→ Funding for Undergraduate Research

→ Summer Mathematics Research Experience for Undergraduates (REU)

→ Interdisciplinary Mentoring Program in Analysis, Computation, & Theory (IMPACT)

The Mathematics Department is one of the top math departments in the nation for undergraduate mentored research, and was recently given a $1.3 million grant from the National Science Foundation to teach other universities across the nation its undergraduate research methodology. In the past four years faculty in the BYU Mathematics Department have been awarded more than $2,675,000 in competitive, national grants for use in teaching and mentoring. In 2008 alone, the Mathematics Department spent over $600,000 funding undergraduate research.

Undergraduate students can pursue research in various exciting topics. For example, Dr. Jeffrey Humpherys is currently working with students who are doing research on option pricing, and Dr. Tyler Jarvis is working with students who are doing research on cutting edge math related to theoretical physics. Many of these undergraduate research projects have led to publication and an opportunity to travel and present at various conferences.

In 2008 alone, there were over 53 students who participated in undergraduate research.

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## Why Undergraduate Research?

- Provide out-of-classroom learning experiences and apply class room knowledge to solve new problems.
- Develop and foster an analytical approach to doing research
- Gain motivation and create new knowledge
- Excellent experience and preparation for graduate school
- Develop oral and written communication skills
- Promote interactions with faculty and graduate students
- Make better informed decisions about your future career

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## Funding for Undergraduate Research

## Origami and Design Mathematics Undergraduate Research

In this research experience, you will help lay the mathematical foundations needed to generalize the principles of origami design in order to be utilized in other settings, particularly the design of compliant mechanisms. Compliant mechanisms are mechanisms that gain their motion from the deflection of flexible components rather than from traditional motion element such as hinges and bearings. Compliant mechanisms are often highly desirable in design because of reduced part count, ease of manufacturing, compactness, low weight, low wear, reduced maintenance, and high precision. In many respects, origami represents the ultimate efficiency in creating sophisticated motions: it is constructed from a single, regular-shaped sheet of paper using one fabrication process (folding). The goal of this research to develop the mathematics needed to establish the principles that unite these two areas of design.

Please visit Origami and Design Mathematics Undergraduate Research Homepage for more information.

## Summer Mathematics Research Experience for Undergraduates (REU)

The REU is an exciting eight-week program to provide undergraduate students with the opportunity to experience doing mathematical research; to encourage undergraduate students to attend graduate school in mathematics; and to prepare participants to be successful in graduate school. Students come from all over the country to be part of the BYU REU.

Please visit the Mathematics Research Experience for Undergraduates (REU) Homepage for more information.

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## Interdisciplinary Mentoring Program in Analysis, Computation, & Theory (IMPACT)

The Interdisciplinary Mentoring Program in Analysis, Computation, & Theory (IMPACT) is an interdisciplinary cooperative of research groups and laboratories committed to mentoring BYU students in high-quality research in the pure and applied sciences.

Each year, a number of students are selected to participate in a year-long program in which they receive a $10,000 fellowship stipend.

Please visit the Interdisciplinary Mentoring Program in Analysis, Computation, & Theory (IMPACT) Home Page for more information.

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## Finding a Research Mentor

#### Darrin Doud

Undergraduate research with Dr. Doud can include topics such as modular forms with connections to Galois representations, diophantine equations, elliptic curves, and LLL-reduced lattices. A prerequisite for all of this research is Math 371, and several topics would require Math 372.

*Winter 2007*

Student: Kevin Powell

Title: Singly ramified Galois extensions

Kevin succeeded in proving a criterion that shows when a degree six polynomial in a certain family of polynomials has Galois group the full symmetric groups. He presented his work in the college Spring research conference, and produced a written report describing his work. He has graduated, and will not be working with me as an undergraduate researcher in the future.

*Winter 2006*

Student: Meghan DeWitt

Identified polynomials defining certain kinds of Galois representations, using the theory of elliptic curves and computational class field theory. Her honors thesis and a paper are almost done.

