The Mathematical Association of America
North Central Section
Spring 2006 Meeting
April 21-22, 2005
Minnesota State University, Mankato
Prior to the Spring MAA-NCS Section Meeting, there will be a meeting of the North Central Section NExT, beginning at 1:00 pm on April 21. Please visit the North Central Section NExT website for additional details concerning this event.
7:00-8:00 Registration, Armstrong Hall
7:00-8:00 Book Sales, Armstrong 123
Friday Evening Program, Armstrong Hall 101
Dr. Larry Pearson, presiding
8:00-8:05 Welcome by Dr. John Frey, Dean of the College of Science and Engineering Technology
Lecture, "Using Distances to Reconstruct Evolutionary History," Professor
Steven Willson, Iowa State University:
Abstract: An important model for evolutionary history is a phylogenetic tree in which the vertices correspond to species and the edges correspond to direct genetic relationships. Often from the DNA of extant species, a distance between the species can be estimated. It is shown how a phylogenetic tree including the inferred ancestors can be reconstructed from the correct distances between extant species. It is also shown how a Markov model leads to estimates for the distances between extant species.
9:00-? Reception, Centennial Student Union Heritage Room
8:15-11:00 Registration, Armstrong Hall
8:15-11:00 Book Sales, Armstrong 123
Armstrong Hall 101
Dr. Pavel Kitsul, presiding
9:00 The Pascal Fractal at the Other End of the Rainbow, Prof. John Holte, Gustavus Adolphus College
9:20 What Complex Variables Students Can Teach You, Prof. David Appleyard, Carleton College
9:45 English Mathematics in the 1400’s, Prof. William Branson, St. Cloud State University
10:00 Viewing Higher Dimensional Objects Through the Lens of a Group Representation, Prof. Michael O’Reilly, University of Minnesota—Morris
10:20 An Odd Checkerboard Problem and Its Solution, Prof. David Wolfe, Gustavus Adolphus College
Armstrong Hall 101
Dr. Mary Wiest, presiding
11:00-12:00 Invited Lecture, "Splendor in the Graphs," Professor Lowell Beineke, Indiana University-Purdue University Fort Wayne
Abstract: Connections between graphs and games go back hundreds of years. In this talk, we will take a look at some of the ABCs of the subject, starting with the game of Asteroid, moving on to Bridg-It, from there to Curious Coins, Dots-and-Boxes, as time permits, showing how graphs and graph theory can be used to win games and solve puzzles.
12:00-1:00 Lunch, Centennial Student Union Heritage Room ($12.50 per person)
12:00-1:30 Book Sales, Armstrong 123
1:00-1:30 Business Meeting, Armstrong Hall 101, President Dale Buske, presiding
Armstrong Hall 101
President Dale Buske, presiding
1:30 Invited Lecture, "Three Good Ways to Compare Apples and Oranges: Central Problems in Bioinformatics," Professor Laurie J. Heyer, Davidson College
2:30 Civic Engagement in a Finite Math Course, Prof. Brian Birgen, Wartburg College
Armstrong Hall 102
Dr. Namyong Lee, presiding
2:55 Displaying Mathematics in Three Dimensions, Prof. Jonathan Rogness, University of Minnesota
Friday Evening Program
An important model for evolutionary history is a phylogenetic tree in which the vertices correspond to species and the edges correspond to direct genetic relationships. Often from the DNA of extant species, a distance between the species can be estimated. It is shown how a phylogenetic tree including the inferred ancestors can be reconstructed from the correct distances between extant species. It is also shown how a Markov model leads to estimates for the distances between extant species.
It is well known that Pascal’s triangle modulo a prime p has a self-similar pattern, and the nonzero residues may be viewed as forming a fractal that Mandelbrot calls a Sierpinski gasket. The nonzero residues correspond to binomial coefficients whose “order of divisibility by p” is zero. What if we look at the binomial coefficients in the initial
p-by-p Pascal’s diamond, marking those binomial coefficients exactly divisible by p? Then dn is the order of divisibility, and d, the “degree” of divisibility, must be between 0 and 1. As n, these “prefractals” for d = 1 (maximum p-divisibility) approach a fractal of the same dimension as the Sierpinski gasket, the d = 0 case. Furthermore, it turns out that these dimensions for d = 0 and d = 1 correspond to the endpoints of a rainbow—a spectrum of fractal dimensions f(d).
The speaker just finished teaching an introductory course in complex analysis and was delighted to see the wide variety of valid solutions students submitted to rather standard exam questions. In this brief talk, he’ll show how they employed results such as the Cauchy-Riemann equations, Liouville’s Theorem, the Maximum Modulus Principle, the Fundamental Theorem of Algebra, Cauchy’s Integral Formula, Taylor’s Theorem, Cauchy’s Residue Theorem, and Picard’s Theorem in ways that were unexpected (at least to him, a non-specialist in the subject) and that might delight you as well.
William Branson, "English Mathematics in the 1400’s"
There was very little English mathematics in the 1400’s; the MacTutor archive lists no English mathematicians between Bradwardine (died 1350) and Tunstall (born 1474). In large part, this gap is explained by the Black Death of 1350. Yet mathematics in France, Germany, and Italy recovered from the plague much faster than it did in England. I suggest that England’s lagging behind the continent in this matter might be due in part to the politics surrounding the deposition of Richard II in 1399.
Michael O’Reilly, "Viewing Higher Dimensional Objects through the Lens of a Group Representation"
If the basis vectors of n-dimensional space can be identified with a basis of a permutation representation of a group G and if that representation has a three dimensional summand, we can project an image of an n-dimensional cube and view it in three dimensions. The image will display symmetries from within the group. We will illustrate this with a six dimensional cube imaged to display symmetries from within A(5).
In how many ways can you place checkers on an n by n board so that each square is adjacent to an odd number of checkers? Adjacent means orthogonally adjacent (so each square has 2 to 4 adjacent squares), and all n squares must satisfy the constraint, not just the empty ones.
Here is one of the ways of placing checkers on a 4 by 4 board.
. x . x
. x x x
. x x x
. x . x
Saturday Morning Invited Address
Connections between graphs and games go back hundreds of years. In this talk, we will take a look at some of the ABCs of the subject, starting with the game of Asteroid, moving on to Bridg-It, from there to Curious Coins, Dots-and-Boxes, as time permits, showing how graphs and graph theory can be used to win games and solve puzzles.
Saturday Afternoon Invited Address
Bioinformatics, the science of analyzing molecular biological data, is a hot new field for mathematicians and computer scientists. Many problems in bioinformatics require the comparison of DNA sequences, entire genomes, or gene expression patterns. I will describe three algorithmic and statistical approaches to these comparison problems: dynamic programming, sorting by reversals, and correlation coefficients. This talk is accessible to undergraduates, and no prior knowledge of biology is assumed.
There have been efforts to teach introductory science classes by finding significant social issues to drive learning (for example, teaching introductory biology by focusing on tuberculosis). Attempts to apply this same notion of Civic Engagement in introductory math courses have been more modest. We will discuss some of the successes we have had using projects in teaching our Finite Math course to show students the relations between math and problems in the real world.
Our colleagues in the earth sciences have developed a stereo projection system known as the GeoWall; it uses two data projectors and inexpensive glasses to show real three-dimensional images to an entire class. They report a significant increase in student comprehension of spatial concepts when using the GeoWall.
Visualizing concepts in three dimensions is always difficult for mathematics students as well, and we can all think of places in our courses where this system could be useful. The goal of this talk is to introduce everyone to the technology and show what is feasible. Along the way we’ll view a variety of mathematical images and movies in 3D.