# Geneseo Mathematics Colloquium Schedule

## Spring 2013

Wednesday, February 13  2:30 - 3:20p
Newton 203
Gillian Galle, University of New Hampshire

### The Trouble with Trigonometry

Students that enroll in algebra-based physics courses for life science may be less prepared mathematically than their counterparts in the engineering, physical science, or mathematics majors.  This means it can be especially difficult for them to develop conceptual understandings of equations that possess both physical and mathematical interpretations within the same context.  Based on such students’ answers to a particular question on simple harmonic motion equations, this study undertook to systematically probe the following questions: What is the range of students’ initial knowledge with respect to trigonometry? Is reviewing trigonometric concepts valuable and/or necessary? Can students see the trigonometric equations describing oscillations as conveying an idea, in addition to being a tool to get “the answer?” In this talk I focus on the efforts of my colleague and I to answer this last question through the design and timely implementation of a trigonometric intervention and motivational activity meant to help these students reason through the underlying connections between trigonometry and modeling simple harmonic motion.  In addition to discussing the research the intervention was based on, I will address the development of our motivational activity, our finding that students can learn to see trigonometric equations describing oscillations as conveying an idea, and what implications this may have for the way we address this topic in both high school and undergraduate physics courses.

Friday, February 15  3:30 - 4:20p
Newton 203
Valentina Postelnicu, Arizona State University

### The Functional Thinking in Mathematics Education: A Cultural Perspective

One of the most important ideas that influenced the mathematics education of the last century is the idea of educating functional thinking, particularly a kinematic-functional thinking. Bringing students up to functional thinking has proved to be a difficult task for mathematics educators. We examine the current state of mathematics education with respect to functional thinking by considering different curricular approaches to functions in the United States and other parts of the world. We closely look to one problem and the way it may appear in different cultural settings. We focus on issues related to the covariational approach to functions, the rise of digital technologies, and the need for symbolic representations

Thursday, February 21 4:00 - 4:50p
Newton 203
Amanda Beeson, University of Rochester

### Title: Did Escher know what an elliptic curve is?

We will give a naïve introduction to elliptic curves. Then we will discuss whether the 20th century Dutch artist M.C. Escher knew what an elliptic curve is. Along the way, we will discover many wonderful things about his piece called "Print Gallery". This talk will be enjoyable if you remember how to add and multiply, but some paper-folding skills never hurt. This talk is based on work of H. Lenstra.

Monday, February 25  4:00 - 4:50p
Newton 214
Carlos Castillo-Garsow, Kansas State University

### Chunky and smooth images of change

Students have well documented difficulties with graphs. In this talk, I discuss recent and current research that investigates connections between these difficulties and student difficulties in forming images of change, the impact that these student difficulties have on more advanced mathematical reasoning at the secondary and undergraduate level, the damage that developing these difficulties can do to the preparation of teachers, and the potential role of technology in developing solutions to these systemic and persistent problems.

Wednesday, February 27  3:30 - 4:20p
Newton 201
May Mei, University of California, Irvine

### Attack of the Fractals!

You may not know it, but you're surrounded by fractals!  They are all around you and even inside of you. In this talk, we will explore the prevalence of fractal structure in the natural world and in mathematics. Then we will construct the standard Cantor set and show you how you can construct your own fractals.

Friday, March 1  3:45 - 4:35p
Newton 203
Emma Norbrothen, North Carolina State University

### Number Systems Base p

Rational numbers can construct the real numbers by using the absolute value norm.  Under different norms, rationals can construct different types of numbers.  In particular, the p-norm evaluates how much a prime, p, is a factor in a given rational.  We will explore some consequences of the p-norm and what kind of numbers it creates from the rationals.

Friday, April 5  2:30 - 3:30p
Newton 204
Sue McMilllen, Buffalo State
President, Association of Mathematics Teachers of New York State (AMTNYS)

### Fibonacci Fun

Explore interesting properties of the Fibonacci sequence.  Look for patterns and make conjectures.  Learn about connections between matrices and the Fibonacci sequence. Bring your calculator.

If you would like to know more about graduate studies at Buffalo State or about AMTNYS, please stay  around after the talk to converse with Dr. McMillen.

Thursday, April 25  2:30 - 3:30p
Newton 203
Arunima Ray, Rice University, SUNY Geneseo class of 2009

### A friendly introduction to knots in three and four dimensions

If you've ever worn sneakers or a necktie, or ever been a boy scout, you know a lot about knots. Knot theory is also an exciting (and young) field of mathematics. We will start from scratch to define and discuss some basic concepts about knots in three dimensions, such as how to quantify the 'knottedness' of a knot and how to tell if two knots which look different are secretly the same. We will also see how a four dimensional equivalence relation reveals a simple and elegant algebraic structure within the set of knots.

