Archive of Past Talks
Archive of Past Topics - Fall 2009
Complete Growth Series and Products of Groups
Daniel Allen
Wednesday, October 28, 2009
3:45 pm in Preble 333
There is a long history of studying combinatorial structures in the context of finitely generated, infinite groups. Notable examples are growth series, where for a given set of generators, one counts the number of elements of length n, and converts this sequence into a formal power series. Recently there has been interest in complete growth series, where instead of counting the elements of the group of length n, one adds them together, forming an element of a group ring. In this talk we show that many of the standard products studied in geometric group theory, direct products, free products (also with amalgamation), and graph products, all preserve the rationality of their complete growth series.
A Model of an Epidemic
of a Contagious Disease
Russell Rainville
Wednesday, September 16, 2009
3:45 pm in Preble 333
We will look at the deterministic SIR model of the dynamics of a contagious disease and what it can tell us about the up coming H1N1 flu epidemic. How do we mathematically describe the movement of people from being Susceptible to the disease to being Infective to being Removed from the process? What does the model tells us we should expect? And of course what are the limitations of the model?
Archive of Past Topics - Spring 2009
Simulating and Analyzing the Coin-Moving Puzzle
Adam Case
Wednesday, April 15, 2009
3:45 pm in Ricker Addition 205
A Look at Braid Groups
Daniel Allen
Wednesday, April 8, 2009
3:45 pm in Roberts 101
How are the complex motions of subatomic particles related to a field as vastly different as cryptography? In this talk, we will explore the mathematical constructs that link these fields: the braid groups. An object as simple as a braid can embody a variety of physical processes, and so we will define the braid groups and examine accessible examples in order to understand why these groups are currently a hot topic in mathematics and the sciences. Along the way we will construct the Cayley graph of braid groups and then calculate the Bieri-Neumann-Strebel geometric invariant, ?1, of braid groups.
The Marriage Between Abstract Algebra and Geometry
Nic Koban
Wednesday, March 25, 2009
3:45 pm in Ricker Addition 205
I will present a relatively new branch of mathematics called Geometric Group Theory. Along the way, such fundamental ideas as group presentations, Cayley graphs and a certain boundary at infinity of a group will be discussed. The goal is to describe two different (but similar) subsets of this boundary, ?1 and ?1, that will help measure “how infinite” certain subgroups of an infinite group are. Once these sets have been defined, we will compute ?1 and ?1 for some interesting groups. I will finish by discussing the relationship between ?1 and ?1 and (time permitting) by discussing the higher dimensional analogs ?n and ?n. This talk is a prequel to Dan Allen’s talk about braid groups and the geometric invariants of these groups.
A Brief Introduction to GeoGebra
Mike Molinsky
Wednesday, March 4, 2009
3:45 pm in Ricker Addition 205
GeoGebra is an award-winning free software program specifically created for use in schools. The program, which runs on the Java virtual machine, can be used to explore a wide variety of mathematical topics and create dynamic learning tools. A brief summary of the capabilities of this software package will be presented, along with some examples.
Archive of Past Topics - Fall 2008
Generalized Musical Intervals
and Transformations
Joshua Case
Wednesday, October 15, 2008
3:45 pm in Ricker Addition 205
The Mathematical Education of a Founding Father:
John Adams (1735 - 1826)
Mike Molinsky
Wednesday, October 1, 2008
3:45 pm in Ricker Addition 205
Although John Adams made no significant contributions to the development of mathematics, his life provides an opportunity to investigate the study of mathematics at the time of the American Revolution. This talk will examine Adams’ education at Harvard, explore some of the mathematical works in his personal library, and present several recreational mathematics problems found in his journals.
Mathematics is One (part I)
Russell Rainville
Wednesday, September 17, 2008
3:45 pm in Ricker Addition 202
All the parts of mathematics are related. The purpose of this talk is to give an example of one part of mathematics helping us understand another. This is an example of the use of descriptive statistics to prove a theorem about tournaments (directed complete graphs). The picture above is a tournament on 7 points and the path from 1 to 4 to 2 and back to 1 is a directed 3-cycle. How many directed 3-cycles does a tournament have? The theorem which answers this question has its roots in the study of the transitivity of preference relations (x ? y means y is prefered to x), so we are also doing political science.
Archive of Past Topics - Spring 2008
A Graphical Approach to Dynamical Systems
Hunter Basselet, Adam Case
and Daniel Jackson
Monday, March 24, 2008
3:45 pm in Ricker Addition 202
Join the presenters as they demonstrate the computer program they designed and coded to aid the study of the mathematical field of dynamical systems. We will explore various function systems resulting in visible strange attractors and see how initial conditions can effect these systems in real time. Also, we will view more complex aspects of the program such as the generation of famous probabilistic systems like Sierpinski’s Triangle and the Fractal Fern.
