inclusions of highly saline seawater, or brine, and pockets of air. The thermal conductivity of brine is close to that of pure ice, but the conductivity of the air phase is quite different. Thus as a first approximation we may treat the sea ice as two component medium, and we plan to apply the mathematical theory of bounds on transport coefficients to estimate the thermal conductivity of the sea ice, about which very little is currently known. In subsequent work we hope to be able to incorporate the effects of brine moving through the ice when it is warm enough that a percolating network of brine exists, and model how the advection of the brine through the ice can enhance the ability of the sea ice to transfer heat from the ocean to the air. Mathematically this will involve the analysis of nonlinear heat equations, as well as percolation theory.

Amy Heaton (Second Semester Only)
Vermillion High School, 2001
Hometown: Vermillion, SD
Majors: Mathematics, Chemistry
Year: Freshman
Mentor: Ken Golden
Project Proposal:
Current modeling of sea ice microstructure consists of 2-D slices taken throughout a portion of sea ice. Although these slices effectively convey information about their particular brine inclusions, these images alone are inadequate for describing the connectedness of the inclusions outside the plane of the slice. To take this imaging to the next step, I would like to work with Professor Golden on developing a sophisticated 3-D computer model which could be used to piece together these 2-D images to create a full-scale model of the microstructure of sea ice. Once accomplished, this model could be used to simulate the changes in brine permeability as it relates to variations in climatic conditions. It could be used to model the transport capabilities of materials through the conected pathways, and biological populations living in the sea ice microstructure could also be studied in new ways. Moreover, this modeling isn't limited to sea ice alone. Geological specimens such as rocks and glacial ice could be studied in much the same way, as the percolation theory underlying the analysis of sea ice microstructure relates directly to them as well.

Tara Henriksen
Cimarron-Memorial High School, 1999
Hometown: Las Vegas, Nevada
Major:
Year:
Faculty Mentor: Fred Adler
Project Proposal:
In a continued study (from the summer), I will attempt to explain why FEV1 is such an unpredictable variable. I will also separate the patients into groups by their predicted survivorship, and determine whether or not FEV1 is dropping yearly. An interesting variable to test its acute exacerbations ­ the only variable that is not directly a measurement of a patientıs physical health. It is almost a judgment call by the doctor admitting the patient to decide whether or not the visit should be counted as an acute exacerbation. For this reason we have decided that I will test the accuracy of the model excluding this variable. After completing the aforementioned tasks, I hope to answer the question about how accurate a one-year assessment of a CF patient actually is.

Jenny Jacobs (Second Semester Only)
Irvine High School, 1996
Hometown: Orange County, CA
Major: Mathematics
Year: Sophomore
Faculty Mentor: Andres Treibergs

David Lindsay (Nov. - Dec. 2001)
Viewmont High School, 1997
Hometown: Centerville, UT
Majors: Mathematics, Physics Year: Junior Faculty Mentor: Wieslawa Niziol
Project Proposal:
For my research project, I intend to study the theory of elliptic curves. This area of mathematics is of interest for many reasons. First, is that the study of elliptic curves brings together many different branches of mathematics: geometry, number theory, and modern algebra. Second, the study of elliptic curves is of interest because Fermat's Last Theorem, on of the most famous conjectures ever, was recently proven using ideas based in this theory. Third, the study of elliptic curves has many useful modern applications; in particular, elliptic curves can be used to construct cryptographic systems.
In my project, I plan on studying various computational aspects of elliptic curves. In simpler terms that means I plan on studying certain invariants of elliptic curves and how complex it is to compute such invariants. Along with this, I plan to research many already existing techniques and algorithms used to compute such invariants.

Josh Stewart
HS
Hometown:
Major:
Year:
Faculty Mentor: Jingyi Zhu
Project Proposal:
For my REU project I plan to explore the utility of mathematical models in finance, particularly in the area of derivative securities. I am certainly interested in studying the basic modern theory behind some models and their real world applications; in addition I will equally attempt to discover their strengths and weaknesses, efficacy and limitations in portraying pertinent information for investors. During my project I will receive help and direction from Professor Jingyi Zhu who has suggested the following list of topics for my studies:

I. Motivations and derivatives of the Black-Scholes model and the formula
II. Application of the Black-Scholes model to interest rate derivatives
III. Consideration of the Vasicek model for the term structure of interest rates
IV. Valuation of some sample portfolios using the Vasicek model
V. Dependence and sensitivity of the Vasicek model to various parameters
VI. Creation of spreadsheets that employ the Vasicek model
VII. Day-to-day tracking of portfolios with the help of the model

One of my main goals for this project is to apply the knowledge I gain to real life, thus to complement my theoretical studies, I have agreement from Sam Stewart, Portfolio Manager at Wasatch Funds Inc., to apply valuation models to the funds and portfolios at Wasatch. I will also complete other projects for the company, for example I have agreed to study certain statistics used to rate or measure the success of a portfolio such as the tracking error, the sharpe ratio and the alpha, in order to discover the assumptions being made in their calculation and to gauge their effectiveness in measuring a funds performance.

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