* Fall 2006*

Student: Kevin Powell

Kevin Powell continued his study of singly ramified Galois extensions of Q. He is attempting to prove that a certain family of degree 6 polynomials which produce extensions of Q ramified at only one prime actually have Galois group isomorphic to the symmetric group on six letters. He has produced a good deal of computational evidence, and has done some computations of resolvent polynomials which has the potential to prove the theorem that he is aiming for. His work will continue in Winter 2007, and he will present his research at the Spring Research Conference.

*Fall 2006*

Student: Wayne Rosengren

Wayne Rosengren began a study of Diophantine equations related to the ABC conjecture. In particular, we are studying the Brocard-Ramanujan Diophantine equation. This equation in two variables, n and y, was studied by Brocard and Ramanujan in the early 20th century. Until 1999, it was known that there were only three solutions with n < 70. In 1999, Berndt and Galway proved that the equation has no additional solutions with n < 1,000,000,000. In November, Wayne succeeded in proving (computationally) that there are no additional solutions with n < 1012. He is also studying methods to bound possible solutions, using standard conjectures. Our ultimate goal is to prove that some effective form of the ABC conjecture will imply that there are no solutions with n < B for some bound B. Hopefully, we will be able to determine B, and have it be less than 1012. Wayne will present his research in the number theory seminar this month, and in the Spring Research Conference in March.

*Summer 2006*

Student: Meghan Dewitt

During the summer, she completed and defended her Honors thesis, “Finding a Galois representation corresponding to a Hecke eigenclass.” We also worked together to prepare her thesis for publication as a joint paper. This paper is nearly finished, and we plan to submit it for publication within the next month. In addition, Meghan attended the Young Mathematicians Conference in Columbus, Ohio, where she presented her research. The web page for the conference can be found at http://www.math.ohio-state.edu/conferences/ymc/

#### Tyler Jarvis

*Winter 2007*

Students: Natalie Wilde, Nathan Grigg, Michael Anderson, Julian Tay

Title: Tropical Algebraic Geometry

This is a continuation of our ongoing project to study the foundations of tropical geometry and to understand the relations between tropical and classical geometry. We have proved tropical analogues of several classical theorems. This resulted in Honors theses for Julian and Nathan, and research presentations by all these students at the Spring Research Conference and at the Mathematical Association of America Meeting in March. (Visit Tyler Jarvis’ Mentoring Page at https://math.byu.edu/tropical/)

*Winter 2007*

Students: Eric Lewis,Chris Challis

Expected values in pari-mutuel lotteries. Eric spoke in the Spring Research Conference and a draft of a paper is close.

#### Chris Grant

*Winter 2007*

Student: Quinten Christensen

Title: Optimizing Valve Geometries for a Small Scale Pump

Quinten was involved in the mathematical modeling of a milliscale pump design that will ventually produce differential equations that will be solved numerically to determine optimal valve lengths and membrane hole geometries. In addition, he conducted preliminary investigation of applicable numerical schemes. This project represents Quinten’s Honors Thesis, the abstract of which follows:

“This research investigates the validity of a milliscale pump design. Primary attention is paid to two membrane vibration models that describe membrane displacement caused by pressure changes in the cavity of the form sin(kt). One model relies on the method of lines while the second is the double Fourier series solution of the partial differential equation. The double Fourier solution is then solved to account for pressure values input by the thermodynamic equations. The failure of this modified model to accurately predict the membrane vibration is contrasted with the previous models. Additionally, the thermodynamic equations adapted from Dr. Aaron Astle’s research are presented in context of the current pump design.”

#### Stephen Humphries

- Research in representation theory: representations of finite groups; in particular topics involving S-rings and fusion of character tables. This is sort of like taking a quotient of a character table of one finite group and getting the character table of another finite group. The techniques are combinatorial and algebraic. You will need to have taken math 371 to participate in such a project.
- Counting in Monoids: given a monoid presentation <x,y|w(x,y)=u(x,y)> and a word q in x,y, we consider the problem of determining the number of words in x,y that are equivalent to q^n. This gives an infinite sequence of natural numbers. For example, consider <x,y|x=y> and you get the central binomial coefficients (2n,n).
- Problems related to certain infinite groups called braid groups. These are related to knots in 3-space and have wonderful combinatorial properties. You will need to have taken math 371 to participate in such a project and a course in topology would also be useful.

