Abstracts of Talks
(following the invited address, abstracts are alphabetical by speaker's last name)

Invited Address Applications of Optimal Control to Various Population Models
Dr. Suzanne Lenhart, University of Tennessee and Oak Ridge National Laboratory

An introduction to the basic ideas of optimal control of ordinary differential equations will be given. The differences between optimal control of ordinary differential equations and optimal control of partial differential equations will be briefly explained. An ordinary differential system for disease model will be used to illustrate the techniques. Bioreactors can be used to transform contaminants into less hazardous substances through bacteria metabolism. An example model for a gas phase bioreactor has a parabolic partial differential equation coupled with an ordinary differential equation.

The Grand Valley State University
Research Experiences for Undergraduates (REU) Program
Dr. Ed Aboufadel, Dr. Will Dickinson, 
Dr. Steve Schlicker, Dr. Jody Sorensen
Talk Level: 1

Grand Valley State University will be hosting a Research Experiences for Undergraduates (REU) next summer. Four faculty will work with 8 students on problems on wavelets, dynamical systems, metric and spherical geometry. This talk will give the details of the program, and will describe some of our past work and problems for the summer of 2004.

Take Me Out to/of the Ball Game
Brandon Alleman and Michael Cortez, Hope College
Talk Level: 1

advisor: Dr. Tim Pennings

When should a person leave a baseball game in order to maximize his/her enjoyment? How does this decision depend on the score of the game? Assuming a modified logistic rate of departure from the stadium, and a constant maximum exit rate from the parking lot, we find optimal strategies for leaving games of various scores. (By the end of the talk, you will know whether you should have left 
early.) 

Existence of (8,8,8,1) Relative Difference Sets
Nicholaus Bauer, Central Michigan University
Talk Level: 1

advisor: Dr. Ken Smith

It has been known that a relative difference set of (8,8,8,1) parameters exists in the group C₄ x C₄ x C₄. I will discuss the underlying difference set and relative difference set definitions. I will show the existence of two non-abelian (8,8,8,1) relative difference sets. I will also show a correlation between the existence of GH(4,2) in various groups and the existence of a (8,4,8,2) relative difference set analog. This is applied further to GH(8,1) and (8,8,8,1) relative difference sets.

When Matrices Go Bad
James Boerkoel, Hope College
Talk Level: 3

advisor: Dr. Aaron Cinzori

You have probably experienced the frustration of completing a math problem, and yet the answer is nowhere remotely close to the correct answer. What if you were told that it might not be your fault; it may be the fault of the problem itself. These “ill-posed” problems need to be fixed so that results are reliable. In our research, we examine a discrete method for regularizing ill-posed Volterra problems. We quantify how much better the condition number of the original ill-conditioned matrix is when using this method in a number of cases. We also provide numerical evidence of the improved condition number in all cases.

The Irredundant Number of Star Graphs
Mindy Bradford, Central Michigan University
Talk Level: 2

advisor: Dr. Sivaram Narayan

An irredundant number of a graph is a vertex labeling problem on simple graphs.  In such a labeling, it is possible for two distinct connected induced subgraphs H₁ and H₂ to have the same label sum.  If a labeling has the property that the sum of the labels of H₁ equals the sum of the labels of H₂ if and only if H₁ = H₂ it is called irredundant and the smallest label sum of an irredundant labeling is called the irredundant number of G, denoted l(G).

Mathematical Careers in Operations Research
Dr. Jim Bradley, Calvin College and the US State Department
Talk Level: 1

A large number of math-related careers are in a field commonly called "Operations Research." This talk will present a brief survey of the kind of work operations researchers do and the kind of preparation students need for OR careers.

The Graduate Program In Mathematics at Purdue University
Dr. Johnny E. Brown, Purdue University
Talk Level: 1

The Department of Mathematics offers rigorous degree programs leading to M.S. and Ph.D. degrees. In addition to degree programs in pure and applied mathematics our department offers degrees in Computational Finance (with emphasis on quantitative finance) and in Computational Science and Engineering (an interdisciplinary degree program between 18 departments at Purdue). We are strong in several areas of mathematics with 70 full-time faculty. Details of our graduate program along with our various fellowship opportunities will be presented.

