Math 4901
Math majors at UMM are required to complete a senior seminar (Math 4901) as part of their major requirement. The senior seminar runs over two semesters, and the choices of faculty advisor and topic should be made early in the semester you first register, if not before.
My goal in creating this page is to help you decide if I would be a suitable advisor for you, and to give you ideas for senior seminar projects. Feel free to stop by and chat if you have any questions.
My Expectations of My Advisees
I will want to set up a weekly meeting time (typically we block out on hour, but early on we may not need all that time). Since this is a one credit course, spanning two semesters, you will want to spend roughly 1.5 hours a week on your project.
Early in the first semester, I will want to see clear evidence that you have a suitable project, and that you have a very clear idea about what your project will entail and what you need to do to complete it. By the end of the first semester, you will need to have begun writing your paper, and have an outline of what you need to do in the second semester. Time moves quickly in the second semester, and for your own benefit I will want to see significant progress by the end of the first semester.
There are four components to the grade a student receives for Math 4901. The web page for Math 4901 contains plenty of information on each, but I will add a few of my own comments here.
Mathematical Foundation
You will need to do a literature search to place your project in context. Also, you need to understand the mathematical foundations of your project. Typically, you need a greater understanding of the area your project is in than will appear in either your paper or your presentation. This broader understanding of your project will allow you to answer questions at your talk confidently.
Typically, the work you do towards context and foundation will occur in the first semester. The work you do may involve going to the library and finding resources, refining the question you want to address, and informal discussions with me. Make sure you note any references you use for possible inclusion in your paper.
The Written Paper
The paper must have: references (in the text of the paper), section headings, a clear introduction to your problem, a clear conclusion, and sufficient theorems/diagrams/tables/formulas to make your paper understandable to the reader. I can (and will!) help you with this if you have any questions as we go along, but you should try to get a head start on writing the paper as soon as you can.
Writing drafts of the introductory sections of your paper early on is also a good idea. At this time, you should also be making choices about your final paper. Not big choices, but little ones. Are you going to use LaTeX or Word or something else to write your paper? If you are going on to graduate school, LaTeX may be a good choice for you. If you haven't used LaTeX before and would like to learn it, I can help you get started. They key if you are going to be learning some new technology is to start early, so possible problems (and there is always at least one!) are not a big issue.
The Presentation
The presentation may not contain all the information that is in your paper. The presentation should be pitched at a level that allows your peers to follow along. It is a good idea to plan on spending ten or fifteen minutes going over basic material, explaining definitions or other concepts, so that there is some context for what will follow.
As far as presentation tools, you should use what is appropriate to get your ideas across. There is nothing wrong with using overheads! Certainly, you can use Powerpoint, but if you do, make sure the content comes across, and that the content is not overshadowed by the bells and whistles of the computer (in fact, I would suggest turning off all the goofy things Powerpoint does, but that is only my suggestion--you are the one giving the presentation, so you should do what you feel comfortable with).
Writing something on the whiteboard can also be a useful technique in a presentation. In fact, using multiple tools (computer, Mathematica, webpages, java, whiteboard, overheads, etc) in your presentation would be a very good thing (I can help you with questions you have about what would be best)--but only if it helps convey content to your audience! A very bad thing would be creating beautiful presentation materials and then simply reading the slides you have created. It is far better to have notes which you refer to periodically and to think on your feet than to have no notes and simply read from the overhead or computer. I am more than willing to watch you practice your presentation, and offer any advice for improvement.
A great presentation is based on a clear understanding of what you are going to talk about, being confident in your ability to discuss your project with others, and spending the time at the beginning to orient your audience.
Participation
Participation means being actively involved in your project over the two semesters (by meeting with me every week and making sure you don't fall behind, as well as following up the answer to questions yourself, working independently, finding suitable resources, etc.), and meeting the other requirements of Math 4901 (meeting with faculty, going to the seminars, etc).
Topics for Senior Seminar
If you already have a topic you are interested in, and you think it fits with my expertise, stop by and we can talk about it.
Projects could be drawn from the following areas (my latest ideas are at listed first) :
Duality of Force Laws
What is the mathematical connection between Hooke's Law and Newton's Law? Are there other beautiful connections between physical laws that could be explored?