*Winter 2007*

Students: Kayla Barnes and Nathan Perry

Title: Braid groups and free groups

This is a continuation of the mentoring that I did with the same two students during Fall 2006. We were studying S-rings, braid groups and free groups. One outstanding problem is the classification of S-rings of dimensions 3 and 4. Another is the application of the S-ring theory to certain problems involving free groups and braid groups that I have been thinking about. These latter problems include that of classifying the composition factors in certain finite quotients of the free group or the pure braid group. This is too general a problem of course and the emphasis was on those quotients which are generated by powers of conjugates of the free generators of the free group or the Dehn twist generators of the braid groups. We are some way along in both of these projects. During Fall semester (when I began working with Kayla and Nathan) the students read the material basic to S-rings (â€œFinite permutation groupsâ€ by Wielandt) and have read some papers that have extended these results. We have applied these results to the problems that I have outlined above and obtained some results that are not yet suitable for publication, but which I hope can be enhanced in such a way as to become good papers. Both Kayla and Nathan spoke at the Spring research conference on these problems. Nathan has aspirations of doing a graduate degree in mathematics at BYU. (Kayla is now pregnant and has different plans.).

The mentoring has continued into Winter 2007 semester during which time we have studied projective geometry as well as those topics mentioned above. We have shown that certain groups cannot occur in the above mentioned composition factors. This involved looking at the conjugacy classes of the groups PGL(2,q) and we hope to extend the results to other groups of the kind PGL(d,q).

I found out during this semester that I was awarded a MEG to cover more mentored research.

#### Kening Lu

*Winter 2007*

Student: Yi Luo

Title: Numerical solutions for pricing of spread options

The focus of this research was to simulate the pricing of spread options. A spread option is an option whose payoff is based on the difference of the prices of two assets. Yi Luo continued his study on a class of stochastic differential equations arising in the pricing of spread options. He obtained numerical solutions by solving the distribution functions and developing numerical algorithms for the stochastic differential equations. He presented his results at the Spring research conference.

*Fall 2006*

Student: Yi Luo

Yi Luo studied a class of stochastic differential equations arising in the pricing of spread options. A spread option is an option whose payoff is based on the difference of preces of two assets. He obtained numerical solutions by solving the distribution functions and developing numerical algorithms for the stochastic differential equations. He planed to present his results at the Spring research conference.

#### Tiancheng Ouyang

*Winter 2007*

Student: Skyler Simmons

Title: Numerical computation of n-body problem

- Computation software: (Skyler Simmons) Study the structure of 3D computation and simulation in Java. Simulate the orbits (including periodic and non-periodic orbits) and simulate bi-collision behavior of orbits.
- Numerical computation of n-body problem in Matlab.(study numerical simulations in Matlab study N-body problem and modify comment the existing programs)
- Study Dynamical system and Hamiltonian system. (took part in the Dynamical system seminar which is hold two hours per week for winter semester)
- The 3D simulator program and numerical simulation of n-body problem was presented in Spring conference.

*Fall 2006*

Students: Jared Duke (from Sep. to Dec. 2006), Skyler Simmons (from Sep. to Dec. 2006 and continue to 2007)

Title: Numerical computation of n-body problem

- Computation software: (Skyler Simmons) Study the structure of 3D computation and simulation in Java. Simulate the orbits (including periodic and non-periodic orbits) and simulate bi-collision behavior of orbits.
- Numerical computation of n-body problem in Matlab.(study numerical simulations in Matlab study N-body problem and modify comment the existing programs)
- Study Dynamical system and Hamiltonian system. (took part in the Dynamical system seminar which is hold two hours per week for winter semester)
- The 3D simulator program and numerical simulation of n-body problem was presented in Spring conference.

