# Undergraduate Colloquium

**Fall 2009**

**More Mysteries of the McKay Correspondence**

*Abstract:*Consider the following task: write down a finite list of rotations in three dimensions so that the composite of any two of them is once again on your list. A moment's thought shows that this is easy to do -- simply fix an axis and an integer

*n*and take the rotations about that axis through an angle of

*2*pi/j*where

*j*ranges from 0 to

*n-1*. So instead it's natural to ask for

**all**possible finite lists.

It turns out that it is possible to organize the answer in a way which suggests all kind of connections with other areas of mathematics (many of which are still quite mysterious). This talk will touch on some of them.

**Introduction to Geometric Group Theory**

*Abstract:*A group is an algebraic object. I will talk about how to associate a geometric object to a group, and use geometry to recover algebraic information about the group. (No prior experience with groups will be necessary.)

**What can you do with a slide rule?**

*Abstract:*Back in the days when people first went to the moon, electronic calculators did not exist. Instead we used slide rules. They were indispensable for professionals, and students were required to own one and know how to use it. There were courses on the proper use of a slide rule. Just like calculators today, slide rules were mostly everyday and commonplace instruments, but some were fancy, expensive, and treasured by their owner.

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.

Here's a couple of home work problems. You can do them before or after the talk. Let me know your answers:

Why is it so hard to find slide rules that can be used for addition and subtraction?

What's the base of the logarithm used for the design of any specific slide rule?

**Gröbner Bases**

*Abstract:*Gröbner bases are an exciting tool in computational commutative algebra that have been applied in a wide variety of problems since their invention about forty years ago. Given their relatively recent development and widespread use, you might expect they're complicated and difficult to understand, but they're not. Basically, all you need to understand is polynomial division! In this talk I'll discuss the problems inherent in dividing polynomials with more than one variable (more precisely, in determining if a given polynomial is in an ideal, which is a term I'll define), and then I'll explain what a Gröbner basis is and how it addresses these problems.

**Science and Engineering: Partners in Creating Advanced Military Communication Systems**

*Abstract:*Dr. Sylvester has 25 years of experience in communication system analysis and design. He has been a lead system analyst/consultant on several software radio based military and commercial wireless communication systems including covert/anti-jam spread spectrum systems, wireless internet/telephony systems including ground and satellite systems, and high-rate RF communications in excess of 4 Gbit/sec. His skills include: network physical layer design, interference mitigation strategy evaluation, multiple access selection and implementation, covert/anti-jam waveform definition and performance analysis, fast signal acquisition evaluation and implementation, channel coding selection and performance analysis, and network entry strategy development.

In his role as Chief Technologist, Dr. Sylvester is responsible for the development of the division technical roadmap, IRAD planning and execution, coordinating the technical exchange of information with universities, and providing intellectual property management and oversight.

Dr. Sylvester has presented or published papers on Military Communication Networks and Multiple Access Communications. He currently holds twelve patents relating to communications. He received BS (highest honors), ME, and PhD degrees in Electrical Engineering from the University of Utah, where he currently serves on the Engineering National Advisory Council and the Industrial Advisory Board. The emphasis of his study was statistical communication theory, digital signal processing, and error-correction coding. He was honored with the Distinguished Young Alumnus award from the department of Electrical and Computer Engineering at the University of Utah in 2007.

**A Dive in the Zeroes of Polynomials**

*Abstract:*Suppose we want to measure how fast a (complex) polynomial F vanishes, say, when we set all the variables equal to zero. In the case of one variable, we look at the multiplicity, namely, the least of the degrees of the monomials appearing in F. However, if the number of variables is more than one, more sophisticated approaches appear to give better information than just looking at the multiplicity. Focusing on the two examples F(x,y) = y^2 - x^2 and G(x,y) = y^2 - x^2 - x^3, I will present three different ways of measuring how fast they vanish, using respectively (a) integrability conditions, (b) vanishing of higher order derivatives, and (c) geometric modifications of the plane xy. I will then discuss the striking fact that these completely different approaches are actually equivalent.

*Abstract:*

**Symmetry and Groups**

*Abstract:*Symmetry is a fundamental concept in geometry. In fact, symmetry plays an important role in helping us understand the nature of matter. Groups are ubiquitous in mathematics, and in particular give us a language in which to describe symmetry and its consequences. In this talk, we will explore the groups controlling the symmetry of the Platonic solids and discuss some applications to the study of crystals. The topics covered in this talk are the basis for the undergraduate research course Math 4800 which I will run in Spring 2010.

**Counting Graphs**

Counting graphs on a given surface is a non-trivial combinatorial feat. However, using the physics of quantum fields, we can approach the problem in an original way: by computing integrals. In this talk we will present the background and maybe we'll get some time to compute some elementary results.

**Quadratic Forms and Topographs**

*Abstract:*A binary quadratic form is a degree-2 homogeneous polynomial in 2 variables: f(x,y) = ax^2 + bxy + cy^2. What are the integral values of such a form? Can f(x,y) = k be solved in the integers? The goal of this talk will be to "see the values" of a quadratic form. We do this by viewing a binary quadratic form as a certain type of function on a rank 2 lattice. In such a lattice we can define the notion of a vector, a basis, and a superbasis, and from these objects, we can form a topograph consisting of vertices, edges, and regions. It is in the regions of this topograph that we will be able to "see the values" of binary quadratic forms.

**Mixed Areas and the Isoperimetric Inequality**

*Abstract:*One convex set in the plane can be linearly morphed into another using Minkowski addition. The mixed area of these two sets is determined from the midpoint of the morphing. Brunn's Inequality relates the mixed area to the regular areas. It implies the Isoperimetric Inequality, which says that any simple closed curve in the plane encloses an area which is no larger than a circle with the same boundary length. The arguments will use the support function of a convex set.

**The Gamma Function**

*Abstract:*The gamma function was defined by Leonhard Euler in 1729. It is defined by means of an integral, and generalizes the factorial of natural numbers. It has many beautiful, elementary properties. In this talk, we will discover some of its basic features; the nice part is that only Calculus II is needed for this.

**The Great Pi/e Debate**

*Abstract:*Colin Adams and Thomas Garrity settle once and for all the burning question that has plagued humankind from time immemorial: "Which is the better number, e or Pi?"

**A Little Bit of the Math Behind Google**

*Abstract:*It is not only how many times a keyword appears or how many pages link to a certain page that will make it important, it is more about who links to that page that will make it important, the page inherits the importance of who links to it. This is one of the ideas of the PageRank algorithm that Google uses for ranking web pages. How can one describe who links to whom, how and how many times? How can one find the importance of a page? Google's PageRank algorithm is mathematically powerful. It views the web as a giant graph, constructs a gigantic matrix and considers surfing a stochastic process.

**What's the arclength of an ellipse?**

*Abstract:*The arclength of a circle is computed with the arcsin function, which has a nice "addition" rule. By contrast, the arclength of an ellipse is a new "elliptic" function that is not computable in terms of classical functions. Nevertheless, it has a very similar addition rule, and indeed there is a whole host of addition rules for integrals that collectively are known as Abel's Theorem. In other words, if we tweak some of those integrals we teach you how to do in calculus, they become uncomputable but can be added in strange ways.