All mathematics majors are required to take the Senior Colloquium course. The Senior Colloquium gives students the chance to do independent study on a topic of their choice, culminating in an academic conference or seminar before students and faculty. Each student works under the supervision of a faculty member. Senior Colloquium is usually a four-hour capstone course, although
a one-hour version is available for students who take another capstone
Topics are negotiable between the student and the faculty supervisor and may be a favorite topic brought up in a course, or something outside the academic classroom that has inspired your mathematical curiosity, or something suggested by a faculty member, etc. The following list of suggestions is not in any particular order, and is not meant to be all-inclusive.
- Google's PageRank Algorithm:
Consider how Google decides the order to present search results. See Google's PageRank and Beyond by Amy Langville and Carl Meyer.
- The 3N+1 Problem:
Generate a sequence of numbers Nk by successively halving
a number if it is even, or taking Nk+1 = 3Nk+1 if it is odd. For every initial number tested so far, the sequence gets to 1, but nobody has been able to prove this always happens.
- Geometry and Billiards:
Follow a billiard ball (or a ray of light) as it endlessly bounces around inside a billiard table of some shape. See Geometry and Billiards by Serge Tabachnikov, which can be found in the Blough-Weis Library.
- Proof assistants:
Research computer programs that check mathematical proofs for logical soundness. Some examples are Coq and Isabelle.
- The number Pi:
Several topics could result from conducting an in-depth study of Pi. Visit the Pi Pages for ideas.
- Inverse Symbolic Calculator:
Consider the question: If you have a numerical value like 31.0062766802998201754763150671, can you find a formula it came from? See the Inverse Symbolic Calculator.
- The PSLQ Algorithm:
Used for finding integer-coefficient linear relationships between real numbers known only in numerical form. Named one of the ten algorithms of the century in 2000. See Recognizing Numerical Constants.
- The Admissions and Recruitment Problem:
Question: If a group of students rank universities in order of preference, and the universities rank the students, what is the optimal assignment of students to universities? See American Mathematical Monthly, May 2003, p. 386.
- The Mathematics of Juggling:
A good start to this topic would be B. Polster�s book by this title, which can be found in the Blough-Weis Library.
- Seashell patterns:
Consider how dueling chemicals diffuse along the growth edges of sea shells to make colored patterns. See The Algorithmic Beauty of Sea Shells by H. Meinhardt, which can be found in the Blough-Weis Library.
- Circles within circles:
Contemplate recent findings concerning circles nesting within a circle, a pattern that has been studied since the ancient Greeks. See Science News, April 21, 2001, p. 254.
- Dots and Boxes:
A good start to this topic would be Elwyn Berlekamp�s book The Dots-and-Boxes Game: Sophisticated Child�s Play, which can be found in the Blough-Weis Library.
- Folding maps:
Consider whether or not it is possible to fold a sheet of paper with given horizontal and vertical creases. See the Folding and Unfolding page.
- Unfolding polygons:
Can a really crinkly plane polygon can always be unfolded into a convex polygon. More information on the Web.
- Differential forms:
Research what div, grad, curl, Green's Theorem, and all that are really about.
- Capillary surfaces:
Consider how shapes are formed by liquid surfaces, due to surface tension, gravity, and contact with walls.
- Calculus of variations:
Find the optimal form of a function�for example, the shape of a suspension bridge cable.
- Sparse matrices:
Study the techniques that solve linear systems with zillions of variables but relatively few nonzero coefficients.
- Voronoi tesselations:
Consider the question: Given a discrete set of points (called nodes) in space, how do you divide up space so each point belongs to the node it is closest to?
- Conformal mapping:
Use complex functions to morph one plane region into another.
- Map making:
Although you can't put a round Earth on a flat piece of paper without distortion, consider which ways are better than others for various purposes.
- Karmarkar's Algorithm:
Research the new way to solve linear programming problems that promises to be faster than the simplex method for large problems.
- Ordinal arithmetic:
Center a topic on the ideas: an ordinal number is just a well-ordered set. Finite ordinals are the familiar whole numbers. But all kinds of infinities are easy to define, and do all the standard arithmetic with.
Quaternions are generalizations of complex numbers. They are of the form a + bi + cj + dk where i^2 = j^2 = k^2 = -1. Multiplication is noncommutative. Quaternions can be used to represent rotations in 3D.
- Lebesgue integral:
Consider how the integral of a function is defined by slicing the y-axis rather than the x-axis, as the Riemann integral does. This is the definition of integral mathematicians really use.
