Title: An Introduction to Diophantine Approximation
Abstract: Diophantine approximation is a branch of number theory that deals with the approximation of real numbers by rational numbers. Our goal will to be to de?fine what it means to be a "good" approximation and then ?find the "best" among these. Along the way, we'll develop the basics of continued fractions and even take advantage of some geometrical properties of lattices. When the smoke clears, we'll see why 22/7 is a good approximation to pi and why 355/113 is even better. We'll also see why the golden ratio is in some sense the most irrational of all the real numbers. Finally, we will reveal the secret to one of Ramanujan's famous approximations of ?pi.
Abstract: How can you tell when a knot can be untangled? This deceptively simple question is at the heart of Knot Theory, a field which emerged with the work of Vandermonde in the 18th century. In this talk, we will introduce the knot classification problem, covering a few (but definitely not all) of the following ideas: knot invariants, Reidemeister moves, Conway and Jones polynomials, Gauss linking numbers, braid groups, Vassiliev invariants, the Kontsevich integral, Khovanov homology, categorification...
Title: The brachistochrone problem and variational methods.
Abstract: The brachistochrone is the problem of finding the path that would take a ball bearing between to points in the least amount of time. We will construct the solution using variational methods.
Other classic problems we may talk about include describing the shape of a hanging chain and finding the shape which encloses the largest area given a set perimeter.
Abstract: Lagrange's Four Square theorem states that any natural number can be written as the sum of four integers. This is the best we can do since, for example, 7 cannot be written as the sum of three squares. We will prove the theorem and take a look at the natural numbers that cannot be written as the sum of 3 squares.
Solitons are solitary waves that are stable to perturbations. This talk will focus on the paper "Korteweg--deVries Equation and Generalizations. VI. Methods for Exact Solution" by Gardner, Greene, Kruskal, and Miura which explores solving the Korteweg--deVries equation using inverse scattering, a method pioneered in this paper. This allows one to write explicit solutions to the KdV equation, including solutions that demonstrate solitons.
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!
