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Banquet Speaker

Colm Mulcahy, Spelman College

Eine Kleine Nachtmagie

We'll explore a variety of magical feats based squarely on mathematical principles.  Starting with simple card forces/prediction tricks--diners are invited to bring their own decks of cards-- we'll move on to an amazing demonstration of mind reading as "Mathematical Idol 2005" is selected at random from the finalists Archimedes, Euclid, Euler, Gauss, Hilbert, Newton, Pythagoras and Riemann.

Colm MulcahyColm Mulcahy earned a B.Sc and M.Sc. in Mathematical Science from University College Dublin, in Ireland, in the late 1970s.  A few years after getting his PhD for research in abstract algebra under  the direction of Alex F.T.W. Rosenberg at Cornell University in 1985, he joined the faculty at Spelman College, in Atlanta, Georgia.

His interests have since expanded to include number theory, geometry, geometric design, image processing and data compression, as well as the math of voting and mathematical biology.  He got hooked on mathematical card tricks a few years ago, and is now is a member of Georgia's Ring 9 of the International Brotherhood of Magicians (the original IBM!).

He likes music, running, eating and cooking, not necessarily in that order.  In his spare time he is the chair of the Department of Mathematics at Spelman.

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Saturday Morning Invited Presentations

Peter Turner, Clarkson University

Overcoming the effects of the Logarithmic  Distribution of Numbers:  Gradual and Tapered Overflow and Underflow

The talk  begins with a discussion, and proof, of the surprising observation that 30% of  numbers have leading significant digit 1. The proof uses elementary ideas and is  based on the effect of repeated multiplications.
This fact, often known as  Benford's Law, has important consequences for computer arithmetic design. The  later parts of the talk will address some of these, most notably the frequency  of overflow and underflow and its prevention through the use of tapered overflow  and underflow.

Turner photoPeter Turner grew up in England – in the Liverpool area in the Beatles era. He received his B.Sc. (Honors) in Mathematics in 1970 and his Ph.D in Pure Mathematics in 1973, both from the University of Sheffield. After a year as a postdoc in Applied Mathematics at Sheffield, he joined the faculty of the University of Lancaster as a Lecturer in Numerical Analysis. During this time his research interests developed to include systems of computer arithmetic. In 1987, Turner moved to the U.S. Naval Academy where he stayed until taking up his current position as Chair of Mathematics and Computer Sceince at Clarkson in 2002. He has published four undergraduate texts in scientific computing, edited three volumes of conference proceedings, (co-)authored more than 40 research papers, and served on national committees for SIAM.

James Tattersall, Providence College

Episodes in the Early History of the Lucasian Chair

In 1663, Henry Lucas, the long-time secretary to the Chancellor of the  University of Cambridge, made a bequest, subsequently granted by Charles II, to endow a chair in mathematics. A number of conditions were attached to the Chair.    Among the more prominent Lucasian professors were  Newton, Babbage, Stokes, Dirac, and  Hawking. We focus attention on the early Lucasians, many of whom were very  diligent in carrying out their Lucasian responsibilities, but, as history has  shown, such was not always the case. In the process, we uncover several untold  stories and some interesting mathematics

Jim TattersallJim Tattersall received his undergraduate degree in  mathematics from the University of  Virginia, a Master's degree in  mathematics from the University of  Massachusetts, and a Ph.D. degree in  mathematics from the University of  Oklahoma. On a number of occasions he  has been a visiting scholar at the Department of Pure Mathematics and  Mathematical Statistics at Cambridge  University. He spent the summer of  1991 as a visiting mathematician at the American Mathematical Society. In  1995-1996, he spent eighteen months as a visiting professor at the U.S. Military  Academy at West Point. He was given awards for  distinguished service (1992) and distinguished college teaching (1997) from the Northeastern Section of the MAA. He is former President of Canadian Society for  History and Philosophy of Mathematics, the Archivist/Historian of Northeastern  Section of the MAA, and the Associate Secretary of the Mathematical Association  of America.

David Poole, Trent University

John F. Randolph Lecture:  Everybody  Wins: Why Mathematics Courses for Elementary School Teachers Matter and How They  Can Benefit All Students of  Mathematics

Drawing on my own experience, I will describe a successful  mathematics course for pre-service elementary school teachers and argue that  such courses need to be more widespread. Even in universities and  colleges where  mathematics-for-teachers courses do exist, they are almost universally perceived  as low-level service courses, while the "real mathematics courses" are on "the  other side of the fence". I claim that not only is this viewpoint flawed, but  there isn't really a "fence" (or doesn't need to be).  With examples from various mathematics courses I have taught, I hope to demonstrate that serious mathematical topics can be fruitfully incorporated into  a course for elementary teachers. Conversely, teaching methods appropriate in a  math-for-teachers course can be successfully used in other mathematics courses.

