Extended departmental description for MATH 236

This course is intended to introduce students to certain features of the mathematical enterprise including: (1) basic structures in mathematics; (2) the nature of formal arguments that establish the validity of theorems; (3) strategies for problems-solving; and (4) analogies that exist among various mathematical concepts. Amidst all of this mathematical formality, you will discover some remarkable facts. In particular, you will learn that when Buzz Lightyear said "To infinity and beyond!", he was being mathematically precise.

Math 236 is the last course in the math sequence that is required of all math majors, and is the first course that suggests what being a math major (as opposed to a math user) is all about. If you are undecided about majoring in math, taking this course before you make the decision might prove helpful.

Extended departmental description for MATH 285

This course will cover the statistical theory and methods used to design and analyze sample surveys. We'll briefly discuss questionnaire design, then move on to basic sampling designs: simple random sampling, stratified random sampling, one- and two-stage cluster sampling, and double-sampling for stratification. We will discuss why complicated sampling designs are used in practice and you will have a chance to implement one of these designs (anything but a SRS) in a group project.

Extended departmental description for MATH 341

Partial Differential Equations (PDEs) is the culmination of the calculus sequence in that we get to solve equations in 3-space plus time. Typical equations are those that arise in physics, such as those involving diffusion of heat, vibrating waves, and fields, and also from other areas such as economics. We're interested in establishing the solutions theoretically as well as practically using such methods as separation of variables, Fourier series, Fourier transforms and Green's functions. Consideration of equations in cylindrical and spherical coordinate systems will give rise to special functions such as Legendre polynomials and Bessel functions.

Extended departmental description for MATH 351

What happens to calculus when you replace the real variable x by the complex variable z = x + iy and the real-valued function y = f(x) by the complex-valued function w = f(z)? For starters, the statement "f is differentiable" becomes more powerful while the idea of integration becomes more flexible---you can now integrate along various paths in the complex plane. Of course this subject is inherently beautiful. But, perhaps surprisingly, a lot of this material can be applied to "real" mathematical and physical problems in which no complex number occurs. For instance, we will see how to compute some important improper integrals of functions such as cos(x^2) that don't have an antiderivative in closed form.

This course has connections with many other upper-level math courses. Those who have taken other courses should enjoy discovering some of those connections. However, Math 211 is really the only prerequisite!

Extended departmental description for MATH 352

So you liked Abstract Algebra I? Then it might well get even better, because you now have the tools to study one or more selected areas in some depth. The choice of topics is not quite fixed yet - there won't be a textbook, and your specific requests or interests might influence what gets done - but the current plan is to spend about five weeks each on the representation theory of finite groups and on Galois theory. Representation theory, which involves describing the structure of groups by using their homomorphisms to matrix groups, is used in classifying and predicting elementary particles (which we won't do) and in chemistry, as well as in mathematics. Galois theory establishes unexpected deep connections between fields and groups - more precisely, between field extensions and groups of automorphisms - and is used widely elsewhere in mathematics, especially within algebra. Both topics have quite beautiful results, but you can't state the results before you really get into the material! However, if you're not sure whether to take the course, feel free to stop by and talk; I might be able to give you some of the flavor of the material by showing an example or two.

Extended departmental description for MATH 354

You're traveling through space, you get tired of driving, so you stop off at a road-side planet named Diner. You order some breakfast: coffee and a doughnut. The waitralien comes by and hands you a chocolate-covered coffee cup with a sopping wet doughnut soaked with coffee. Turns out the Dinerians, in addition to only being able to see in black and white, lack the ability to perceive or measure exact distance (or length)--but they do have a very good sense of relative distance. (As in, "You're getting warmer... warmer...nope, you're getting colder now...") Put another way, they're endowed with a keen sense of topological structure, so they can't tell the difference between cups and doughnuts, which are identical from the point of view of topology. What other things can the Dinerians see? If you want to find out you'll have to take the course.

Extended departmental description for MATH 395

In the early 1980s, mathematicians William Mills, David Robbins, and Howard Rumsey were studying an old method of computing determinants called ``Dodgson Condensation'', which is named after Charles L. Dodgson (aka Lewis Carroll), who had studied it nearly a century before. Mills, Robbins, and Rumsey soon realized that to understand Dodgson condensation, they needed to understand a new collection of objects, called alternating sign matrices, which generalize permutation matrices. They conjectured a remarkable formula for the number of n by n alternating sign matrices, which they were unable to prove. This conjecture soon became known as the alternating sign matrix conjecture, and it is one of the conjectures about which David Robbins wrote in the early 90s, ``These conjectures are of such compelling simplicity that it is hard to understand how any mathematician can bear the pain of living without understanding why they are true.'' While they were trying to prove the alternating sign matrix conjecture, Mills, Robbins, and Rumsey discovered deep connections between alternating sign matrices and plane partitions (which can be thought of as stacks of blocks in a corner), which allowed them to prove a longstanding conjecture about certain symmetric plane partitions. The story of these two conjectures, along with twelve other related conjectures, is told by David Bressoud in his book Proofs and Confirmations. We'll use this book as our text, as we learn about these conjectures and their proofs.