This talk will be very visual with lots of pictures and will be accessible to students at all levels.

## For a taste of the type of talks to expect, here are quite a few previous talks.

Xiao Xiao, Utica College

### The Frobenius Problem

Let a, b, . . . , c be positive integers greater than 1 with no common factors. The Frobenius problem studies the largest non-negative integer g(a, b, . . . , c) that cannot be expressed as a non-negative linear combination of a, b, . . . , c. In this talk, I will review some of the old and not so old results of solving the Frobenius problem for g(a, b) and g(a, b, c). If time permits, I will also describe how a closed formula of g(a, b, c) in a special case can be used to solve problems in the classification of F-crystals over algebraically closed field. This talk is accessible to undergraduates.

Katherine Socha

### Sea battles, Benjamin Franklin's oil lamp, and jellybellies

"During our passage to Madeira, the weather being warm, and the cabbin windows constantly open for the benefit of the air, the candles at night flared and run very much, which was an inconvenience.  At Madeira we got oil to burn, and with a common glass tumbler or beaker, slung in wire, and suspended to the ceiling of the cabbin, and a little wire hoop for the wick, furnish'd with corks to float on the oil, I made an Italian lamp, that gave us very good light...." (Benjamin Franklin, December 1, 1762 letter to John Pringle)

Observations of real phenomena have led to mathematical modeling of surface water waves, interfacial waves, and Lagrangian coherent structures among other examples.  This expository talk will provide a quick tour of the (mostly advanced undergraduate level) mathematics needed to describe idealized versions of the rings formed by striking a surface of water with a large object (like a bomb), the oil-water waves observed by Founding Father Benjamin Franklin on his voyage to Madeira, and the motion of nutrient laden water being swept into the underbelly of a swimming jellyfish.

Ron Taylor, Berry College

### The Difference Between a Small Infinity and a Big Zero

Can two people have a different answer to the same question and both be right?  Is there room for perspective in mathematics?  Most often we find that any given mathematical question will have a single answer, though there are usually many different methods that can be used to find that answer.  In this talk we will discuss the Cantor set, a remarkable object that seems to leave room for perspective to play a part in mathematics.  Given time we will discuss generalized Cantor sets, a class of sets with interesting properties of size.

Emilie Weisner, Ithaca College

### The mathematics of bead crochet

Creations in the fiber arts are often based in pattern and symmetry.  Because of this, the fiber arts and mathematics are a natural pair.  In this talk, I'll talk about some of the mathematics related to bead crochet.  In particular, I'll discuss the work of Susan Goldstine and Ellie Baker, who use wallpaper groups to understand symmetries in bead crochet patterns.  I'll also talk about work on additional mathematical aspects of bead crochet, being carried out by IC juniors Rachel Dell'Orto, Sam Reed, and Katie Sheena.

Ryan Gantner, St. John Fisher College

### The Stochastic Voter Model

In this talk, the stochastic voter model will be introduced and we'll see how it works.  After deriving some results about its long-term behavior, we'll turn to some examples of how it can be applied.  Some examples include the spread of diseases, the evolution of zombie attacks, ... and elections!  The talk will conclude with a simulation to predict the outcome of the 2016 presidential election.

Prerequisites:  The major mathematical proof in this talk should be accessible to anyone who has had calculus 2.  All other aspects of the talk involve only intuitive aspects of probability, and should be accessible to all mathematically inclined students.

Kalyani Madhu, SUNY Brockport

### Periodic Points in Finite Fields: A Question in Arithmetic Dynamics

In Arithmetic Dynamics we study dynamical systems from a number theoretic point of view. This talk will introduce basic ideas in this relatively new branch of Mathematics that will enable us to understand an interesting question concerning the dynamics of ï¬nite ï¬elds.

Laurel Miller-Sims, Hobart & William Smith Colleges

### When Not Knowing is Enough

It's impossible to do mathematics without running into things we don't know. We will look at logic puzzles where the key to the solution lies in a lack of knowledge. It's sometimes surprising where not knowing will get you!

Rebekah Yates, Houghton College

### Numerical Range: Counting Necklaces and Looking for Symmetry

The numerical range of a matrix A is the set of all complex numbers resulting from mapping a vector in the unit sphere to the dot product of that vector with A. We will consider the question of when, under certain conditions, this set is symmetric about the origin. Along the way, we'll see several properties of the numerical range, encounter many familiar matrix characteristics, and find ourselves counting how many different necklaces can be made from a fixed number of beads of two different colors.

Prerequisite:  Math 233 (Elementary Linear Algebra).