- Sphere packing:
Determine how tightly spheres can be packed. It has recently been claimed to have been solved in 3 dimensions, but is still unknown in higher dimensions. See The Pursuit of Perfect Packing by Tomaso Aste and D.L. Weaire.
- Mathematics of Elections:
Consider the strange paradoxes that sometimes arise in an election scheme. See Chaotic Elections: A Mathematician Looks at Voting by Donald Saari, which can be found in the Blough-Weis Library.
- Error correcting codes:
How you can tell when data has been transmitted correctly? Or a credit card number typed in correctly? See Identification Numbers and Check Digit Schemes by Joseph Kirtland, which can be found in the Blough-Weis Library.
- Asymptotic series:
A power series can diverge and still be very useful to calculate a function value.
There is a generalized notion of "function" that permit all functions, including discontinuous functions, to have derivatives.
- Convex hulls:
The convex hull of a set of points is the smallest convex body containing the set. Computer algorithms to find convex hulls are important.
- Surreal numbers:
If real numbers aren't strange enough for you, try surreal numbers, which are generalizations of the idea of �number� to include infinitesimal, infinite and much weirder numbers.
- History of mathematics:
Trace the development of some important mathematical concept like "function" or "number", and see how thinking has changed from ancient to modern times.
- Catastrophe theory:
As you increase the load on a column, all of a sudden it buckles. Catastrophe theory is the study of sudden changes that can happen to a system.
- The four-color map problem:
How many colors you need to color a map so no adjacent countries have the same color? If the map is on a sphere? On a doughnut?
Wavelets are a new way of decomposing functions into a standard set of functions (a "basis" in linear algebra terms). They are used in audio compression, video compression, and image analysis.
- Steiner networks:
Find the shortest network connecting a set of points.
- Fast multiplication of very long numbers:
Determine which method of multiplication produces the fastest results. Grade-school long multiplication isn't the best way to do it if you have million-digit numbers!
- Fast matrix multiplication:
Determine which method of matrix multiplication produces the fastest results. For really large matrices, it can be done faster than the row-times-column method you learned in linear algebra.
- Prime number testing:
Consider fast ways to show a number is probably prime, and slow ways to show it is certainly prime.
What does it mean to be infinite? Are there different sizes of infinity? How can you construct infinities?
- Public key cryptography and prime numbers:
Research the schemes that are used to keep private info private.
- Algebraic topology:
How the concepts of algebra, like groups, can be used to distinguish different kinds of two-dimensional surfaces.
- Nonstandard numbers:
Nonstandard numbers are an extension of real numbers to include infinite numbers and infinitesimal numbers. Not the same thing as surreal numbers.
Spinors are strange things which halfway between scalars and vectors. But our universe has a great fondness for them. Spinors are used in physics to describe electrons, protons, and other spin-1/2 particles.
- Continued fractions:
Continued fractions are fractions within fractions forever. These are to ordinary fractions as power series are to polynomials.
- Fourier transforms:
How to decompose any function into sines and cosines.
- CAT scanning:
How to reconstruct a 3D image from X-rays sent through an object.
- Cellular automata:
Cellular automata are sets of cells that change their state according to a set of rules and the states of their neighbors. An example is the Game of Life.
- Braess' Paradox:
Opening another road can actually increase traffic congestion.
- Zero-knowledge proofs:
How to convince somebody you know something they don't without giving anything away.
- Riemannian geometry:
Riemannian geometry is the intrinsic geometry of curved surfaces. Used in general relativity.
- Elliptic functions and AGM:
Elliptic functions are a generalization of trig functions (also called the circular functions), and they can be calculated using the Arithmetic-Geometric Mean algorithm.
- Tilings and patterns:
Consider ways of filling the plane with repeating patterns.
Zonohedra are polyhedra with parallel faces and sides.
- Random number generators:
Consider how computers generate numbers that seem random.
- Can you hear the shape of a drum?
Determine whether or not two differently shaped drumheads can vibrate at the same set of frequencies, given that the frequencies a drumhead vibrates depend on the drumhead's shape.
- Mathematics of musical instruments:
Consider why different types of instruments produce different sounds.
- Parrondo's Paradox:
It is possible to have two games of chance that when played separately are both losing games, but when played alternately the net outcome is winning!
- Famous unsolved problems:
See this Web site for topic ideas. Odds are you won't be able to solve any of these, but many are easily stated and deceptively simple looking.