The fourth edition of the American Heritage Dictionary has a pithy description: a slide rule is a device consisting of two logarithmically scaled rules mounted to slide along each other so that multiplication, division, and other complex computations are reduced to the mechanical equivalent of addition or subtraction.

But there is more! I'll describe how slide rules work, why they work, and what you can do with them. A typical slide rule has anywhere from ten to thirty scales, rather than just two, and there are thousands of mathematical expressions that you can evaluate just as easily as you can multiply or divide two numbers. On the other hand, you can't use a slide rule to add or subtract two numbers, and you need to understand your problem well enough to be able to figure out on your own the location of the decimal point in your answer.

You'll be able to examine several slide rules, and I'll tell you what's involved in being a slide rule collector. Apart from pizza, you will be given a basic but genuine, mint, NIB ("new in the box") forty years old and never used slide rule to start your own personal collection.

If you are curious, before the colloquium you may want to explore the slide rule exhibit in our library donated to our department by Chris Smith of the University of Utah School of Music.

Here's a couple of homework problems. You can do them before or after the talk. Let me know your answers:
1. Why didn't they build slide rules that can be used for addition and subtraction?
2. What's the base of the logarithm used for the design of any specific slide rule?

Representation theory seeks to get these same advantages for group actions. I'll explain what that means, and look at a series of (closely related) examples: the permutation group S_n acting on the set {1, ..., n}; the group GL(n) of invertible matrices acting on the projective space of lines in n dimensions; S_n acting on k-element subsets of {1, ..., n}; and GL(n) acting on the Grassmann variety of k-dimensional planes in n dimensions.

Here are two warmup questions: (1) How many k-dimensional element subsets are there in {1, ..., n}? (2) How many k-dimensional planes are there in an n-dimensional space? (The second question is actually in two parts: first, how can this question possibly make sense? Next, what's the answer?)

I was also interested in what other network properties might be studied to further understand the interactions students have with one another. Some linear algebra made popular by the website Google is quite interesting. Google uses some linear algebra to rank websites in terms of how 'authoritative' they are. I have used this same bit of linear algebra to rank University of Utah students. This gives us some indication of how popular any given student is. Who is the most popular person on campus? Come and find out.

I will discuss some basic definitions from network theory and then present my results. You may be as surprised as I was.