Elizabeth Wilcox, Colgate University

### The Enormous Theorem

Have you ever thought about the work that goes into proving mathematical theorems? Take, for instance, the Fundamental Theorem of Calculus - who proved it? How did he (or she?) even come up with the idea to prove such a thing? And how long did that mathematician spend thinking before a proof materialized?

During the talk we'll learn about a theorem that was nearly eighty years in the making, taking mathematicians from 4 continents nearly 15,000 pages to prove - in a time before FaceTime, Skype, or even cell phones! I will introduce you to mathematical groups and show you that, even though you've been working with groups since grade school, these structures are so complex that after hundreds of years we are still learning new things about groups every day.

Jim  Matthews, Siena College

### The Twenty Locker Problem

In many games, an important goal is to maximize your chances for success. This is true for the locker problem where teams of players attempt to maximize the chances of every team member being successful. We will describe the locker problem, invite participants to suggest solution strategies (possibly offering one or two of our own), and then using some basic probability (along with a bit of calculus and computer programming), determine the chances of success for each.

This is a version of a search problem for succinct data structures that appeared in a paper written by Peter Bro Miltersen and Anna GÃ¡l which won the best paper award at the 30th International Colloquium on Automata, Languages and Programming in 2003.

The main ideas for this exposition are accessible to the general public and the solutions that will be discussed can be carried out by average middle school level students. The applications of the college level material to this problem are delightful and probably surprising.

Olympia Nicodemi, SUNY Geneseo

### How many real numbers are there?

In this talk we will find out if infinity comes in sizes and we will find out how many really real numbers there are.  We hope that it will be accessible and interesting to all students, majors or not.

Douglas Haessig, University of Rochester

### An introduction to number theory, from a p-adic point-of-view

Many number theorists are interested in the solution sets of multivariable polynomial equations. For example, Fermat's last theorem asks about integer solutions of the polynomial equation x^n + y^n = z^n. In this talk, we will see how the solution set of a conic (quadratic equation) depends on the domain of the variables. This will lead us into defining "p-adic numbers." After discussing a few theorems to see how p-adic analysis works, and how p-adic geometry looks, we will employ the p-adics to count the number of (finite field) solutions to a famous cubic equation using differential equations.

Jane Cushman, Buffalo State University

### Which cup material holds hot water the hottest, the longest?

Using a temperature probe, TI-84, boiling water and various cup material (Styrofoam, paper and ceramic), data will be collected.  The data will be shared with the participants and everyone will analyze the data to determine which cup material held the hot water the hottest and longest.  Which do you think will?

This talk is aimed at pre-service teachers and mathematics majors; there are no pre-requisites.

Jonathan Hoyle, Eastman Kodak

### A Mathematician in the Real World

"How much Mathematics is really used out there in the real world?"  The answer to that question of course varies, depending on your definition of "real world".  Obviously, a different level of Mathematics would be used by a statistician than would be by the owner of a pizza parlor.  My "real world" is one as a software developer, and I will talk about some of the mathematics with which I have been involved.  These include work at DuPont, Kodak, and writing forensic DNA software used to identify victims of the World Trade Center attacks on 9/11.

Ryan Gantner, St. John Fisher College

### It's time to play Hackenbush!

Let's play a game.  The only things we?ll need are several red and blue sticks.  Oh, and some interesting mathematical theories.  The game is called "Hackenbush", and it is fairly simple to play.  In this presentation, we'll develop some theory of partisan combinatorial games to help us better understand what is the best way to play the game.  It turns out that the game positions can be given number values in a meaningful way.  Then, we can ask what happens when randomness is added to the game, which is an ongoing area of research.

Bonnie Jacob, Rochester Institute of Technology

### A discrete challenge based on optical imaging

Diffuse optical imaging is a method of imaging that consists of reconstructing the optical parameters inside a medium using data gathered at the boundary. Optical imaging has wide appeal in medical applications in particular because the light that is used does not harm the patient. Despite the appeal of optical imaging, however, difficulty in obtaining high quality images has prevented it from being widely used in clinics. One way to get a better reconstruction is to choose the source cleverly.

Motivated by this problem, we consider a discrete version of the typical optical tomography model: a network in which each node is designated as either "scattering" or "absorbing." We consider how the structure of the network, the location of the different classes of nodes in the network, and the initial location of the signal --- the "source"--- all influence our ability to recover each node's classification.

Helpful co-requisites: This talk will relate well to topics from an elementary linear algebra course.

Matt Koetz, Nazareth College

### Coding Theory: Yeah, We've Got That

Coding theory is the study of transmitting information efficiently across noisy channels.  It aims to reduce the number of transmission errors, detect and correct errors, and do these things as quickly and cheaply as possible.  In the search for better codes, we use many branches of mathematics, including linear algebra, combinatorics, graph theory, geometry, probability, and number theory.  We will explore the ways in which coding theory uses each of these fields, from its basic definitions to its most beautiful results.