David Poole photoDavid Poole received his B.Sc. in Mathematics from  Acadia University and his  Ph.D. from McMaster  University. Since 1984 he has been in  the Department of Mathematics at Trent  University,  Peterborough,  Ontario. He served as Chair of that  Department for six years and is currently  Trent's Associate Dean of Arts &  Science (Teaching & Learning).  Professor Poole has been a frequent participant in education sessions and  working groups of the Mathematical Association of America (MAA), the Canadian  Mathematical Society (CMS), the Canadian Mathematics Education Study Group  (CMESG), and the Mathematics Education Forum of the Fields Institute. For the  CMS, he has also served on the Education Committee and chaired the Human Rights Committee. In Ontario, he is a  member of the provincial advisory panel for the high school mathematics  curriculum. Professor Poole's research interests are in ring theory, discrete  mathematics, and mathematics education. He is the author of the textbook Linear Algebra:  A Modern Introduction  (Brooks/Cole), now in its second edition. While at Trent  University, Professor Poole has  received three merit awards for excellence in teaching as well as the  university's highest award for teaching excellence, the Symons Award. In 2002,  he was awarded an Ontario  Confederation of University Faculty Associations  Teaching Award and in 2003, he received a 3M Teaching Fellowship, the highest  award for university teaching in  Canada.

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Saturday Afternoon Contributed Talks

Cristina Bacuta,  SUNY Cortland

Proof: A process strand for in-service teachers and an activity for pre-service teachers

This presentation will outline some important results of our efforts to assess proof abilities at SUNY Cortland.  This work involves collaboration between the mathematics department faculty and a group of in-service teachers, and is supported by our PMET mini-grant "Assessing pre-service teachers' abilities to do proofs".

Howard Bell,  Brock University

Some Ruminations on Wedderburn’s Theorem

This year is the hundredth anniversary of the publication of Wedderburn's seminal theorem on the commutativity of finite division rings. We comment on the historical context in which the theorem appeared, the various proofs, and some applications - old and new.

Daniel Birmajer,  Nazareth College

The standard identities and the Quantum planes

Since the standard polynomials play an essential role in the Polynomial Identity (PI) theory of the matrix algebras, it is natural to look for similar situations in other contexts. Let F be a field and q an arbitrary nonzero scalar. The quantum plane is the associative ring generated by the variables x  and y, whose product satisfies the following relation yx=qxy. In this talk we explore the role of the standard identities in the PI-theory of the quantum planes.

David Brown,  Ithaca College

Experimental Mathematics and Writing for First Year Math Students

In this talk, I will describe a course that combines computer experimentation, mathematical exploration, and intensive writing in a second semester course for  mathematics majors and minors.  Using open-ended problems and even some famous unsolved problems, such as the Collatz Conjecture, simple questions motivate students to investigate topics.   Students produce weekly reports, complete with supporting computer and mathematical evidence.   Emphasis is placed on a clear style of mathematical writing.  In this talk, I will discuss how this approach helped foster a deeper understanding of why we study mathematics and how mathematicians tackle ideas. I will illustrate how student understanding improved during a semester full of writing.  I will also indicate how student attitudes toward mathematics changed over the course of the semester.  Many students did not initially care for the open-ended approach, but ultimately came to value it.

Allen Emerson and Kris Green, St. John Fisher College

How a writing assignment transformed a writing intensive mathematics course for business students

Five years ago we were approached to develop an alternative mathematics course for business students that introduced spreadsheets and approached mathematical topics in a way that we were later able to identify as being in line with the CRAFTY report from the MAA. A few minutes from the start of the very first class, we discovered that we had no idea what to assign for the students to do. By nature, the incorporation of the spreadsheet led us to need an assignment allowing students to make arguments involving the development and interpretation of data. This required the development of a communicative context and data for analysis Years later, we find that the course is now based almost solely on such argumentation and analysis assignments. These assignments give insight into student approaches to problem solving and have led us to a completely new system of grading that focuses on the standards we set for student learning.