The Graduate Program in Mathematics
at the University of Kentucky
Dr. Russell Brown, University of Kentucky
Talk Level: 1

The department of mathematics at the University of Kentucky offers a friendly environment for the teaching and learning of mathematics. Many of our doctoral students accept jobs at smaller colleges and universities that value teaching. We have active research groups in algebra, topology, analysis, numerical analysis, partial differential equations and discrete mathematics. We offer a traditional Master's in mathematics. This degree is often earned on the way to a Ph.D. In addition, we offer the Master's of Science in Applied Mathematics which includes courses in statistics and computer science as well as mathematics.  

Most of our students our supported by teaching assistantships. Fellowships are available for outstanding entering students.

The Singular Value Decomposition of a Composition Operator
Michael Dabkowski, University of Michigan-Dearborn
Talk Level: 3

advisor: Dr. John Clifford

We find the Singular Value Decomposition of a Composition Operator acting on the classical Hardy Space (H²) with linear fractional inducing maps. Such a decomposition yields two very interesting results- First, we find the norm of the operator, as the solution to an eigenvalue problem. Secondly, we generate two orthonormal bases for the Hardy Space in question.

Central Michigan University: 
Opportunities for Graduate and Undergraduate Students
Dr. Lisa DeMeyer, Central Michigan University
Talk Level: 1

Central Michigan University offers a mathematics Ph.D. degree with an emphasis on the teaching of college mathematics. Our students take courses in a broad range of mathematics, including mathematics education. Our program started in 1994. Teaching assistantships, doctoral fellowships, and GAANN fellowships are available.  

For undergraduates, Central Michigan University hosts a National Science Foundation Research Experiences for Undergraduates (NSF-REU) program. Undergraduates spend eight weeks during the summer working on a problem related to algebra, matrix theory, or graph theory. Students are paid a stipend and recieve housing and a meal allowance. 

In this talk, we will give more details about both the graduate program and the NSF-REU program.

Knotted Spheres in Four Dimensions (Part 2)
Jon Dent, Calvin College
Talk Level: 1

advisor: Dr. Gerard Venema

Groups can be used to describe geometric objects. Given certain groups, a diagram of a 4-dimensional object (4- manifold) can be constructed. These diagrams represent 4-manifolds by linked closed curves in 3-dimensional space. Ideas behind these diagrams will allow a theorem to be presented concerning knotted spheres in 4-dimensional space.

Mathematics in Banking: a Changing Industry
Larry D'Haem, Fifth Third Bank
Talk Level: 1

We will have a brief discussion of the current trends in the financial services industry and particularly in Banking. Many of these trends have been precipitated by the mathematical power of the computer and are opening opportunities for mathematicians in fields previously dominated by the bean-counters.

Centralizers of Straight-Line Functions
Giovanni Dimatteo, Albion College
Talk Level: 3

advisor: Dr. Mark Bollman

The set A of linear functions from R to R forms a group under composition. However, it is known that composition of functions is not in general commutative, and hence we cannot say that A is abelian. Specifically, when given a linear function, we also cannot know which other linear functions will commute with it. The “centralizer” of a particular group element is the set of all other elements within the same group that commute with that element. We give a complete characterization of the centralizers for each type of linear function. Results include a proof that the centralizer of a particular linear function is in fact an abelian group, under appropriate restrictions.

Distribution of Power in Weighted Voting Systems with
Multiple Voters and Multiple Candidates
Kevin Dufendach, Taylor University
Talk Level: 1

advisor: Dr. Matt DeLong

Weighted voting systems arise in many everyday situations, giving proportionally more votes to CEO’s, high stock holders, or large states. Oftentimes, however, the ratios of the votes are not directly proportional to the ratios of influential power. For example, consider a majority-wins voting system with weights (100, 100, 100, 100, 1). We will analyze and order combinations for four-player weighted voting systems, then we will take and expand these results to see the effects of adding more voters as well as multiple options in a plurality system.

Comparing Systems of Gambling
Aaron Dull and Remington Steed, Calvin College
Talk Level: 1

Advisor: Dr. John Ferdinands

A gambler repeatedly plays the same game of a fixed probability, always starting with a bet of $1. He plays until he either wins, or loses twice in a row. After a win, he starts back at his original bet. After each loss, he multiplies his bet by (k+1). What is his probability of gaining N dollars?   Is it better for him to choose a higher k?

Knotted Spheres in Four Dimensions (Part 1)
John Engbers, Calvin College
Talk Level: 1
advisor: Dr. Gerard Venema

Thinking about higher dimensional manifolds, or spaces, is often a difficult task.  This talk focuses on how to understand some simple higher dimensional manifolds by showing analogies from two and three dimensions.  Some of the ideas of this first talk are explored further in the second talk, in which Jon Dent speaks about a theorem on four-dimensional manifolds that we proved this summer.

