Math 4901 Senior Seminar

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 two credit course, spanning two semesters, you will want to spend roughly three 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) :

Parameters in Time Series

I am thinking this would be a good MAP, but someone might also wish to do it as a senior seminar so I am including it here. You would definitely want to have an application to apply this to if it was a senior seminar.

References:

HIV Infection Models and Lyapunov Functions

I am thinking this would be a good MAP, but someone might also wish to do it as a senior seminar so I am including it here.

References:

Finite Element Method

Key Ideas: numerical methods, partial differential equations, multivariable calculus, calculus of variations.

The Finite Element Method (FEM) is a an important numerical technique used to solve certain classes of boundary value partial differential equations. FEM is used extensively in engineering, and although in practice many engineers would use one of the various "black box" algorithms to implement the FEM, the focus in the senior seminar would be on understanding how it works rather than writing efficient code to implement it.

For this project, you would show how optimizing a double (or triple) integral is equivalent to solving a partial differential equation. This uses ideas from multivariable calculus, but also has some connections to calculus of variations so you might choose to discuss a bit of calculus of variations. Depending how deep you go in the theory, you might also choose to implement the method for a simple example (understanding how FEM works is not trivial, so including an example could be an important part of your paper).

References:

Fast Fourier Transforms

Key ideas: Fourier Transforms, Cooley-Tukey Algorithm, Chinese remainder theorem.

Computing Fourier Transforms of discrete data sets with N points using the definition is simple to write down, but takes O(N2) arithmetical operations--which can take an extremely long time to implement. Fast Fourier Transforms are able to perform the transformation (using ideas from group theory, number theory, or other mathematical areas) substantially quicker, and thus are extremely useful and important. The FFT is used in signal processing, as well as in chemistry when you want to convert from the time to energy using correlation functions.

For this topic, you would provide some of the history of FFT, the mathematical details of one or more algorithm, and code the algorithms to provide examples.

References:

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:

Dynamical Systems

Key ideas: dynamical systems, numerical methods, periodic orbits, Duffing oscillator.

If you enjoyed differential equations, you could look at dynamical systems. This is a huge topic! After a general introduction to the concepts of dynamical systems (see portions of Strogatz's book), you would want to pick one advanced topic to focus on. It could be more theoretical (like the existence and uniqueness, Poincare-Bendixson theorem), an application, or maybe even a numerical algortihm.

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, Lie algebra, Lie group, 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(3) algebra to study angular momentum, 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.

This is a challenging topic, so you would want to build your paper carefully. A good place to start with this topic would be an investigation of the Special Orthogonal Group SO(n) (see Stillwell's book), followed by an example. Once you have an understanding of those concepts, you could branch out to other areas (there are plenty of directions to go depending on your interests).

References:

Mathematical modeling

There are a multitude of systems which can be modeled using advanced mathematics.


Previous Senior Seminar Topics I Have Supervised

  • Edge Coloring of Graphs to Model Sporting Tournaments (co-advisor Mark Logan) (2011)
  • Crossing Number in Complete Graphs (co-advisor Peh Ng) (2011)
  • Dual Systems: Hooke's Law and Newton's Law (2010/11)
  • Lie Algebras in Physics(2010/11)
  • Finite Element Method (2010/11)
  • Dynamical Systems with Applications in Neuroscience (2009/10)
  • Modeling Tumor Cell Population with Stochastic Noise Variables (2009/10)
  • An Application of Markov Chains (2009/10)
  • The Billiards Problem(2009)
  • The Arnold Cat Map and Cryptology of Images (2008/09)
  • How Flat Can a Soap Bubble Be? (2007/08)
  • Deterministic Problems in Genetics Using Dynamical Systems (2006/07)
  • Fininte Markov Chains and Processes (2006/07)
  • Modeling Catalytic Converters (2005/06)
  • Modeling Traffic Flow (2005/06)
  • Pade and Algebraic Approximants Applied to the Quantum Anharmonic Oscillator (2005)
  • Population Landscapes in Evolutionary Computation (2004/05)
  • Fractional Derivatives: The Mystery of What's Between Integer-Order Derivatives (2003/04)
  • Summation of Divergent Series (2003/04)
  • Linear Algebra Techniques for Solving Differential Equations (2003/04)
  • Finite Difference Approximation Method of Solving Partial Differential Equations (2003)
  • Getting Around That $%$#% Corner (2002/03)
  • Cyclotomic Polynomials at a High School level (2002)
  • Loess: A Nonparametric Data Smoother (2002)
  • Multidimensional Numerical Integration (2001/2002)
  • Solving the Time Independent Schrödinger Equation (2001/2002)
  • Pendulums Through the Years (2000/2001)
  • Using Mathematical Models to Describe Epidemiology (2000/2001) (coadvisor with Peh Ng)

Office Hours Sci 1380:
Drop-in Office Hours (no appointment needed) are listed on google calendar.

Appointment:
UMM students may sign up for an appointment using google calendar.

Email:
mcquarrb@morris.umn.edu

Phone:
589-6302
(I do not use voicemail)