Semisimple Lie Algebras
Peter Hardy
Wednesday, March 12, 2008
3:45 pm in Ricker Addition 202
The theory of semisimple Lie algebras has many interesting applications in other areas of mathematics as well as quantum physics. What happens when radicals, annihilators, and killing forms meet upon the fields of Lie algebras? Is there anything left when the members of nilpotent Lie Algebras operate upon each other? Any questions left unanswered will be examined in more detail during my May term course: Mat 477 Topics in Mathematics - Lie Algebras.
Archive of Past Topics - Fall 2007
Searching for Maximal Chaos
Hunter Basselet
Adam Case
Daniel Jackson
Monday, November 19, 2007
3:45 pm in Ricker Addition 205
Iterative systems often display sensitivity to initial conditions. In this case, one says that the system behaves chaotically. Perhaps the most famous illustration of this penomenon is called the “butterfly effect.” This refers to the idea that the weather in Europe a month from now may be drastically affected by a butterfly flapping its wings today on some tropical island. We will demonstrate a computer program written by the first two speakers which is intended to find examples of systems possessing maximal chaos.
Cavalieri’s Principle
Mike Molinsky
Wednesday, October 17, 2007
3:45 pm in Ricker Addition 205
The diagram above shows:
(a) An amoeba kickboxing match (b) A Piet Mondrian artwork worth $14 million
(c) A precursor to modern integral calculus (d) Casper the Friendly Ghost in a police lineup
To find the answer to this pressing question, you will have to attend the talk.
The Steiner Tree Problem
Russell Rainville
Wednesday, September 26, 2007
3:45 pm in Ricker Addition 205
How did Delta Airlines save money on its telephone bill? For calls between corporate offices AT&T billed its corporate customers on the total length of the edges in the smallest tree that connected all of the company offices. By adding an office Delta’s phone bill was decreased by over a million dollars a year.
Steiner’s Problem: Given a set of points in the plane, what is the smallest graph (graph with least sum of edge lengths) whose vertex set includes the given set of points?
Archive of Past Topics - Spring 2007
What is a Matroid?
Lori Koban
Tuesday, April 3, 2007
11:30 AM in CC 006
Vector spaces have minimal dependent sets of vectors. Graphs have circuits. But in the world of matroids, minimal dependent sets of vectors and circuits in graphs are the same. I’ll introduce matroids in general as well as my favorite matroids, which come from labeling the edges of graphs with group elements.
The Steiner Problem on the Cone
Jamie Burwood, Bowdoin College
Monday, February 5, 2007
5:30 PM in Education Center 106
This talk is based on a paper coauthored by Caroline Nielson of the University of Southern Utah. The paper investigates the n-point Steiner problem on the thin cone. The Steiner problem involves finding the minimal path between a set of points, adding additional vertices if necessary. This problem has been investigated extensively on the plane and solved on the sphere, but no one has previously explored this question on a surface with a sharp point.
In order to reduce the problem on the cone to the problem in the Euclidean plane, the cone will be “cut” in such a manner that it collapses to a circular wedge in the plane. For the 3-point problem, one cut is sufficient to determine the minimal Steiner tree; however, for n-points, many more cuts must be made in order to find the solution. The talk will present an algorithm for making these cuts and constructing the Steiner tree.
Archive of Past Topics - Fall 2006
Fractal Images of Cremona Maps
Dustin Gage
Nov. 29, 2006
3:00 PM in Roberts 105
This talk is based on work done during the summer of 2006 by Dustin Gage†, Dan Jackson, and Dan Savage. We will discuss the single variable dynamics of a parameterized set of maps and focus on how visual representations of mappings on C2 can facilitate a better understanding of their hyperbolic components. We developed a Java applet to assist our research by producing fractal images of Cremona maps, as well as images of other quadratic rational maps of interest. I will describe the hyperbolic components - the Escape Type in particular - and discuss the utility of visualization for more research in the field. The aim is to understand the variation of the dynamics - in particular hyperbolic dynamics - of Cremona maps. We will discuss the progress made towards this goal.
† The continuation of the work on visualization of quadratic rational maps is supported by a Wilson Scholarship.
A Preview of Eulerpalooza
Mike Molinsky
Wednesday, November 8, 2006
3:00 pm in Ricker Addition 205
Next year will mark the 300th anniversary of the birth of Leonhard Euler. A wide variety of publications, conferences and other events are being planned in honor of this great mathematician. This talk will briefly address the importance of Euler’s work, present a few examples of his proofs, and also summarize some of the upcoming events of Euler’s tercentenary celebration.
Coordinate Functions of Plane Cremona Maps I
Dan Jackson
Wednesday, October 18, 2006
3:00 pm in Roberts 105
This is part 1 of a series of 3 talks based on work done during the summer of 2006 by Dustin Gage, Dan Jackson, and Dan Savage. In this talk we will introduce rational mappings of C2 and their extensions to P2. This is a large class of mappings whose global dynamics are still not well understood. We will focus on a special class of these mappings and a parameter space formed from their coordinate functions. The aim is to understand the variation of single variable dynamics - in particular hyperbolic dynamics - in this parameter space. We will discuss the progress made towards this goal.