Classifying Semigroups where Gamma(S) is a
Star Graph or the Refinement of a Star Graph
Amanda Geiser, Susquehanna University
Talk Level: 3

advisor: Dr. Lisa DeMeyer, Central Michigan University

A semigroup with zero is a zero divisor semigroup. To each commutative zero divisor semigroup S, we associate a simple graph Gamma(S). The vertices of the graph are the nonzero zero divisors of S and there is an edge between two vertices a and b if and only if ab = 0. Via examination of the relationship between the semigroup theoretic properties and graph theoretic properties of Gamma(S), we classify those semigroups where Gamma(S) is a star graph or the refinement of a star graph.

Local Maximums at Infinity and the Generalized Maximum Principle
Kyle Glashower, Calvin College
Talk Level: 2

Advisor: Dr. David Yang, Tulane University

If a function, f, is twice differentiable and has a local maximum at a finite point, p, the calculus implications are fairly obvious: f '(p) = 0 and f ''(p) <= 0. But what if there is a local maximum at infinity? What does it even mean for there to be a "local maximum at infinity"?

The Graduate Program at Michigan Tech
Dr. Mark Gockenbach, Michigan Tech University
Talk Level: 1

The Department of Mathematical Sciences at Michigan Tech offers Masters and PhD degrees with concentrations in Applied Mathematics, Discrete Mathematics, and Statistics. Faculty members carry out cutting-edge research in such areas as statistcal genetics, coding theory and cryptography, multiphase flow and combustion, financial mathematics, computational science, and wildlife statistics. Most of our students are fully supported as Graduate Teaching Assistants, and we have one of the most thorough GTA training programs in the country. Michigan Tech is located on the scenic Keweenaw Peninsula about 10 miles from Lake Superior.

Fermat’s Last Theorem: Advances in the 19th Century
William Green, Albion College
Talk Level: 3

advisor: Dr. Mark Bollman

Fermat’s Last Theorem puzzled many great mathematicians from its first proposal in the margins of Pierre de Fermat’s copy of Arithmetica in 1637 until it was finally proved by Andrew Wiles in 1994. The history and mathematics that have come out of attempts to solve Fermat’s Last Theorem have led to such notable work as the theory of ideals in abstract algebra and numerous advances in the theories of modular functions and elliptic curves. In this talk, particular attention will be paid to the work of Ernst Kummer and other notable 19th century mathematicians. The cyclotomic integers, the idea of unique factorization, and some aspects of Galois theory will be explored in their connection to the work towards proving Fermat’s Last Theorem.

An Interdisciplinary Course for Biology and Mathematics Majors
 Henry Gould and Mike Ross, Hope College
Talk Level: 1

 advisor: Dr. Janet Andersen  

Mathematical Biology is an ever-expanding field that benefits greatly from its interdisciplinary nature. At Hope College, we have created a mathematical biology course co-taught to a mixed audience of biology and mathematics students. The course is based on biology research papers and includes wet labs. We will discuss the format of the class, details of research papers and labs, student reactions, and outcomes from the course. The biology topics include Ecology and Neuroscience. The mathematical topics include Linear Algebra and Differential Equations.

k-Alternating Knots
Philip Hackney, Central Michigan University
Talk Level: 1

advisor: Dr. Leonard Van Wyk, James Madison University

A projection of a knot is k-alternating if its overcrossings and undercrossings alternate in groups of k as one reads around the projection (an obvious generalization of the notion of an alternating projection). We show every knot that admits a k-alternating projection also admits a (k+1)-alternating projection. We prove the surprising result that every knot admits a 2-alternating projection, which partitions nontrivial knots into two classes: alternating and 2-alternating.

The University of Notre Dame Ph.D. Program
Jacob Heidenreich and Dr. Liviu Nicolaescu
Talk Level: 1

Our very friendly faculty and graduate students are pursuing exciting research projects in a variety of areas of pure and applied mathematics. Our Ph.D. students all receive a full fellowship in the first year. This allows plenty of time to explore various areas of mathematics, and to get acquainted in the department. Teaching duties are introduced gradually. Our goal is to allow students to sample the greatest possible variety of professional experiences, creating mathematics, speaking about mathematics (in seminars, and also at regional and national meetings), writing about mathematics (in papers and in the thesis), and teaching mathematics.