References:
Gravitational Lenses
Gravitational wells, such as our sun, act to bend light. Something so intimately related to the nature of the cosmos must surely be a complicated subject? Actually, a study of gravitational lenses can be carried out at a senior undergraduate level.
References:
Perturbed Differential Equations
References:
Multidimensional integrals of highly oscillatory integrands
Key ideas: Monte Carlo integration, integrand smoothing, numerical analysis.
Have you ever tried to integrate a wildly oscillating integrand over six variables? That is what you sometimes have to do if you want to take a phase space average to calculate the autocorrelation function of a wavefunction in three spacial dimensions. Performing the six interals is difficult enough, but add in that oscillation, and the problem becomes almost impossible to solve numerically.
Impossible, that is, unless you use some interesting techniques. Like Monte Carlo integration to speed the integration, and a smoothing of the integrand to eliminate the oscillation.
Discovering ways to perform difficult integrations numerically is a benefit to many of the sciences.
References:
- McQuarrie, B.R., and Brumer, Paul (2000) 'Semiclassical initial value representation techniques for chaotic systems', Chemical Physics Letters, 319.
- Press, W.H., Tuekolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992) Numerical Recipes, Cambridge University Press, Cambridge.
Chaos and classical trajectories in Hamiltonian systems
Key ideas: chaos, classical trajectories, Lyapunov exponents.
Chaos is apparent in many physical systems--but chaos can occur to a variety of degrees. Measuring the degree of chaos present in a system is an important concept. This can be done by computing the Lyapunov exponent for the system.
Many physical systems are modeled using classical trajectories for particle dynamics. The accurate computation of the trajectories is greatly affected by the degree of chaos present in the system. Simply throwing smaller time steps and a simple numerical routine at a chaotic system is neither expedient nor wise. The calculation of trajectories in highly chaotic systems is difficult, and following trajectories for long times may be all but impossible numerically. An analysis of the chaoticity of a system is therefore essential to any calculation which is based upon classical trajectories.
Refrences:
- Gutzwiller, M.C. (1990) Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York.
- McCauley, J.L. (1997) Classical Mechanics: Transformations, Flows, Integrable and Chaotic Dynamics, Cambridge University Press, Cambridge.
Lie algebras in quantum mechanics
Key ideas: group theory, quantum mechanics.
Quantum mechanical systems can sometimes be represented by Lie algebras, where the Lie product is taken to be the quantum commutator. The quantum system is defined through raising and lowering operators. These operators move the quantum system through the basis set by raising or lowering the eigenvalue.
Examples of these algebras are the so(4,2) algebra to study Hydrogenic atoms, and the so(2,1) algebra to study the Klein-Gordon equation. There are many others.
The use of group theory and algebraic structures to represent quantum mechanical systems is a powerful tool in the advanced study of physical systems.
References:
- Bednar, M. (1973) 'Algebraic Treatment of Quantum-Mechanical Models with Modified Coulomb Potentials', Annals of Phys., 75, 305.
Mathematical modeling
There are a multitude of systems which can be modeled using advanced mathematics.
Previous Senior Seminar Students I Have Supervised
- Amy Johnson: Modeling Catalytic Converters (2005/06)
- Danielle Metcalf: Modeling Traffic Flow (2005/06)
- Chris Orth: Pad\'{e} and Algebraic Approximants Applied to the Quantum Anharmonic Oscillator (2005)
- Dan Harms: Population Landscapes in Evolutionary Computation (2004/05)
- Marit Gilkey: Fractional Derivatives: The Mystery of What's Between Integer-Order Derivatives (2003/04)
- Chris Zabinski: Summation of Divergent Series (2003/04)
- Chad LaForge: Linear Algebra Techniques for Solving Differential Equations (2003/04)
- Jeremy Kallstrom: Finite Difference Approximation Method of Solving Partial Differential Equations (2003)
- Phillip Westby: Getting Around That $%$#% Corner (2002/03)
- Jessica Stenberg: Cyclotomic Polynomials at a High School level (2002)
- Brian Gluth: Loess: A Nonparametric Data Smoother (2002)
- Daniel Enderton: Multidimensional Numerical Integration (2001/2002)
- Zach Heinen: Solving the Time Independent Schrödinger Equation (2001/2002)
- Amy Simon: Pendulums Through the Years (2000/2001)
- Judy Brown: Using Mathematical Models to Describe Epidemiology (2000/2001) (coadvisor with Peh Ng)