Quincy Loney, Binghamton University

### The Octonions: An Alternative (Algebra) To The Reals.

In this talk we will discuss some of the history and the properties of the octonions, the 8-dimensional normed division algebra, discovered by John T. Graves in 1843. We will begin with the real number system and use the Cayley-Dickson process to construct this exciting alternative algebra.

Candace Schenk, Binghamton University

### A brief overview of the conjugacy problem in Thompson's group V.

This is an introductory talk where I will define all terms used! I will explain Thompson's group V, what the conjugacy problem is and the solution.  There will be dynamics!  There will be functions!  There will be trees!

Brandt Kronholm, St. Mary's College of Maryland

### Integer Partitions: 1 + 1 = 2 and the Subtleties Therein.

The partitions of a number are the ways of writing that number as a sum of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1 and we write p(4) = 5. Whatâ€™s the formula? Itâ€™s less than 100 years old and you wouldnâ€™t believe it even if you saw it.

Around the same time that the formula for p(n) was formulated, it was observed that p(n) had unexpected divisibility properties:
p(5n+4)â‰¡0 (mod5)
p(7n+5)â‰¡0 (mod7)
p(11n + 6) â‰¡ 0 (mod 11)
Fifty years later one more divisibility property modulo 17 was discovered. Are there any others?

The restricted partition function p(n,m) enumerates the number of partitions of a non negative integer n into exactly m parts. For example, the two partitions of 4 into exactly 2 parts are 3 + 1 and 2+2 and we write p(4,2) = 2. p(n, m) is like a little brother function to the unrestricted partition function p(n) in that p(n) = p(n,1) + p(n,2) + ... + p(n,n).
This talk will introduce the theory of partitions from the ground up and segue into a discussion of recent results on divisibility properties for p(n, m).  Time permitting, we will consider future research regarding p(n, m) and formulate some formulas!

Jobby Jacob, Rochester Institute of Technology

### Towers of Hanoi and Rankings of Graphs

The Towers of Hanoi is a famous puzzle that has been studied for centuries. In the Towers of Hanoi, the idea is to move a stack of disks from one pole to another without placing a larger disk on top of a smaller disk. The goal is to do this in the fewest number of moves possible. It is known that this puzzle involving k disks can be solved in 2k âˆ’1 steps. The puzzleâ€™s optimal solution relates to some interesting mathematical problems.

An optimal solution to this puzzle is related to rankings of graphs, which are generalized vertex colorings. In this talk we will look at the relationship between rankings of graphs and an optimal solution of the Towers of Hanoi problem, as well as some properties of rankings.

Jeff Johannes, SUNY Geneseo

### Game:  SET - Math

SET is a popular game for many student math groups (including PRISM).  In this talk, we will see many good reasons for this.  From a simple game enjoyed (and excelled) by children, we will find geometry (the fourth dimension), topology (tori), algebra, probability, and combinatorics.  Aside from all of this, there will be times to play along, and takeaway prizes.

Bogdan Petrenko, SUNY Brockport

### Probabilistic Algebra and Number Theory

At the beginning of my talk I plan to give an intuitive explanation of how to find the probability that 2 integers are relatively prime.  This should be interesting and quite accessible to any student who has taken Calculus 2. Next I plan discuss some related explicit examples from my recent joint work with R. Kravchenko and M.Mazur posted at http://arxiv.org/abs/1001.2873.  For this part of my talk I will assume that the audience knows matrix and modular arithmetic.

Aaron B. Luttman, Clarkson University

### The Mathematics of Your Lifetime: Mathematical Advances of the Last and Next 20 Years

Throughout our mathematical training from elementary school through the first few years of college, we're taught mathematics as facts and ideas that were invented or discovered by people hundreds or even thousands of years ago.   This makes it easy for us to think of mathematics as being complete, as if there's nothing left to uncover or create. The reality is that we live in a wonderfully exciting time of mathematical innovation and development, and new breakthroughs are occurring almost daily. In this presentation we will look back at some of the most exciting mathematical developments of the last 20 years - from the solutions of centuries-old problems in pure mathematics like Fermat's Last Theorem and the Poincare Conjecture to mathematical transformations of applications such as medical imaging and quantum computing - and we'll look forward to the possible advances that today's students will have the opportunity to make in the next 20 years.

Level/Background Required: This talk is aimed at any college students with an interest in mathematics. A first course in calculus will be helpful, but no prior knowledge of the problems discussed will be assumed.