Jeff Johannes,  SUNY Geneseo

Mathematical Teaching Soundtrack

For the past two years I have begun every single class session by playing a song that has a connection (at times extremely tenuous) to the content of the day.  I will share a ten-minute mix of songs that I have used for this purpose.

John Maceli ,  Ithaca College

Fairness, The Talmud and Pascal

Topics of fairness make terrific subject matter for a contemporary mathematics course. We will illustrate a few problems related to fairness and equity. In particular, we will discuss a bankruptcy problem arising in The Talmud and show a connection with the solution of this problem and Pascal’s solution of the "Problem of Points".

Neils Hanson, Matthias Youngs, Anthony Macula**, SUNY Geneseo

Pooling Designs for DNA Code Validation

We describe a random group testing method used to design laboratory experiments to validate a DNA code. A DNA code is a collection of synthetic DNA strands closed under complementation. DNA codes serve as universal components for biomolecular computing and biomolecular nanotechnology and also have been used in other areas of molecular biology (e.g., single nucleotide polymorphism genotyping, gene expression profiling and DNA microarray development).

Facilitator:  Anthony Macula, SUNY Geneseo

Discussion of Undergraduate Biomathematics

An informal discussion about biomathematics at our institutions as presented by participants. Some topics for discussion could be:
1. What is a biomathematics program?
2. How (not) to organize interdepartmental programs.
3. External funding opportunities.
4. Organization of a local biomathematics group.
5. Organization of a local undergraduate/faculty meeting.

James Marengo,  RIT

The Noiseless Coding Theorem

The Noiseless Coding Theorem gives a lower bound in terms of entropy on the expected length of a binary coding of the values of a random variable or vector. The concept of entropy will be discussed and a proof of this theorem will be presented. The author will then examine some implications of this result with the help of some examples. 

Donald Muench,   St. John Fisher College

A Mathematical Tour of St. Petersburg

In the summer of 2004, my wife and I were in Tallinn, Estonia at a Bridges for Education English Conversation “Camp” for 4 weeks.  This led to taking a 5-day tour of nearby St. Petersburg and spending one morning visiting the sites of Euler’s tomb and apartment house.  A nearby house was the residence of a number of other famous mathematicians.  We will mention a few facts about each of these mathematicians and those “descendents” connected to St. Petersburg. We will show pictures of these sights and the plaques on the walls of these two buildings.  This should whet one’s appetite for the MAA Mathematical Tour in 2007 to celebrate Euler’s 300th birthday.

Colm Mulcahy,  Spelman College

Wavelets made easy
We will give a Fourier-free introduction to wavelets by presenting a linear algebra approach to the simplest example, Haar wavelets, motivated by a real world application to data/image compression.  Anybody who can add, subtract and divide by two will be able to sing along in no time at all.

Moderator:  Olympia Nicodemi,  SUNY Geneseo

Whither the Textbook? [90 minute roundtable discussion]

A round table discussion wherein we discuss the changing role of the textbook in the college classroom. Questions to be addressed include:   What should a good textbook do for a course?  Must authors and publishers be at odds with the students and the used book market? How can authors shape the college math curriculum in the current publishing environment?  What are or should be the alternatives to textbooks?  What about the cost to the students?  Representatives from all sides of the issue will attend and contribute to the discussion.

Sam Northshield,  SUNY Plattsburgh

Summing across Pascal’s triangle and its continuous analog.                                   
It is well known that parallel row’s of Pascal’s triangle sum to 2^n and that sums along another family of parallel lines sum to the Fibonacci numbers (asymptotic to p^n where p is the Golden ratio). The generalization to families of parallel lines with a fixed rational slope are also known to be asymptotic to c^n for some c dependent on the slope;  we shall discuss a proof.  The entries of Pascal’s triangle are the binomial coefficients and can be extended, via the gamma function, to have positive real arguments.   We shall discuss the analog of the results above involving integration, over families of parallel lines, of the extended binomial coefficients.

Morris Orzech,  Queen's University

 Matter and mindset in the pre-service classroom

There is much current discussion, but little consensus, about the role of our courses and programs in turning out teachers with mathematical and intellectual attributes that will make them effective. My approach to courses for pre-service mathematics teachers is conditioned by a goal selected from a palette of current prejudices about teacher training, namely that it is desirable to intertwine mathematical, pedagogical and career goals. My exercise of this prejudice is constrained by the context of the courses in our program: the material should be appropriate for an advanced course without duplicating other courses, and should introduce new mathematical ideas while keeping an orientation more for prospective high school teachers than for prospective graduate students. One strategy is to use material that is accessible at high school level (but new) as a vehicle for engagement and surprise, and to follow up with reflection about knowledge, beliefs, and habits that students have developed in previous courses. The classroom interaction that supports the introduction of this material is intended to help develop students’ pedagogical sensitivity and (if I am lucky) their later professional practice.