Statistics and the Environment
Sarah Hession, Michigan Dept. of Environmental Quality
Talk Level: 1

A statistician specialist with the Michigan Department of Environmental Quality will describe how she puts her math degree to work to protect and improve environmental quality. She will describe typical problems in environmental statistics and important skills that can help you be successful in this field.

Photographic Mathematics
Jason Hill, University of Michigan-Flint
Talk Level: 1

advisor: Dr. Larry King 

It seems very common to discuss how arts experience can be of great benefit when doing mathematics. Yet, the topic of how mathematics can aid the arts is rarely discussed except when formal technicalities are involved. In other words, we know that creativity can flow from the arts to mathematics, but we seem hesitant to let the same sort of creativity flow from mathematics to the arts.

In my talk, I will discuss how being a mathematician has influenced my creative photographic skills. I will give a short introduction to several areas: color filters and polarization filters, the base-2 system of photography, digital versus film, and lens construction / lens correction.

The Graduate Program in Biostatistics
Dr. Jack Kalbfleisch, University of Michigan, Ann Arbor
Talk Level: 1

The Department of Biostatistics in the School of Public Health at UM has graduate programs leading to the Master of Science, Master of Public Health and PhD degrees.  Biostatistics concerns the study and development of statistical methods for the design and analysis of studies in life sciences.  We have a strong theoretical component in the Department and many links with researchers in public health, medicine, biology, genetics and bioinformatics. If you enjoy the mathematical sciences and have an interest in applications of mathematics and statistics to the life sciences, you should think about biostatitistics.  Our Department is internationally known for research in survival analysis, clinical trials, statistical genetics, image analysis, clinical trials and many other areas. Our graduate programs are very successful and lead to a high demand market with interesting jobs, both at the Masters and PhD levels. In this talk, I will describe some background to the theoretical work done in analysis of survival and life history data, and also provide a preview of our Department and its activities.

Overview of Oakland University's
Graduate Program
Raymond Kleinberg, Oakland University
Talk Level: 1

Oakland University has been called one of the best kept secrets in Michigan. A mid-sized public university of about 16,000 students offering undergraduate and graduate degrees in many fields, including a recently established Ph.D. program in Applied Mathematical Sciences in addition to a Master of Arts in Mathematics, a Master of Science in Industrial Applied Mathematics, and a Master of Science in Applied Statistics. This talk will give a brief overview of our programs and placement of recent graduates.

Calculating Equivariant Homotopy Groups
Bree Koehler, Kalamazoo College
Talk Level: 3

advisor: Dr. Michele Intermont

For this talk, we will focus first on developing, through examples, the concept of homotopy groups, which is a part of the study of topology. Then, we will define G-spaces and equivariant maps, for the purpose of approaching the problem of computing the equivariant homotopy groups of spaces, objects similar to homotopy groups but which describe a different type of structure. Finally, we will compute a few short examples of equivariant homotopy groups.

Mathematics Majors in Engineering
Wendy Kooiman, Smiths Aerospace
Talk Level: 1

We will discuss employment opportunities for Mathematics majors in the Engineering field. In particular, we will talk about systems engineering in the aerospace field at Smiths Aerospace and what is involved in being a systems engineer.

The Generalized Area Principle
Kristina Lund, Grand Valley State University
Talk Level: 1

advisor: Dr. William Dickinson

Come discover how the volume of a special tetrahedron can be used to generalize the area principle to spherical and hyperbolic geometry. We use this tool to generalize certain cyclic product relations, such as Menelaus' and Ceva's theorem, to polygons in these geometries.

Professional Masters Degrees
Dr. Charles R. MacCluer, Michigan State University
Talk Level: 1

The Science Masters degree is being rehabilitated nationwide as a professional degree. This movement is in response to the need of science-trained managers in the increasing technical American workplace. I will discuss various industrial mathematics and financial mathematics programs nationwide, and in particular, the proMSc program in industrial mathematics at Michigan State University.

An Analysis of Linear Recurrence Relations
Audrey Maness, Central Michigan University
Talk level: 2

Advisor:  Dr. George Grossman

Using previous research, we explored a set of linear recurrence relations.  Our goal was to identify new relationships and/or equations using a combination of known equations through a series of summations.  We were able to employ Pascal's Triangle with other known identities to arrive at a conclusion for a summation of a set of mathematical statements. Our presentation explores the methods used to arrive at our conclusion, and the new mathematical relation formed as a result of our work.