Walker White, Cornell University

### Scaling Games to Epic Proportions

An important aspect of computer games is the artificial intelligence (AI) of non-player characters. Currently in games, developers or players can create complex, dynamic behavior for a very small number of characters.  However, neither the game engines nor the style of AI programming enables intelligent behavior that scales to a very large number of non-player characters; the languages that define character logic are typically very expensive to process.

In this talk, I will show how solve this problem by modeling game AI as relational queries.  Instead of processing characters independently, we can combine all of their behaviors into a single logical query which can then be optimized.   The talk will include an overview of the formal framework for specifying character behavior, as well as highlight some of the mathematics behind the ways that we optimize this behavior.

Gary Towsley, SUNY Geneseo

### What does Dante have to do with Mathematics?

This talk is an introduction to the rich treasury of Mathematics to be found in Dante's La Commedia.

Tom Bleier, Syracuse University

### Introduction to the Projective Plane

Come see an introduction to one of the most beautiful and fascinating areas of mathematics, Algebraic Geometry!  This talk will be accessible to students of all backgrounds.

Paul Seeburger, Monroe Community College

### Making Multivariable Calculus Come Alive using Dynamic Visualization Tools

A tour of an NSF-funded project that seeks to develop geometric intuition in students of multivariable calculus.  This online exploration environment allows students to create and freely rotate graphs of functions of two variables, contour plots, parametric surfaces, vectors, space curves generated by vector-valued functions, regions of integration, vector fields, etc.  This tool is designed to improve student understanding of the geometric nature of many of the concepts from multivariable calculus.  A series of assessment/exploration activities has also been designed to help students "play" with the 3D concepts themselves, and to assess improvements in geometric understanding gained from these activities.  3D glasses will be provided.

Phong Le, University of California at Irvine

### Using Modulo Arithmetic to Find New Solutions to Old Polynomials

In this talk we will rebuild the integers using arithmetic modulo prime numbers.  This will allow us to find solutions to polynomial equations such as x2+1=0 in an entirely new setting: the finite field.  This new algebraic construction will raise many interesting questions with some surprising answers.

Kristin Comenga, Houghton College

### Polytopes: Generalizing Polyhedra to Higher Dimensions

Most people remember working with polyhedra in elementary and high school: cubes, prisms, tetrahedra, pyramids, etc.  Euler's relation states that if V is the number of vertices, E the number of edges and F the number of faces, V + F = E  + 2.  In this talk we will survey variations on this result for generalizations of polyhedra called polytopes.  The polyhedra most of us have experience with are three-dimensional.  Polytopes can be any non-negative dimension, with the simplest example beyond polygons and polyhedra being the four-dimensional hypercube.  How can Euler's relation generalize to polytopes in any dimension?  How can we generalize this to look at angles of polytopes instead of the number of faces?  We will look at a number of examples of polytopes as we explore some of the answers mathematicians have found to these questions.  We will end with a brief glimpse of open questions about angles in polytopes.  No specific math background will be assumed, but curiosity is expected!

Emilie Wiesner, Ithaca College

### Clever Counting Strategies in Sudoku

Have you ever played Sudoku?  Have you wondered what makes one puzzle harder than another? what the minimum number of clues could be? how many puzzles there are?  So have other mathematicians!

I'll talk about these questions and, in particular, how mathematicians have tried to count the number of puzzles.  This turns out to be  a tough count to make, and mathematicians have used clever counting strategies from Combinatorics and Abstract Algebra to do it.

Olympia Nicocdemi, SUNY Geneseo

### Wavelets and Elementary Linear Algebra

Wavelets are in use everywhere, from deep inside a little digital camera to big telescopes that help us find out what's out there. The name wavelets sounds so user friendly. And they are, but the theory behind them is not always accessible to undergraduates in their early studies.  In this talk, we will make that theory a little friendlier by linking what we learn in elementary linear algebra to the theory and practice of wavelets.
Mark Steinberger, University at Albany

### Topological Equivalence of Matrices

Two n x n real matrices A and B are said to be topologically similar if there is a continuous, 1-1, onto function f: Rn ->  Rn whose inverse is continuous, such that  f(Ax)=Bf(x) for all x in Rn.

If f were the function induced by a matrix P, i.e., if f(x)=Px for all x in Rn, then P must  be invertible and PAP-1=B, so A and B are similar in the usual sense. Similarity of matrices can be thought of as saying that the transformations induced by A and B differ by a linear change of variables. So topological similarity means that A and B differ by a topological change of variables.

De Rham, in 1935, conjectured that topological similarity implies linear similarity.   He was wrong. We discuss the geometry underlying the existence of a topological similarity between linearly nonsimilar matrices and connect it to the algebra used in modern topology.

Patrick Rault, SUNY Geneseo

### Arithmetic Geometry

How many rational points are on the unit circle?  That is, how many points on the circle x2+y2=1 have coordinates which are rational numbers?  In answering this question, we will find an algorithm that gives all Pythagorean Triples!