Joseph Petrillo,  Alfred University

Counting Subgroups of a Direct Sum of Cyclic Groups

Recall, a cyclic group Zn of order n is an abelian group which is generated by a single element and has exactly one subgroup for each divisor of n.  For example, Z18 has 6 subgroups.  By The Fundamental Theorem of Finite Abelian Groups, a finite cyclic group can be decomposed into a direct sum of cyclic groups of relatively prime orders.  Thus, Z18 = Z2   Z9, and the number of subgroups of Z18 can be obtained by calculating the product of the number of subgroups of Z2 and Z9.  Unfortunately, this method for counting subgroups fails for Z3   Z9.  The number of subgroups of Z3   Z9 is actually 10 (not 6).  We will derive a formula for calculating the number of subgroups of the direct sum of cyclic groups of orders pm and pn. 

Jamar Pickreign and Bob Rogers, SUNY Fredonia

Right Question, Wrong Answer, or Wrong Question, Right Answer? 
Teacher Evaluation of Constructed Response Test Items.

Much speculation is given to what New York State's new  mathematics tests for their new standards will look like. Assuming  that the types of questions remain essentially the same, this talk will present samples of actual teacher evaluations of student  constructed responses on past Math A exam items. We will discuss the  challenges teachers face in evaluating these responses and the  implications for mathematics teacher preparation.

Gabriel Prajitura,  SUNY Brockport

What we talk about when we talk about Calculus

We will present some examples of differentiable functions having properties which do not look like what we teach in Calculus.

Jennifer Roche, Hobart and William Smith Colleges

Radices and Matrix Rings

Work has been done on characterizing normal subgroups of symplectic groups via nonassociative algebra.  The approach is entirely algebraic and hinges on Jordan ideals and specifically the structure called a radix in matrix rings.  We have generalized previous results about Jordan ideals and radices in matrix rings over graded algebras in future hope of a result for normal subgroups.

Hossein Shahmohamad, Rochester Institute of Technology

Coefficients of  Flow Polynomial of Kn

While the coefficients of the chromatic polynomial of the complete graph Kn is known, the coefficients of the flow polynomial of Kn is not known. We determine some of these coefficients and offer some conjectures on the rest.

Xunyang Shen and Peter Turner,  Clarkson University

 A Hybrid SLI System with Taylor Approximation

Symmetric level-index (SLI) arithmetic was introduced to overcome the problems of overflow and underflow in the floating-point system.  Our current research is to improve the algorithm performance and to promote its implementations.  In this talk, I will first give a brief summary of the SLI system and its arithmetic algorithms, and then introduce an approximation scheme using Taylor’s expansion, together with its software implementation in a hybrid system.  Finally, two applications will show how the system would facilitate some real life computational problems.

Joseph Smith, SUNY Binghamton

Lattices and Groups
We will begin with a brief discussion of what a lattice is and then look at some small subgroup lattices and the lattice of normal groups within a group.  We will conclude by looking at the normalizers in a group and see that in general the set of all normalizers do not form a lattice.  Then briefly discuss some cases where they do.

Barbara Stewart,  SUNY Geneseo

What's New? The Entire State Math Curriculum!
In March 2005, the New York Board of Regents approved a new Mathematics Core Curriculum for all students in pre-kindergarten to high school.  The document outlines math requirements for each grade from pre-K to 8.  Courses in Algebra, Geometry and Algebra 2 will replace Math A and B.  This session will be an overview of the "who, what, where, when and why" of the new program.  This update should be useful to college educators who work with pre-service teachers and may also be of interest to anyone who has children in school in New York State.

Karen Wells,  Monroe Community College

Sand, Water, Chocolate Chips: Infusion of mathematics in an early childhood curriculum.

I will discuss how to create an effective early childhood math program through both unplanned and intentionally planned math learning activities across most areas of an early childhood curriculum.

