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04-05 AR

# 2003-2004 Academic Year REU Projects

Ryan Behunin
Alta High School, 1998
Hometown: Sandy, UT
Major: Physics
Year: Senior
Faculty Mentor: Andrej Cherkaev

Project Proposal:
Studying optimal design in unkown environments will help us to understand more reasonable optimization techniques. In typical situations standard optimization techniques fail to account for realistic loading. These techniques require that one optimize the structure against a specified load. By concentrating the load in one area and direction, the resistance in other directions decreases This process is useful and accurate according to the specific load, but real life situations require a more general optimization. Realistically loads can be exerted in any direction or distribution, and optimizing with respect to any possible load will give a more versatile structure.

To remedy this weakness, I plan to study the problem of optimal design with unkown load. Being that the forces are unkown the structure must necessarily optimize everywhere. Given a constrained load, one can formulate a worst-case scenario for which one must optimize against. Therefore, this method will provide a structure suitable to bare any load consistent with the constraints. For some cases, this scheme will provide a more stable design. Throughout the next semester, I plan to work with Dr. Cherkaev to develop these ideas. Development of these ideas would directly open this method to be used in engineering and biology.

Project Report

John Faust
Olympus High School, 1998
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Paul Bresslof

Project Proposal:
My project is to develop a model of direction selectivity in visual cortex. Suppose that a bar is moved across the receptive field of a neuron in one of two directions orthogonal to the bar's orientation. A direction selective neuron only responds to one direction but not to the other. We will analyze a nonlinear dynamical network model that achieves direction selectivity through spatially asymetric recurrent excitation and inhibition. We will then investigate how direction selectivity can be incorporated into the pinwheel structure of the orientation map.

Project Report

David Groulx
Forest Hills Central High School, 2000
Hometown: Grand Rapids, MI
Major: Mathematics
Year: Senior
Faculty Mentor: David Hartenstine

Project Report

Major: Biology
Year:
Faculty Mentor: Ken Golden

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.

Michael Hofmann
Highland High School, 1999
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Gordan Savin

Project Proposal:
I would like to continue the research that I began last spring with Gordan Savin. The project that I am currently working on started last April when I read a paper by Benedict H. Gross which postulated that numbers of the form pÖ3, where p is a Mersenne prime, have continued fractions with a particularly large period. I wrote a computer program using MATLAB to quickly and accurately calculate the period of numbers of the form (pÖq)/r and found there were a number of interesting patterns, particularly when looked at prime p, q = 3, and r = 1 the plot in figure #1 resulted. The way that the periods seperated into different rays was extremely interesting. I found that by splitting up primes modulo 24, the resulting plots would include only some of the rays. As an example in figure #2 shows primes congruent to 7(mod 24), which happens to be where all the Mersenne primes are. This plot shows that the first ray appears, but the next three do not. The conjecture about Mersenne primes basically asks if they are all in this top ray.

Professor Savin and I had problems trying to figure out any other patterns in these numbers, but finally we determined what could be causing these rays to form. Namely by using the class number (h(p)) of the order Z+ZpÖ3 in the field Q(Ö3). In quadratic forms with positive discriminant (the discriminant in our case is 12p2) the length of the continued fraction is the length of the cycle of the units in the field, while the class number is the number of such cycles. We found using computational methods that all the numbers in the first ray have class number 2, that all the numbers in the second ray have class number 4, etc. If you notice the slope of the first ray is approximately 1/2, for the second ray it is approximately 1/4 and we conjecture that the period of the continued fraction pÖ3 is approximately p/h(p).

Our current goal in this REU is to find proofs and make all this mathematics rigorous, since now it is basically the results of computation.

Project Report

Zsuzsanna Horvath
Brighton High School, 2002
Hometown: Salt Lake City, UT
Major: Mathematics
Year: Junior
Faculty Mentor: Davar Khoshnevisan

Project Report

Les Kartchner
South Summit High School, 1995
Hometown: Kamas, UT
Major: Mathematics
Year: Senior
Faculty Mentor: Gordan Savin

Project Proposal:
This will be a second semester of research that Dr. Savin and I have been pursuing. We have researched various areas of cubic reciprocity, p-adic analysis, and representation of binary quadratic forms. We will use the extended period of time to continue researching binary quadratic forms and extend the knowledge to deal with representation of binary cubic forms.