More generally, arithmetic geometry is the study of integral and rational points on curves.  In this talk, we will generalize each of the following concepts:  fractions, discriminants of polynomials, and the aforementioned Pythagorean Triple method.

We will end with several related unsolved problems, and an upcoming research and travel funding opportunity to participate in a 2009-2010 research course.  This small 3-credit course would be ideal for those who have taken Math 319 or 330 and are planning to attend graduate school.

Corequisites:  Any students who have taken or are taking Elementary Linear Algebra (Math 233) should enjoy this talk.

Bronlyn Wassink, Utica College (SUNY Geneseo Alumna)

### Functions, Rubber Bands, and Trees

There is a group of special functions, Thompson's Group F, that  can be represented in many different ways.  After showing exactly which  functions are in Thompson's Group, we will explore both the rubber band  model and the tree pair model for this group.  This exploration will  involve defining these models, showing how to quickly and easily change  from one model to the next, and how to compose two functions in this group  using only trees.  No background knowledge is required to understand this talk!!

Michael Starbird, University of  Texas

### To Infinity and Beyond

Infinity is big. For thousands of years, people also thought it was incomprehensible--an idea so vast that understanding it was beyond the scope of people's finite minds. But a child's method of sharing "'one for me, one for you", an Infinite Inn, a barrel containing infinitely many Ping-Pong balls, and a game called Dodge Ball combine to take us to infinity. And beyond.

Vijay Sookdeo, University of Rochester

### The Exotic World of p-Adic Numbers

Under the usual notion of an absolute value, we may "complete" the set of rational numbers (fractions) to obtain all the real numbers. In 1902, Kurt Hensel found different ways of taking absolute values on the rational numbers, and thereby discovered the "p-adic numbers". This talk will introduce some of the bizarre and wonderful properties in this strange new world. Among them are: all triangles are isosceles, every point in a disk is the center, and two disks can only (non-trivially) intersect in one way. We will also sketch the proof of an important theorem concerning recurrence relations (the Skolem-Mahler-Lech Theorem) which exploits the exotic features of p-adic numbers.

Keary Howard and friends, SUNY Fredonia

### Teachers' Masters Capstone Projects in Secondary and College Mathematics

Does cash money motivate in a college math classroom?  Whatâ€™s the â€˜rightâ€™ amount of high school geometry homework?  Can your iPod improve math performance?  Are ninth graders really faster than a calculator?  Join us as we attempt to explore these questions in mathematics education from a quantitative perspective.

Scott Russell, SUNY Geneseo Computer Science

### Beyond Confidentiality: Cryptographic Applications of Homomorphic Encryption Schemes

Most people are aware that encryption schemes such as RSA and DES are useful tools for protecting the privacy of confidential information, e.g. your credit card or bank account number.  Homomorphic encryption schemes provide additional algebraic properties beyond those necessary for confidentiality.  Consequently, these schemes are oft-used building blocks in solutions to a variety of other cryptographic problems.  We will briefly explain the exclusive-or-homomorphic Golwasser-Micali encryption scheme whose security is derived from the Quadratic Residuosity Assumption.  Then we'll demonstrate how to use Golwasser-Micali encryption to construct solutions to a couple of cryptographic problems.

Chris Leary, SUNY Geneseo

### Fractals, Average Distance, and the Cantor Set

After briefly introducing fractals and some of their properties, we introduce the idea of the average distance between points of a set.  We will construct the famous Cantor Set and use it as an example of a fractal and finish by computing the average distance between the points of the Cantor Set.  The material of this talk was developed over the summer with Dennis Ruppe, a math major here at Geneseo.

Prerequisites:  Calc I would be nice, but not essential.  A trusting personality is both nice and essential.

Daniel Birmajer, Nazareth College

### The arithmetic of formal power series over the integers

We study the arithmetic (units, irreducible elements, unique factorization, etc.) in the ring of formal power series (in one variable) with integer coefficients, and discuss some irreducibility criteria. We will examine in some detail whether or not a quadratic polynomial is irreducible as a power series.

Bob Rogers, SUNY Fredonia

### New Tricks for Old Curves

Even though conic sections have been studied for at least 2400 years, they still prove to be useful in our modern world.  This talk will provide examples of current uses in medicine, global positioning systems, and optics and explore potential uses in quantum computing.

Claudiu Mihai, Daemen College

### Generalization of Some Optimization Problems

In this talk we present several optimization problems from Calculus books that can be more easily solved by generalizing them.  For example, maximizing the area inside a triangle, or maximizing the area inside a parabola region. Similar results will be shown for problems involving volumes. Some surprising results will be presented.