Julia Wilson, SUNY Fredonia

 Music as a Mathematical Subject:   the Backstory

By the early Medieval period, music was considered a subfield of mathematics, along with number theory, geometry, and astronomy.  How does this make sense?  We will discuss the origins of this classification among the ancient Greek philosophers, and trace the influence of their theories through the early Christian era and into the Renaissance.  This will shed light on how early scholars viewed math and its role in their quest to understand the universe.

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Student Program

Megan Cullen, Tim Ellis and Laura Jenkins, SUNY Geneseo

Student Workshop:  Mathematical Designs - Pretty Pictures Produced Painlessly

Just using a ruler and a pencil, you can use basic concepts to create geometric shapes and complex designs that are visually stimulating.  You will learn a fun, easy and exciting way to make cool designs with which you may decorate your home. The foundations of these beautiful pictures are simple patterns and lines, but they appear more difficult to make than they are.  Anyone can do it!

Jeremy Entner,  SUNY Brockport

Photography and Numerical Interpolation

In our work, Photography, and Numerical Analysis have something in common.  Photographs can be reconstructed, in a reduced form, using parametric equations of curves found within images. With Maple, Spline and Lagrange Interpolating Polynomials are in used for a better description of a  landscape, a fish and a portrait. A comparison of different methods of interpolation/extrapolation will also be discussed.

Ryan Grover, SUNY Geneseo

Finding the Perfect Set

We define a perfect set in Rn as a closed set in which every point is a limit point.  In this talk we will search for a nonempty perfect set in
R1  that contains no rational numbers. 

Sandra Anita Lacea, SUNY Brockport

Mathematical Methods in Modeling Landslides

Based on the information that we have on a certain site, different mathematical methods are discussed in order to determine the best strategy for evaluating the slope instability. Matlab, Maple and GIS software are incorporated in our approach in order to get the mathematical model and the landslide hazard map.

Diane Marie Lunman, Nazareth College
Applications of covering maps to spectral theory of graphs

The spectrum of a graph is the set of roots of the characteristic polynomial of its adjacency matrix. A covering map between graphs relates the spectrum of the base and the cover through divisibility of characteristic polynomials.  Using this and iso-spectral graphs we produce infinite sequences of graphs with nice properties. This talk is based on research carried out during the summer 2005  "Geometry and Physics on Graphs" REU at Canisius College.

Joseph McCollum,  SUNY Albany

Walks on the Dihedral Group

This talk will give a brief overview of the Upper Bound Lemma of Diaconis and Shahshahani, representation theory, and probability on finite groups.  We will focus on using these ideas to study various random walks on the dihedral group with 2n elements where n is odd.  In particular, for a random walk generated by one rotation and one flip it will take order n squared steps for the random walk to get close to uniformly distributed.

Brian Meagher, SUNY at Buffalo

Properties of zigzag products of graphs

We give a general categorical description of zigzag of graphs in analogy  to the algebraic description of semidirect products of groups. This allows us to extend the zigzag construction to infinite graphs. In the second part of the talk, we estimate certain spectral values of the generalized zigzag products.

Tricia Profic, Canisius College

Covers, Laplacians, and Kesten's Theorem.

Kesten's Theorem is a classical result that estimates the spectral radius of regular random walks on graphs.  It also characterizes the two extremes in the estimate: the lowest value is obtained by regular trees and the highest by amenable graphs.  We provide a proof of Kesten's theroem using combinatorial covers, topological covers, and normalized Laplacians.  In the process, we analyze the heat kernel of the weighted ray, the dynamical properties of amenable graphs and calculate the spectrum of the regular tree using combinatorial methods.  *This is joint work with Jack Wessel, Binghamton University, and it is part of the REU program at Canisius College under the direction of Dr. Terrence Bisson and Dr. Stratos Prassidis.

Matthias L Youngs, SUNY Geneseo

Generalizing Kirk-Livingston Type 1 Link Invariants

A link is the disjoint union of simple closed curves in space.  Link invariants are used to distinguish links. Kirk and Livingston introduced an invariant λ for 2-component links, which they called the enhanced linking number [Topology 36 (1997), 1333-1353]. They observed that lambda is a link invariant of ``type 1'' in an appropriate sense, and proved that every other type 1 invariant of 2-component links is of the form aλ+b for some constants a and b. Our research focused on attempting to generalize this result to 3-component links. The tools that we used included an algebraic and geometric homotopy classification of 3-component links with singularities and the coefficients of a power series expansion of the multivariable version of Conway's potential function, a well known polynomial link invariant.

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