Pejman Mahboubi
Major: Mathematics
Faculty Mentor: Nat Smale

Project Proposal:
Real Analysis is one of the most brilliant evidence of human power of reasoning, and it may be true to be considered the core of mathematics.

In spring 2003, the course math 5210 acquainted me with Lebesgue integral that, as the author of the book believed, "stands at the doorway to the twentieth century mathematics."

In the same course we used the Lebesque integral, among plenty of other applications, to analyze Fourier series in a preliminary level, which I find it very tempting to work on in a rigorous way.

Unfourtunately, the math department at the UofU, except a course in applications of Fourier series, which deals with solving some standard forms of differential equations, doesn't offer any abstract and pure course in Fourier analysis and wavelets.

In this research I want to acquire a well-developed knowledge of measure theory, Lebesgue integral, and finally concentrate on Fourier analysis and wavelets. At the end of this research I will be one step closer to the starting point of an original research.

Project Report

Hyrum Nielsen
Hillcrest High School, 1996
Hometown: Idaho Falls, ID
Major: Mathematics
Year: Senior
Faculty Mentor: Florian Enescu

Project Proposal:
Algebraic geometry and commutative algebra represent an important area of research in today's mathematics. It consists of analyzing solutions of polynomial equations from a geometrical perspective. For example, given a polynomial f in C[x1,..., xn] one can look at the set of its solutions, more precisely all x = (x 1,..., xn) such that f (x) = 0. This set is naturally contained in the Cn and can also be investigated from a geometrical point of view. Algebra, topology and geometry interact in this broad area of mathematics and I have been already exposed to them. I would like to learn in depth this rich interplay in algebraic geometry by being involved in a research project.

The project will study the jet schemes associated to an affine hypersurface X = V(f) Ì CN. Namely, I want to find a systematic and computable way of describing the jet scheme of a hypersurface starting with its equation. This research will consider f = 0, a hypersurface in the N dimensional complex affine space as described above, and I will look at the collection Xn of jet schemes of order n, for n >. The project will focus on common examples of hypersurfaces with isolated singularities.

I would like to be able to compute the jet schemes directly by looking at the higher order partial derivatives of f and investigate the structure of Xn by using classical algebraic methods. The notion of jet schemes, while introduced by Nash decades ago, has been a current topic of research in algebraic geometry connected to other deep and abstract parts of mathematics, including motivic integration. Some recent remarkable results, such as Mustata's theorems, relate important geometric properties of the hypersurface to classical algebraic features of jet schemes. His analysis relies on motivic integration. What would I like to do is to examine this connection by using classical tools while looking at various examples of hypersurfaces. It is known that a surface admits at most rational double points if and only if all its jet schemes are irreducible if and only if Xm is irreducible for m = 5. I would like to prove results of this type for hypersurfaces of arbitrary dimension. If result can be extended to arbitrary dimension, can one predict what value of m will work ( as m = 5 does in the surface case)?

As opposed to the current techniques that rely on abstract mathematics, I will use traditional methods and stress computation in my study. I hope this research will help make the notion of jet schemes more approachable, lead to the discovering of new properties and settle some of the open questions relating to the subject.

This analysis will examine whether it is possible to concretely describe the relation between the geometry of f = 0 and the algebraic properties of the jet schemes f = 0, at least in many useful examples. The research was suggested by Assistant Professor Florian Enescu and I regard it as a great opportunity for both research and actively learning about an important area of mathematics.

Project Report

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

Project Proposal:
My research project will be to investigate some of the many different kinds of networks. I will study come of the most interesting networks around us. My goal is to understand the way they work and behave and to see what are some of the fundamental things that effect how a network will behave. Once I have studied the network closely and can clearly break it up into its simple components and know what they mean, then I can simulate it under different conditions to see how it effects the network as a whole and its behavior. Once I have some simulation and result from it then I can try to construct the spectral function. This function will give me information about the structure of the network. It would be interesting to see if there are other ways to characterize the structure of a network.

Project Report
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