Darwyn Cook, Alfred University

### Geometry and Art

We will look at the relationship between geometry and perspective art. In particular we will show why vanishing points exist and how to figure out where they occur. We will use that information to create some simple pictures using Microsoft Excel and we will also develop a method for viewing art pieces.

Patrick Rault, University of Wisconsin

### Mathematical game theory

Mathematical strategies and solutions of various games will be discussed.  Recent developments in the game theory of Checkers and Rubik's Cube will be presented within a historical context.

Sharon McCathern, University of Illinois at Chicago

### The Triangle Game, Symmetry, and Dihedral Groups

Using a simple arithmetic game as an introduction, we will discuss the symmetries of an equilateral triangle. I will introduce the dihedral groups, which consist of the symmetries of regular polygons, and briefly mention some of their nice properties.

Palalanivel Manoharan, Penn State University

### The Angel of Algebra and the Devil of Geometry - or is it the other way around?

We will discuss the history of cordial (or uneasy?) relationship between Algebra and Geometry, two ancient pillars of mathematics. We will look into some specific incidents in mathematical history when unexpected bridge developed between Algebra and Geometry to create duality among them.

Patrick van Fleet, University of St. Thomas

### Basic Image Processing with Wavelets

On my desk sits a digital image of my children. The camera my wife used to take the picture allows the user to save the image to disk in either raw format or as a JPEG file. We saved the image using both options. The raw format produced a file whose size is 861KB while the JPEG version of the image was stored on disk using 46KB. The difference between the two images are inconsequential.  So how did the JPEG format produce a file that so accurately represented the original image but required substantially less disk space? This is a question that is paramount in the minds of anyone who wants to make effective use or enjoy fast transfer of digital images in today's world.

In this talk, we will give a very elementary introduction to a tool that finds itself at the center of many image processing applications. We will introduce the Discrete Haar Wavelet Transform (HWT) and discuss how it can be used to process digital images. While the HWT is not the best wavelet transform for processing images (that is the subject of Friday's talk!), it serves as a perfect tool for introducing the use of wavelets in applications.  During the talk, we will take some digital pictures (audience participation is thus required!) and use the HWT to compress the images. We will also show how to use the HWT to search for edges in our digital images.

Patrick van Fleet, University of St. Thomas

### Wavelets and Lossless JPEG Compression

The JPEG format, developed in 1992 by the Joint Photographic Experts Group, is used by over 80% of all images that appear on the internet. Despite the popularity of the image format and the impressive compression ratios it attains, there is room for improvement.  In particular, JPEG is capable of only compressing images in a lossy manner. That is, the size of the compressed file is significantly smaller than the raw format, but the savings was gained by discarding portions (typically deemed insignificant) of the original image. Thus it is impossible to recover the original image from a compressed JPEG image. In 1997, JPEG introduced a new format called JPEG2000. This format corrects several flaws in the original JPEG format and also provides many enhancements.  In particular, JPEG2000 allows the user to compress a digital image in a lossless manner. We get the best of both worlds - the size needed to represent the image is reduced and the compressed version can be used to recover the original image!

At the heart of the JPEG2000 compression standard are two wavelet transformations. One transformation is used to perform lossy compression while the other allows users to compress images in a lossless manner. In this talk we will consider the wavelet transformation used by JPEG2000 to perform lossless image compression.  Incredibly, the mathematics behind this transformation is quite straightforward - the perplexing part of the process is realizing that the algorithm \undoes" the rounding operator to exactly recover the original image!

Ding Feng, University of Virginia

### General Concepts of Point Estimation

One very important application of statistics is in obtaining point estimates of population parameters such as a population mean, population variance, and a population proportion.  Given a parameter of interest, the objective of point estimation is to determine the plausible approximate value of the parameter on the basis of a sample statistic. In this talk, we first introduce the general concepts of a point estimate and a point estimator for a population parameter.  Since we may have several different choices for the point estimator of a particular parameter, to decide which point estimator is the â€œbestâ€ one, we need to examine their statistical properties and develop criteria for comparing estimators. Two extremely important criteria, the principle of unbiased estimation and the principle of minimum variance unbiased estimation (MVUE), will also be introduced.

Lingji Kong, Union College (Kentucky)

### Beta-Power Distribution and Applications

A class of generalized power distribution, namely Beta-power distribution, is proposed. Properties of this distribution including limits, modes and moments are presented. Graphs of the density functions are presented to examine shapes of the distribution for various combinations of parameters. The beta-power distribution is shown to be four kinds of shapes: increased, decreased, bathtub or reverse bathtub. Reliability and hazard functions are derived; in the end parameter estimations and the test for Beta-power distribution are also discussed.