#### Vianey Villamizar

Project 1. This project is concerned with the development of 3-D grid generators with nearly uniform cell volume and surface spacing, respectively. The proposed algorithm will be based on recently developed 2-D quasi-linear elliptic grid generators with similar features. It requires knowledge of boundary value problems of partial differential equations (Math 347), numerical iterative methods for linear and non-linear systems, interpolation techniques (Math 311), and good programming skills.

Project 2. We propose to obtain numerical solution for the Helmholtz equation in locally perturbed half-plane with Robin-type boundary conditions. This problem is motivated by a system sea-coast where each media is represented by a half-plane. Knowledge about partial differential equations (Math 347), numerical solution of partial differential equation (Math 511), and numerical methods in general is desirable.

*Winter 2007*

Student: James Taylor

Title: Elastic Scattering from High Reynolds Number Fluid-Filled Cavities.

We continue our work on elastic scattering from spherical inclusions. The governing equations in the fluid region were derived and the matching with the elastic medium was started. We expect to complete the results for the spherical and cylindrical inclusions and submit a paper to SIAM Applied Math Journal by the end of the summer 2006.

Currently, James Taylor is working with me in his Honor Undergraduate Thesis. I would like to continue mentoring him during the Fall 2007 if I can get funding for him.

*Fall 2006*

Student: James Taylor

Title: Elastic Scattering from High Reynolds Number Fluid-Filled Cavities.

We were able to find approximations for spherical inclusions. Now, we are working with cylindrical inclusions. Since there are two small parameters the inverse of the Reynolds number and the wavenumber, the method of matched asymptotic expansion needs to be applied in two different regions of the physical domain. We expect to complete the results and submit a paper to SIAM Applied Math Journal by the end of the summer 2007.

#### Gregory Conner

Over the last few years I’ve had several undergraduate students work with me on research projects in low-dimensional wild homotopy groups. Topics range from geometric — understanding how “fractal-like” objects in the plane can be deformed in to others, to algebraic — understanding infinitely stranded braid groups, to analytic — understanding how to prove very delicate continuity arguments on wild subsets of our universe. These undergraduate research projects have all turned into masters’ theses at BYU and have lead each of the students into a high-quality mathematics Ph.D. program such as Vanderbilt, Tennessee and BYU.

*Spring/Summer 2006*

Students: Keith Penrod and Curt Kent

Keith has been studying background material on combinatorial group theory automorphism groups of free groups to prepare himself for our research on mapping class groups of wild spaces –which generalizes both the theory of free group automorphisms and the theory of braid groups. Since there is a lot of background it will be some time before he will have original results — we’re hoping for good things by winter.

Student: Curt Kent

Curt is picking up the theory of low-dimensional homotopy theory at light speed. He’s very close to a really good result on fundamental groups of planar sets. We were hoping to have a paper ready by the end of the summer but that didn’t quite pan out. He’s working about 60-70 hours a week on construction to save money to be able to study full-time in our graduate program this fall. He turned down a very good offer from Purdue to stay at BYU, and he’s paying the price by having to work construction this summer. He’s certainly one of the smartest and hardest working students I’ve had.

*Winter 2006*

Student: Jeremy West

Lossless compression techniques: developing a mapping between fixed-length binary strings in the source file and shorter fixed-length in the compressed file, together with recovery methods.

Student: Curt Kent

Proved that two homotopy-equivalent planar Peano continua have “bad sets” (sets of points fixed under every self-homotopy) that are homeomorphic, and that each point in the bad set corresponds to a Hawaiian earring retract of the space.

Student: Micah Croft

Investigated applications of Zermelo Pair-Wise ranking, to Google’s PageRank algorithm and to ranking US Senators. Google did not cooperate with enough information, but the Senator ranking proved to model voting behavior very well.

Student: Brent Kirby

Brent invented a combinator theory in quantum computing and investigated making it work in reversible systems.

