The h*-vector of a given lattice polytope contains information about the polytope that may not be immediately obvious. When the h*-vector is particularly nice, it is a sign of additional structure worth investigating. An open problem is how to completely characterize when an h*-vector is unimodal, and we will discuss progress that has been made in this direction.

It’s time for March Madness, and college basketball fans are excited about the NCAA Tournament, so it’s a perfect time to learn about the graph theory version of tournaments.  In my talk, I will prove the formula for the Vandermonde determinant using a sign-reversing involution on a particular set of tournaments.  I will also use a similar technique to prove a formula that relates the Pfaffian, which is an analogue of the determinant, of a skew symmetric polynomial with the Vandermonde product.

For years, scientists have worked together to solve the art gallery problem.  Past results focused on how many guards are needed to cover an art gallery with a certain number of vertices.  We will cover recent results which determine the number of security guards necessary and sufficient to cover a polyomino which consists of a certain number of unit squares.  We will be using logic and counting arguments.  If time is permitted, we will relate some of the results to graph theory.

p-adic numbers are very applicable in the fields of number theory, algebraic geometry and even quantum physics.  We will cover some of the basic properties of the p-adic numbers including their topological properties.  Only basic facts about algebra and metric spaces will be used.

We will look at some properties of curves of constant width in the plane and talk about why the Reuleaux triangle minimizes area among all such curves. We will then briefly look at the conjectured minimizer for 3-dimensions. Time permitting, we'll also look at the problem of minimizing/maximizing the Mahler volume among all centrally symmetric convex bodies. Again there are conjectured minimizers, but still no proof!

The Kakeya problem was proposed in 1917, by the Japanese mathematician Soichi Kakeya. The problem states,

In the class of figures in which a segment of length 1 can be turned around through 360˚, remaining always within the figure, which one has the smallest area?



In this talk I will give a very brief introduction to the Kakeya problem. I will give connections with harmonic analysis and some other field. I will also talk about the recent progress on the problem.

Lill's Method is a geometric approach for finding roots of polynomials with real coefficients. We will prove Lill's method, with examples, and then we will prove some familiar polynomial results in this new unfamiliar way. If there is time, we will also prove a generalized Lill's Method.

Any two distinct points define a unique line, and any two distinct lines intersect in a unique point.  Have you ever wished that this statement were true?  Has the idiosyncratic behavior of parallel lines always troubled you?  Then fret no longer!  This brief introduction to projective geometry will put parallel lines in their proper place and set your mind at ease.  [Warning: it may also raise questions equally as distressing or more so.

Imagine for a moment that you are sitting through a boring lecture and you start playing with your calculator. You input a number into the calculator and then hit the cosine key. Then you keep hitting the cosine key and look at what happens to the numbers. Do they converge to a ?finite number or do they tend to in?finity? Do the numbers start to repeat in a pattern, or do they continue to change without any pattern? This is an example of a discrete dynamical system. Discrete dynamical systems have important applications in biology and other sciences as well as being interesting on their own. In this talk, we will discuss the properties of discrete dynamical systems and some tools that can be used to determine the behavior of discrete dynamical systems. No previous knowledge of dynamical systems is required to understand the talk, and there will be only one theorem.

Self-dual codes show good error-correcting performance.  The Gleason-Pierce-Ward Theorem is useful for telling us when self-dual codes may exist.  We will present the theorem and outline its proof.