# Math Senior Colloquium

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 course.

**Possible topics:
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**Topics are negotiable between the student and the faculty supervisor and may range anywhere from a favorite topic brought up in a course to something outside the academic classroom that has inspired your mathematical curiosity. The following list is not in a particular order, and is not meant to be 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 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:**

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.**Mathematical Games:**

Games like Tic-Tac-Toe and Dots-and-Boxes have complete knowledge by each player of the state of the game. There is a general theory of such games, and how they can be analyzed. See**Winning Ways**by Berlekamp, Conway, and Guy, in our library.**Proof assistants:**

Investigate 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:**

Look into 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 classical simplex method for large problems.**Ordinal arithmetic:**

An ordinal number indicates the place of an object in an ordered list. Mathematically, 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:**

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 that the integral of a function may be 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.**Distributions:**

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 and complex 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:**

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.**Infinity:**

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:**

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. See*Tilings and Patterns*by Grunbaum and Shepard, in our library.**Zonohedra:**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.