Student: Audrey Kearl

Analyzed voting patterns of Supreme Court Justices, 1993-2006, to establish rankings that proved to be more accurate then win-percentage rankings.

#### David Cardon

*2006*

Student: Adam Rich

Title: Turan inequalities and subtraction free expressions

By using subtraction-free expressions, we are able to provide a new proof of the Turan inequalities for the Taylor coefficients of a real entire function when the zeros belong to a specied sector.

*2007*

Student: Steven Adams

Title: Sums of entire functions having only real zeros

We showed that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we constructed sums of exponential functions having only real zeros, constructed polynomials having zeros only on the unit circle, and obtained the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discussed a similarity of this result with the Lee-Yang circle theorem from statistical mechanics. Also, we stated several open problems.

*2006*

Student: Sharleen de Gaston Roberts

Title: An equivalence for the Riemann Hypothesis in terms of orthogonal polynomials

*2005*

Student: Sharleen de Gaston Roberts

Title: Differential Operators and Entire Functions with Simple Real Zeros

*2002*

Student: Pace Nielsen

Title: Convolution operators and entire functions with simple zeros.

#### Jeffrey Humphreys

*Spring/Summer 2006*

Student: Blake Barker

Blake wrote and executed numerical methods to explore the stability of traveling waves. He is currently exploring the stability of viscous shocks in the p-system (i.e., isentropic Navier Stokes) using finite difference methods which are a variation on the Crank-Nicholson Method and the work of Beam and Warming.

This work, which is in collaboration with a previously supported student Keith Rudd, will turn into a section in a paper that I am currently writing which uses energy methods and Evans function computation to further explore the stability properties of this system.

#### Denise Halverson

*Winter 2006*

Student: Keith Penrod

Proved that the minimal Steiner tree for three points on a flat torus is contained in one rectangular fundamental domain.

Students: Melissa Mitchell,Katie May

Compared lune paths and their competitors for minimal Steiner trees, proving that a lune path is minimizing under certain conditions.

Student: Greg Miller

Completed a simulation program that calculates and displays solutions to the three-point Steiner problem on a flat torus.

#### Lennard Bakker

Undergraduates in this program work in dynamical systems, specifically in the areas of celestial mechanics, complex dynamics, or toral automorphisms. Students who have successfully mastered the concepts in Math 314, 332, 334, and 343 are sufficiently prepared to engage in research in these areas.

*Winter 2006*

Student: Jared Whitehead

Completed and defended his honors thesis in complex dynamics.

*Winter 2009*

Student: Steven Flygare

Completed a project titled *Dynamics of the Collinear Three-Body Problem*

#### Paul Jenkins

We study problems in number theory related to modular forms and their coefficients. Students who have successfully mastered the concepts in Math 371 and 352 will be better prepared to do research in these areas. Problems in computational elementary number theory are also available. More information on papers written by students in this group is available here. Interested students are invited to attend meetings of the Computational Number Theory research group at 10 AM on Thursdays in TMCB 301 during fall and winter semesters.

#### Scott Glasgow

Undergrads in this program either work in Mathematical Finance, including Extremal Events in Insurance and Finance, or in certain components of mathematical physics—symmetries, conservation laws, integrability. These topics require interest in probability theory, differential equations, and/or complex variables, and students will have had success in courses 334, 343, and/or 332.

Shiul Khadka, BYU electrical engineering, is trying to prove the following conjecture: For each of the distinct “unary” free energies1 for a specified passive electrical circuit, there exists a Darlington realization of the circuit2 for which the free energy is, at any given time and for all admissible voltage time series inputs, precisely the same as the total energy stored in each of the lossless elements of the realization, and for which the unique dynamical notion of loss associated with the specific free energy is precisely the same as the energy dissipated in the one and only lossy element of the relevant Darlington realization. Besides making statements about the design and synthesis of electrical circuits, which are man-made devices, the conjecture, if true, gives insight into the lossy energetics of natural electromagnetic media such as dielectrics: the questions there are a) Which of the family of free energies of a natural medium is the one whose loss corresponds to the time-resolved rate of heat production in the medium? and b) What are the significances of the other free energies and their losses if they don’t correspond precisely to heat production?

