Department of Mathematical Sciences

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 a colloquium talk before students and faculty. Each student works under the supervision of a faculty member. Required of mathematics majors. Usually 1 credit hour, although 2 credits may be granted if a paper is also written, as when the course is also used to meet the Capstone requirement or the University Honors Program senior research requirement.

Possible topics

This is a list of possible topics. It is not meant to be complete. You may want to dig deeper into a favorite topic from a course, or you may have seen something somewhere that inspires your mathematical curiosity. The topic and outline for the course must be negotiated between the student and the faculty supervisor. The list is not in any particular order.

Google's PageRank Algorithm.: How does Google decide the order to present search results? New book coming out in June 2006: Google's PageRank and Beyond by Amy Langville and Carl Meyer.
The 3N+1 Problem.: Generate a sequence of numbers N_k by successively halving a number if it is even, or taking N_k+1 = 3N_k+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.: Following a billiard ball (or ray of light) as it endlessly bounces around inside a billiard table of some shape. The book Geometry and Billiards is on order for our library.
Proof assistants.: There are computer programs that check mathematical proofs for logical soundness. Some examples are Coq and Isabelle.
The number Pi.: See The Pi Pages.
Inverse Symbolic Calculator.: 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: 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.: 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.: See the library book of that title by B. Polster.
Seashell patterns.: Duelling chemicals diffuse along the growth edges of sea shells to make colored patterns. See The Algorithmic Beauty of Sea Shclls by H. Meinhardt in our library.
Circles within circles.: Circles nesting within a circle have been studied since the ancient Greeks, but there have been new results recently. See Science News, April 21, 2001, p. 254.
Dots and Boxes.: This classic game is the subject of a new book by Elwyn Berlekamp (in our library).
Folding maps.: New results let one decide if it is possible to fold a sheet of paper with given horizontal and vertical creases. See See http://theory.lcs.mit.edu/~edemaine/folding/.
Unfolding polygons.: Can a really crinkly plane polygon always be unfolded into a convex polygon? See http://theory.lcs.mit.edu/~edemaine/linkage/.
Differential forms.: What div, grad, curl, Green's Theorem, and all that are really about.
Capillary surfaces.: The shapes formed by liquid surfaces due to surface tension, gravity, and contact with walls.
Calculus of variations.: Finding the optimal form of a function, for example finding the shape of a suspension bridge cable.
Sparse matrices.: Techniques to solve linear systems with zillions of variables but relatively few nonzero coefficients.
Voronoi tesselations.: 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.: Using complex functions to morph one plane region into another.
Map making.: You can't put a round Earth on a flat piece of paper without distortion, but some ways are better than others for various purposes.
Karmarkar's Algorithm.: A new way to solve linear programming problems that promises to be faster than the simplex method for large problems.
Ordinal arithmetic.: 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.: A generalization of complex numbers. They are of the form a + bi + cj + dk where i^2 = j^2 = k^2. Multiplication is noncommutative. Quaternions can be used to represent rotations in 3D.
Lebesgue integral.: 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 mathemeticians really use.
Sphere packing.: How tightly can spheres be packed? Recently claimed to be solved in 3 dimensions, but still unknown in higher dimensions. Reference: The Pursuit of Perfect Packing by Aste & Weaire.
Mathematics of Elections.: It can be shown that no election scheme is perfect, and some strange paradoxes can arise! See the book Chaotic Elections by Donald Saari.
Error correcting codes.: How can you tell when data has been transmitted correctly? If you've copied down your credit card number correctly? See the book Identification Numbers and Check Digit Schemes by Joseph Kirtland.
Asymptotic series.: A power series can diverge and still be very useful to calculate a function value.
Distributions.: A generalized notion of function. Permits all functions to have derivatives, even discontinuous functions.
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.: A generalization of the idea of number to include infinetesimal, 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 4-color map problem.: How many colors do 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.: A new way of decomposing functions into a standard set of functions (a "basis" in linear algebra terms).
Steiner networks.: The problem is to find the shortest network connecting a set of points.
Fast multiplication of very long numbers.: Grade-school long multiplication isn't the best way to do it if you have million-digit numbers!
Fast matrix multiplication.: For really large matrices, it can be done faster than the row times column method you learned in linear algebra.
Prime number testing.: Fast ways to show a number is probably prime, and slow ways to show it is certainly prime.
Infinity.: A big topic. What does it mean to be infinite? Are there different sizes of infinity? How can you construct infinities?
Public key cryptography and prime numbers.: Schemes that are coming into use 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.: An extension of real numbers to include infinite numbers and infinitesimal numbers.
Spinors.: A spinor is the square root of a vector. Spinors are used in physics to describe electrons, protons, and other spin-1/2 particles.
Continued fractions.: Fractions within fractions forever. These are to ordinary fractions as power series are to polynomials.
Fourier transforms.: Decomposing functions into sines and cosines.
CAT scanning.: How to reconstruct a 3D image from X-rays sent through an object.
Cellular automata.: A set of cells which 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.: The geometry of curved surfaces. Used in general relativity.
Elliptic functions and AGM.: Elliptic functions are a generalization of trig functions (which are also called the circular functions). The Arithmetic-Geometric Mean method is a way of calculating them.
Calculation of pi.: How do they calculate pi out to billions of digits?
Tilings and patterns.: Ways of filling the plane with repeating patterns.
Zonohedra.: Polyhedra with parallel faces and sides.
Random number generators.: How computers can generate numbers that seem random.
Can you hear the shape of a drum?: The frequencies a drumhead vibrates depend on the drumhead's shape. Can two differently shaped drumhead vibrate at the same set of frequencies?
Mathematics of musical instruments.: Why do different types of instruments produce different sounds?
Parrondo's Paradox.: It is possible to have two games of chance that when played seperately are both losing games, but when played alternately the net outcome is winning!
Famous unsolved problems. Odds are you won't be able to solve any of these, but many are easily stated and deceptively simple-looking. Good fodder for a topic.