This is the home page for the Mathematics Department at Saint Michael's College.
This stuff applies to our department as a whole.
We took our annual trip to the Hudson River Undergraduate Mathematics Research Conference, held this spring at Mount Holyoke College. Here is a group photo. If we didn't have the largest group it must have been close! (Click on the thumbnailed picture for the full size version.)
Abstract The particular sociological set-up of an aboriginal tribe native to Northern Australia, known as the Warlpiri tribe, can be modeled by the dihedral group of order 8. The group structure can then be used to analyze the Warlpiri`s interpersonal relations, showing that it is inherent in their structure to avoid incestuous relationships. This serves as an interesting contrast to the arbitrary laws of the sociological set-up in the United States, where in some states it is legal to marry one`s own cousin.
Abstract The Tutte Polynomial was discovered by Tutte in 1954. It is a generalization of the Chromatic Polynomial, which counts the number of ways to color a graph. A contraction-deletion algorithm is used to determine the Tutte Polynomial of a graph. During this talk we will discuss the universality of the Tutte Polynomial, which essentially allows any graph invariant involving a contraction-deletion reduction to be expressed as an evaluation of the Tutte Polynomial. This talk will provide preliminary information for the following talks regarding the Potts Model and Kaufman Bracket, as well as discuss other evaluations of the Tutte Polynomial. We will also showcase a Maplet program which provides a user with a GUI for easy computation of the Tutte Polynomial as well as other graph invariants.
Abstract I will discuss the correspondence between knot theory and signed planar graphs. The Kauffman bracket is a way of encoding knot information into polynomial form. This polynomial is a graph invariant under Reidemeister moves II and III. Reidemeister moves are moves that change a projection of a knot that will change the relation between the crossings of that knot. The Kauffman Bracket of an alternating knot diagram can be expressed in terms of the Tutte polynomial. This illustrates the universal property of Tutte polynomial, namely that any graph invariant, in this case the Kauffman Bracket, that reduces using deletion and contraction, must be an evaluation of the Tutte polynomial.
Abstract The Potts Model, an evaluation of the Tutte Polynomial, is used frequently in statistical mechanics surrounding phase transitions. It is often necessary to determine the probability of a certain state occurring in a particular system, such as equilibrium. The Potts Model is the partition function which determines the measure of the entire system, and is thus used as the denominator when determining probability. An explanation of how the Potts Model is an evaluation of the Tutte Polynomial will be provided in the talk, along with some discussion of various applications of the Potts Model.
Abstract We are considering which knots and links can be realized from a simple strand of hydrocarbons, and if any structure can be made that is considered a "stuck unknot." Can these structures be realized solely by using alkanes, or are alkenes needed to get the desired structures? The necessary basic elements of the chemistry and knot theory will be described in this talk. Connections between knot theory and applications in biology and chemistry are currently a highly active research area - come get your start now!
Abstract James R. Munkres', Topology: A First Course (1975) has served as a primary resource for my recently completed senior seminar project in mathematics, during which I took a particular interest in the Cantor Set. I intend to discuss what I learned in this "first course on topology", and in particular the Cantor Set and its properties.
Abstract There are many interesting relationships between Math and Music. The first musical theory was developed by the Pythagoreans in ancient Greece during the Golden Age of Mathematics. The Golden Ratio, the "most pleasing" ratio of two parts to the whole, appears so frequently in Mozart`s piano sonatas that it is tempting to believe that he used it purposely. Also, does anyone know why the harmonic series is called the harmonic series? Come find out!!
Abstract One of the main problems involved in the design of computer chips consists of finding an optimal layout for the gates and wires so as to evenly distribute the heat created by the chip's use, while minimizing the length of the more important wires. However, as the numbers of gates and wires increases this problem rapidly becomes impossible under most methods. A Spring Embedder is a graph-drawing algorithm that subjects the various gates and wires to a physical analogy consisting of masses and springs, allowing the system to be affected by physical laws and to push and pull itself to equilibrium. The resultant graph is an approximate solution to the optimization problem. In this presentation I will cover the basics of the layout problem and of force-directed graphing, as well as provide a demonstration of a sample Spring Embedder program. This work has been done in response to an industry problem presented by Cadence Design Systems, a company that develops chip layout tools.