Jonathan Christensen, BYU stats and mathematics double major, is trying to develop the fair price for any (reasonable) contingent claim written in a certain specific type of incomplete market3, namely that of single risky asset undergoing a geometric Cauchy-Process. Buyers and sellers prices have already been developed for such a market4, giving lower and upper bounds for all prices disallowing arbitrage. One the other hand, Jonathan will develop the analog of the unique mean-variance hedging price,5 which is made difficult by the fact that the Cauchy process not only has no second moment, but also does not have a first. (Here we view the Cauchy process as the limit of other processes with both moments.)

#### Ken Kuttler

Student: Josh McKinney

Research on the general version of Green’s theorem for a simple rectifiable closed curve will be attempted. The history of the problem will be researched and all the mathematics leading to its proof will be proved. This includes the Jordan curve theorem and relevant results which make the proof possible. Generalization to Stokes theorem will then be obtained. Pathologies of simple rectifiable closed curves will also be explored along with why certain arguments cannot work because of these pathologies. If there is time and inclination, applications to complex analysis will be established leading to some very good versions of the theorems in this subject.

The QR algorithm is one of the most amazing ways to compute eigenvalues and eigenvectors. This project will involve extending the basic results on the algorithm to include some of the many techniques for improving the algorithm. Also, the history of methods for finding eigenvalues will be studied along with a few other methods like the shifted inverse power method.

This project is to study the topological degree and its applications to topology. This will include proofs of the Jordan separation theorem, the invariance of domain theorem, the Brouwer fixed point theorem and many others. The study of various methods for producing the degree will also be considered along with the uniqueness of the degree. Extensions to the Leray Schauder degree will be explored which will lead to yet more applications.

This project will involve producing a short book on undergraduate differential equations which can be completely understood by undergraduates who have not studied the Jordan canonical form. This means, for example, avoiding the stuff about generalized eigenvectors and the reliance of the existence of the Jordan canonical form which is really at the heart of the typical undergraduate approach to first order systems. Abel’s formula for first order systems will be considered as well as the usual techniques. Geometric theory will be considered slightly beyond what is done in math 334. Existence and uniqueness theorems will be studied and proved completely. This will include a presentation of uniform convergence and a correct definition of the integral including the theorem that continuous functions can be integrated. If there is time, power series methods and regular singular points will be included.

#### Michael Dorff

Minimal surfaces and complex-valued functions:

We investigate minimal surfaces in R^3. In some ways, minimal surfaces can be thought of as soap films that form when a wire-frame is dipped in soap solution–they tend to minimize the surface area for a given boundary condition. Images of minimal surfaces can easily be displayed by using computers, and this lends itself nicely to student explorations. We will use results about analytic functions from complex analysis (Math 332) to investigate minimal surfaces. To help introduce students to this topic and begin to do research, we have received a grant to write two chapters in a book on this topic along with exploratory problems using applets.

#### Rodney Forcade

Students would study point lattices — discrete, full-rank subgroups of R^n, as applied to physics and cryptography. Possible topics include lattice algorithms and periodic colorings of lattices. Some minimal knowledge of group theory may be helpful.

*Winter 2007*

Students: Heather Farley, Kevin Powell and Ben Warner

Title: Cryptography

The nominal subject of our research was cryptography, but we worked specifically on algorithms for integer relation finding and lattice reduction. After comparing the workings of those algorithms, we devised unusual ways to use them. We are taking a second look at the algorithm of Viggo Brun, which was mainly applied to the simultaneous linear approximation problem, and known to work infallibly for integer relation finding only in dimension three or less. We may be able to show that it fails in higher dimensions only on a set of measure zero. Since it is far simpler to execute and has much less overhead than infallible algorithms like PSLQ, it may be of practical use in this regard. We are presently working to finish and write up our results with an eye to possible publication.