Using Group Theory to Examine the Art of Change Ringing
Rana Mikkelson, Kalamazoo College
Talk Level: 3

advisor: Dr. Michele Intermont

Change Ringing is the art of ringing tower or hand bells according to a specific set of rules defined by the British in the 1600s. In particular we examine the Group Structure in the “songs”, which are composed of permutations of the bells, which gives us one application of group theory; the symmetric group on n letters or, here, n bells. In this talk we will examine where subgroups of the symmetric group appear...and where they don’t! Another example is when group structure is present in certain places, the task of composing “songs” is simplified. But is this group structure necessary to compose the “songs”?

Two Families of Randomly Decomposable Graphs
Erin Militzer, Central Michigan University
Talk Level: 1

advisor: Dr. Ken Smith

A graph G, is Randomly H-decomposable if any subgraph isomorphic to H is part of an H-Decomposition. The set of all randomly H-decomposable graphs is denoted by RD(H). We examine RD(H) where H is one of the following: (1) H = K_mP_e, a graph constructed by identifying a vertex of the complete graph K_m with an end of the path P_e, OR (2) H = H₀+ P₁ where RD(H₀) is known.

Counting Quadratic Forms of Rank 1 and 2
Nina Miller, Valparaiso University and
Andrew Wells, Hope College
Talk Level: 3
advisor: Dr. Darin Stephenson, Hope College

Quadratic forms are polynomials in several variables where each term has degree 2. This talk will discuss vector subspaces of quadratic forms with special attention given to counting the forms of rank less than or equal to 2. We will include a brief introduction to the theory of quadratic forms followed by results of our research in 2, 3, and 4 variable cases.

The TRUE Permutations of the Bells
Aileen Murphy, Kalamazoo College
Talk Level: 3

advisor: Dr. Michele Intermont

Change ringing is the art of ringing tower bells, dating back to the early 17th century. The "songs" (methods) rung are composed of permutations of the bells. Each permutation can occur only once in a method, or else the method is said to be "false". For the most common class of methods, falseness has been partitioned into "types". The question we will address here is: for different classes of methods, is the partitioning into types of falseness isomorphic to that of the most common class?

Symmetry Detection in Boolean Functions
Dan Steffy, Oakland University
Talk Level: 3

advisor: Dr. Debatosh Debnath

Symmetry of variables in Boolean functions is an important structural property. Many problems in engineering can be solved more efficiently in the presence of this property including circuit design, Boolean matching, digital circuit testing, and building binary decision diagrams. This talk will discuss the use of Boolean satisfiability (aka SAT) solvers as a tool to detect symmetry in Boolean functions. I will describe the details of our method and give experimental results.

Employment Opportunities in Mathematics
Dr. Ralph Svetic, Michigan State University
Talk Level: 1

Mathematics majors often wonder (and are frequently asked) "What can someone do with a degree in Mathematics?" This talk will answer that question by describing a few of the many employment listings that have appeared during the last few months on the EIMS web site (Employment in the Mathematical Sciences, a production of the American Mathematical Society). The web site is a searchable data base of job announcements by employers seeking mathematicians.

Separable Preferences and Admissible Characters
Micah TerHaar, Grand Valley State University
Talk Level: 1

advisor: Dr. Jonathan Hodge

In situations when multiple decisions are made simultaneously, it is often the case that an individual’s preferred outcome on one or more of the decisions depends on the outcome of the others. We will formalize this notion of interdependence and study its relationship to similar ideas from economics.

Introducing Actuarial Mathematics
Christian R. Veenstra, Watkins, Ross & Co.
Talk Level: 1

We will discuss the nature of actuarial mathematics and its application. Also, we will discuss the steps necessary for becoming an actuary. Finally, we will explore employment opportunities in actuarial careers and associated professions.

How Big can a Polynomial Be?
Ben Vugteveen, Grand Valley State University
Talk Level: 2

advisor:  Dr. Matt Boelkins

One way to measure the size of a function is by its supremum norm,

Geometry, Algebra, and Hilbert Series
Yumi Watanabe, Calvin College
Talk Level: 2

advisor: Dr. James Turner

The Hilbert series is a fascinating infinite series which arises when studying algebraic objects called ideals and geometric objects called varieties. Basic ideas related to these objects are introduced, then the talk will focus on one particular property, the lower bound property, of the Hilbert series for an ideal generated by homogeneous polynomials. An open conjecture regarding this property will be stated and the status of this conjecture will be discussed.