Talks
Undergraduates
"The infinitude of primes: the beauty of many proofs,"
The University of Texas,
Undergraduate Math Club, (2010).
- Abstract
- Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving calculus, infinite series, and topology. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers.
"Legendrian Knots: A New Twist on an Old Favorite,"
Luther College and Clark University, (2009).
- Abstract
- Knot theory began as an attempt to understand the structure of the universe and dates back to the late 1800s. Though its origins are in the physical world, knot theory has been an active area of mathematical research for over a century. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
In this talk we will explore both regular knot theory and Legendrian knot theory. As we will see, both theories seem tantalizing easy and yet both have complexities that have baffled even the best mathematicians
"Imagining the Universe,"
University of Michigan - Dearborn, (2009).
- Abstract
- The concepts of "shape" and "dimension" have fascinated scientists, mathematicians, and daydreamers for millennia. In the past, seafarers looked at the ocean horizon and wondered what lay beyond. Today, stargazers dream of a spacecraft that will allow them to explore the depths of the universe. And all the while, mathematicians build sophisticated theories to try to explain our physical world without ever leaving their desks.
In this talk we will create and investigate shapes of dimension 1, 2, and 3 (and possibly 4!). We will explore Abbott's famous Flatland and consider the possible shape of our universe. Along the way we will come to understand the fundamental questions in the fields of Knot Theory and Low-dimensional Topology.
As a warm-up exercise to get your imagination going, consider the following:
Suppose you had a spaceship that could travel much faster than the speed of light and suppose you left Earth and flew towards a very distant star. Would you be surprised if you arrived at that star only to find that it was the Sun and you were back at Earth? What would this suggest about the shape of the universe?
"Mathematical Elegance in an Art Museum,"
Washington University Undergraduate Math Club, (2004) and (2006)
- Abstract
- Suppose the manager of a museum wants to make sure that at all times every point of the museum is watched by a guard. The guards are stationed at fixed posts, but they are able to turn around. How many guards are needed? We will see how a single clever idea can take this possibly difficult problem and make it easy. We will also give an introduction to "mathematical induction," which will help us turn the clever idea into an elegant proof.
"Properties of the Group of Symplectic Matrices,"
Mathfest, PME Student Presenter, (2003) and
Illinois MAA Conference - Jacksonville, IL, MAA Student Presenter, (2002).
- Abstract
- This presentation was the culmination of a semester of undergraduate research under the direction of Prof. Thomas Bengtson at Augustana College. The research was done as part of the Earl Beling Scholars program at Augustana.
"The Illustrated Analyst,"
Mathfest, MAA Student Presenter, (2002).
- Abstract
- The Illustrated Analyst was a project I completed while participating in the VIGRE REU in Geometric Visualization at the University of Illinois at Urbana/Champaign in the summer of 2002. The program was directed by Prof. George Francis and my mentor was Prof. Karen Shuman. The Illustrated Analyst is a piece of software that provides 3-dimensional, graphical information on a variety of functions, Fourier transforms and convolutions.
