Mathematics

 

Hudson Colloquium Series - 2010

Fall 2010

November 10
William Trentham

Krull Dimension
Abstract: Krull dimension is a notion of central importance in commutative algebra. One can think of Krull dimension as a measure of "how high" one can continue to stack up prime ideals in a ring. In this sense, we get an idea as to the size of the ring. In Dr. Trentham's doctoral thesis, he presented a generalization of the idea of Krull dimension. This generalization allows us to assign a unique cardinal number to every commutative ring with identity. Thus, in the infinite dimensional case, we are able to say what the Krull dimension of a ring actually is instead of being cornered into referring to what it is not, i.e., finite. In this discussion we will consider Krull dimension in both its traditional sense as well as this generalization. We will also perform some computations using this generalization.

October 27
Tricia Muldoon Brown

The Partial n-Queens Problem
Abstract: The n-queens problem asks: Given a positive integer n, can n non-attacking queens be placed on an n x n chessboard? This problem and and its extensions have been studied by mathematicians for over 200 years. In this talk we discuss the history and several of the variations of the n-queens problem. We focus on the problem of finding a maximum solution given a partial arrangement of k non-attacking queens and demonstrate a computer algorithm which solves the partial problem for small n.

October 6
Sungkon Chang

The Algebra of Grand Unified Theory
Abstract: The Standard Model representation in particle physics describes the three forces: the strong force, the weak force, and the electromagnetic force via the actions of the Lie algebras of Lie groups. A grand unified theory (GUT) is an attempt to describe the three forces via the action of the Lie algebra of a simple Lie group, thereby offering an elegant unified mathematical description of the action of the three forces. In this talk, the mathematics behind SU(5)-GUT will be introduced. Although SU(5)-GUT is no longer accepted in physics, it is an excellent prototype of GUTs, from which one can see how modern algebra plays a role in some fundamental part of physics.

September 15
Paul Hadavas

The Power of Proofs
Abstract: In this talk, we will see why we need mathematical proofs, and then discuss various proofs for one of the most popular theorems of all time. A bonus "proof without words" for another well-known theorem will also be included.

September 1
Selwyn Hollis

Building Dynamic Demonstrations with Mathematica
Abstract: With Mathematica, a remarkably small amount of code can result in a small interactive application, or "demonstration," with a sophisticated user interface. This talk will focus on the basic ideas involved in building demonstrations, using a few elementary classroom topics as examples.



Spring 2010

April 21
Sungkon Chang, William Nathan Hack, Selwyn Hollis, Scott King, Tim McMillan

Reflections on the 70th Annual Putnam Mathematical Competition
Abstract: On December 5, 2009, an intrepid team of four Armstrong students spent most of their Saturday working on a dozen frighteningly challenging mathematical problems. Why? Just another installment of the notoriously difficult William Lowell Putnam Mathematical Competition. Members of the team and faculty members will discuss solutions of their favorite problems from the most recent competition.

April 7
Dr. Joshua K. Lambert

Coloring Checkerboards
Abstract: The squares of an m by n checkerboard are alternately colored black and red. Two squares are called neighbors if they belong to the same row or the same column and there is no square between them. It is conjectured that we can place coins on some of the red squares such that every black square neighbors an odd number of coins. We shall discuss how this checkerboard problem relates to the newly conceived concept in graph theory of modular chromatic numbers.

March 24
Moshe Cohen, Louisiana State University

Domino Tilings, Perfect Matchings on Graphs, and the Alexander Polynomial of a Knot
Abstract: The goal of this talk is to investigate how well-understood problems in combinatorics interact with a polynomial from knot theory. Combinatorics (the art of counting) asks questions like "How many ways can we cover a checkerboard with dominoes?". A knot is a circle embedded in three dimensional space. Knot theory asks "How can we tell two knots apart?". The Alexander polynomial is one example of a knot invariant (that is, if the Alexander polynomials of two knots are not the same, the knots must be different). This polynomial is the determinant of a matrix, and we'll construct this matrix using techniques from combinatorics.

March 3
Dr. Sabrina Hessinger

Differential Galois Theory: An Area for Mathematicians Loving the Pure and the Applied!
Abstract: In this talk we will introduce the basic constructs and ideas behind the field of Differential Algebra with a specific focus on Differential Galois Theory. This relatively young field of mathematics was formally developed by E.R. Kolchin et.al. in the mid 1970s. It involves differential equations, abstract algebra, computer algebra, linear algebra and topology. We’ll explore the various avenues of inquiry in differential Galios theory and discuss example results.
 
February 10
Dr. Jared Schlieper

Free, As In Beer
Abstract: With all the talk of budget cuts, "free" has such a nice ring to it. In fact, there are many free open source programs that we can use to enhance our teaching. We will discuss using various open source mathematics software available and spend more time focused on two programs in particular: SAGE and WeBWorK. We will give a brief overview of each one along with classroom experiences from the past couple semesters.

January 27
Dr. Patricia Brown

How Deep Is Your Playbook?
Abstract: Math Awareness Month is coming up in April and this year's theme is Math & Sports. In order to support MAM, this talk will include algebra, combinatorics, and football. We will discuss 4 basic defensive formations utilized in the National Football League and consider how many options a coach has when calling his play. No advanced knowledge of math (or football!) is necessary for this talk.