San Gaku and Other Problems in Various Geometries
Amanda Taylor
Wednesday, October 4, 2006
3:00 pm in Roberts 105
Japanese San Gaku problems are Euclidean geometry theorems colorfully inscribed on tablets and hung on shrines in ancient Japan as a form of worship. In this presentation, we explore how some of these theorems and others are transformed when reformulated in spherical and hyperbolic geometry. The basics of both geometries will be explained.
Amanda did this work with another student, Christy Hediger from Muhlenberg College, at a Summer Research Experience for Undergraduates Program at Grand Valley State University. She will also talk about the Experience and the presentation of their work that they did at MathFest this past August.
Archive of Past Topics - Spring 2006
“A Few Mathematical Models of Infectious Disease”
Christina Hayes
3:30 pm Friday, February 17, 2006
Ricker 217
We will briefly discuss the pathology of influenza in general, as well as avian influenza in particular. I will then present some simple models used in mathematical epidemiology to study the spread of diseases such as influenza, smallpox, and typhoid fever. Assuming only basic understanding of what a derivative is, I will present a qualitative analysis of the SIR model. Using this approach, we will address the questions: What is the final size of the epidemic? What proportion of the population escapes the epidemic? For the SIR model in particular, we will use parameter values associated with the Hong Kong flu outbreak which occurred in New York City - in the late 1960’s.
“What Do Fractions, Completing the Square and Derivatives Have In Common?”
Lori Koban
3:30 pm Monday, February 13
Roberts 207
The Dwarf who prepares these posters (Russell Rainville) has no additional insight into the talk than that provided by the title. You will need to come to find out!
“Vertices, Faces and Edges: An Intuitive Approach to Euler’s Formula”
Nathan Carlson
3:45 pm Wednesday, February 8, 2006
Roberts 205
If you divide the plane or the surface of a sphere into “faces”, what can you notice about the number of vertices, faces and edges and how might you notice this?
“Detecting Errors in Codes”
Nicholas Koban
3:30 pm Monday, February 6, 2006
Roberts 207
Codes are used to transmit information; ciphers are used to hide information. “Words” are encoded in an electric signal. The signal travels along wires, through the air, and maybe through space. At the receiving end the electric signal must get decoded into “words”. As you may know from your cell phone, along the way there is noise and interference. How can the receiver recognize errors in the signal?
“Chaos Games”
Eva Curry
3:30 pm Friday, February 3, 2006
Computer Center 102
A chaos game is a probabilistic algorithm for visualizing certain self-similar sets, including many fractals. This talk will present some background necessary for understanding what a chaos game does. You will also have the opportunity to explore chaos games, probability, and some fractals that can be generated by a chaos game.
Prime Numbers and Hairy Fractions
Steve Bies
3:30 pm Monday, January 30, 2006
Computer Center 006
What does it mean that 21,649 × 513,239 = 11,111,111,111? What does this have to do with converting fractions to decimals? We will work out the decimal form of
- it’s easy!
Archive of Past Topics - Fall 2005
“Multi-Million Dollar Baby:
A Tale of Mystery, Misappropriation, Money and Mathematics”
Mike Molinsky
3:40 pm Monday, October 24, 2005
Computer Center 006
This talk will cover the history of the Archimedes Palimpsest, which sold at Christie’s auction house for two million dollars in 1998. The talk will also discuss new insights the document is giving us into the work of one of the greatest mathematicians of all times.
Finding All Integer Solutions To
Russell Rainville
3:40 pm Monday, October 17 2005
Computer Center 006
Subtitle: “How Geometry solves Problems from Algebra”
This fall Dan Jackson joined the Mathematics Faculty and his mathematical interests are in “Birational Correspondences between Algebraic Varieties”. To introduce these terms to students I decided to give a concrete example. I will use a birational correspondence between two hyper-surfaces to find all the integer solutions to the equations above. In particular we will find all integer solutions to:
“1-dimensional Cellular Automaton using Rule 30″
Dustin Gage
3:40 pm Monday, October 3, 2005
Computer Center 006
This could be subtitled “What I did at Summer Camp”. Dustin will talk about the work that he, Elizabeth Lamb, and Briana McGarry did at the Undergraduate Mathematical Research Experience program at Central Michigan University. They were investigating the use of an Automaton to produce pseudo-random sequences of “0″’s and “1″’s for use in encryption.
“A Sporadic Finite Simple Group”
Paul Gies
3:40 pm Monday
September 12 and September 19, 2005
Computer Center 006
The classification of finite simple groups is one of the great monuments of twentieth-century mathematics. The finite simple groups lie mostly in a few infinite families—An, the projective unimodular groups—but exactly 26 of the groups on the list are not members of an infinite family. The smallest of these “sporadic” groups is the Mathieu group M12, a simple group of 720 elements that act sharply 3-transitively on the projective line over the field with 9 elements. These talks will attempt to explain what that all means and how it is known.