*Winter 2006*

Student: Heather Moore

Simplified the search for Wall primes and found a new way to support the standard heuristic. Hope to publish a paper during Spring/Summer.

#### Xian-jin Li

1. Research on spectral theory of automorphic forms:

In 1956, A. Selberg introduced trace formulas into the classical theory of automorphic forms, a theory whose origins lie in the work of Riemann, Klein, and Poincar\’e. The theory of automorphic forms is intimately connected with questions from the theory of numbers, and is one of the most powerful tools in number theory. The discrete spectrum of the non-Euclidean Laplacian for congruence subgroups is one of the fundamental objects in number theory. My research interests are Selberg’s trace formula, Selberg’s eigenvalue conjecture, and the multiplicity of the discrete eigenvalues.

2. Research on Beurling-Selberg’s extremal functions:

In 1974, A. Selberg used the Beurling-Selberg extremal function to give a simple proof of a sharp form of the large sieve. By using the large sieve, E. Bombieri proved in 1965 a remarkable theorem on the distribution of primes in arithmetical progressions that may sometimes serve as a substitute for the assumption of the generalized Riemann hypothesis. The large sieve is closely related to Hilbert’s inequality. An open problem is to prove a weighted version of H. L. Montgomery and R. C. Vaughan’s generalized Hilbert inequality. A weighted large sieve can be derived from the weighted Hilbert inequality, and is fundamentally more delicate than the large sieve. It has important arithmetic applications. My research interest is to attack the open problem on the weighted Hilbert inequality. ”

#### Roger Baker

Certain Diophantine approximation assertions can be proved ‘almost everywhere”, that is, except for a set of numbers having zero Lebesgue measure. Undergraduate students who have taken, or are taking a course, on Lebesgue measure can prove new results about this after quite a brief introduction to the subject. Another facet of the work is the study of such exceptional sets which have Hausdorff dimension less than 1. Ryan Coatney, currently an undergraduate, is writing up such a paper (jointly with myself and another mathematician, Harman, who works in London).

#### Jessica Purcell

We will work on projects in knot theory, 3-manifolds, and hyperbolic geometry. Knot theory deals with questions about knots in space. One way to create a knot would be to take a giant ball of string, shake it up for a while in a box, and then tie the two ends together. Knot theorists ask: given such a ball of string, can it be untangled without cutting? If not, can it at least be untangled to something simpler? How do you tell two knots apart?

Rather than study the knot itself, sometimes it is easier to study the complement of the knot. That is, think of the knot as living in space, and study all of space except the knot. This object — space minus the knot — is an example of a 3-manifold, called a knot complement. Many knot complements admit a hyperbolic structure, which is a special type of metric. That is, a hyperbolic structure gives us a way to measure distances, areas, and volumes inside a 3-manifold. If we start measuring such things in the knot complement, we can often determine interesting information about the knot.

We will study families of knot complements and their generalizations, as well as other hyperbolic 3-manifolds, using geometry. For certain projects, it may be helpful to have taken complex analysis (math 332), and abstract algebra (math 371).

#### Sum Chow

Pressure recovery problem: Numerical solution of non-Newtonian fluid flows is a theoretically and computationally challenging subject that also has much application in many practical areas. One approach to handle such problems is to employ divergence free finite elements to compute the velocity field. With the velocity is known, it is also of interest to find the pressure. How one recovers the pressure leads to several interesting problems that will be addressed in this senior undergraduate or graduate level research project.

GUI for sincpack: The sinc method is a theoretically advanced method for finding approximate solutions of differential equations. Dr Stenger from the University of Utah has recently released a set of matlab routines called sincpack. The project, which is accessible to any undergraduate with a background in programming and basic numerical methods, is to develop a GUI to aid novice users to solve classical problems without having to know the details of matlab or sincpack.

Image processing problems: Several interesting computational problems arise from image deburring and edge detection that are of interest in several application areas. One approach is to apply alternating direction methods to compute the solution of a nonlinear partial differential equation related to image processing. The research project is at senior undergraduate or graduate level.

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