Shubiao Li, Central Michigan University

### Random Walk and the Ruin Problems

The basic conception of random walk process is introduced from several real life examples.  A classic ruin problem is used to illustrate modeling techniques for a random walk process. Some properties related to the problem such as expected duration and expected gain are discussed; the techniques of obtaining solutions of difference equations are also addressed.

Amy Stornello, Rochester Institute of Technology

### Obtain your Master's in Education at RIT/NTID

Why RIT/NTID?  Well, some benefits we offer are: small class size, personal instruction with faculty and dual certification in grades 7-12 (in your topic area) and Teacher of the Deaf.  If you have a bachelor's in Math or Science, we are even offering \$10,000 scholarships to encourage more Math and Science teachers in the teaching profession.  If you've ever wanted to work with deaf/hard of hearing students, this is the perfect opportunity to do so!  Find out how our two-year master's program works, what classes we offer, internship opportunities and more information about this fantastic scholarship!

Matthew Rashford, SUNY Geneseo

### Exponential Stability of Dynamic Equations on Time Scales

A time scale is an arbitrary nonempty closed subset of the real numbers.  Two of the most common examples of calculus on time scales include differential calculus and difference calculus.  This talk will look at some of the background regarding time scales, conditions for exponential stability, and then will show examples of time scales, including an application on population dynamics.  This talk is strongly recommended for anyone who has taken or is taking Differential Equations.

Patti Fraser-Lock, St. Lawrence University

### Marijuana Use, Goldfish, and Knee Injuries

Effective statistical analysis of data requires, first, that we are able to obtain valid data from a sample.  We will discuss and illustrate some interesting new sampling methods and give examples of some recent thought-provoking results obtained using statistical experiments.

Gary Towsley, SUNY Geneseo

### What is a Ph.D. dissertation in Mathematics?  An Example: Conformal Deformation of Meromorphic Functions

Have you ever wondered what it would take to get a Ph.D. in Mathematics?  This sequence of talks will share with you personal experiences.  Although they will present sophisticated mathematics, no background is assumed beyond calculus.  In this example, we explore the question: when are two continuously homotopic functions from a compact surface to the two sphere joined by a homotopy that ranges through the meromorphic functions? What does such a question mean and what kind of an answer can one get?

Jim Conklin, Ithaca College

### Sudo Latin Squares

Sudoku puzzles have a rich pre-history in recreational and applied mathematics as well as presenting some interesting mathematical issues of their own. Sudoku grids are special cases of Latin Squares, a source of mathematical puzzles since at least the 1620's.  This talk will look at some of the mathematical prehistory of Sudoku-like puzzles and the applied mathematics that grew out of them, and then look at some of the mathematical issues related to the solution and creation of the puzzles.

Christopher Andrews, University at Buffalo

### An Introduction to Opportunities in Biostatistics

Biostatistics, the science of statistics applied to the analysis of biological or medical data, has a large and growing demand for qualified researchers.  In this talk I will describe, through examples, what biostatisticians do.  This includes methodological research, collaborative research, consulting, statistical programming, bioinformatics, and epidemiological research.  Finally I will discuss how you can prepare for a career in this exciting, rewarding field.

Michael Fisher, CSU Fresno

### Iterated Function Systems (How to Grow Your Own Fractal)

In this talk I will introduce the notion of an iterated function system (IFS) and take a look at common types of fractals which are easily described by an IFS.  Specific examples include the Sierpinski gasket, the Cantor set, and Barnsley's fern.  If time permits, I will also talk about graph-directed sets (a generalization of a self-similar set).

Alison Setyadi, Dartmouth College

### Can you hear me now?

Suppose you work for a cell phone company, and your job is to determine where to place cell phone towers in a certain area.  Given that there is a limit to how many calls each tower can handle at once and that each tower has only a finite range, how do you determine where to place the towers so that the company's customers stay happy and the company stays within its operating budget? Once you decide where to place the towers, is there a way to increase the area of the company's cell phone service without having to rearrange the existing towers? By using graphs to model the locations of the towers, we consider ways to answer both of these questions.

Aaron Heap, University of Rochester

### The Fascinating World of Knots

We will discuss some of the basic ideas in knot theory and its history. We will see how knot theory is used in some other sciences besides mathematics. We will also discuss a few examples of knot invariants and how knot theory may be used as a tool in low-dimensional topology.

Sharon Garthwaite, University of Wisconsin

### The Sum of the Parts is â€¦

The sum of the parts is... more interesting that the whole!  In this talk we'll see how a seemingly simple idea -- expressing a number as the sum of smaller numbers -- leads to really interesting patterns.  We'll then discuss various methods for proving our observations, seeing that this simple idea is accessible at many different levels.

David Perkins, Houghton College

### An immortal monkey may have already given this talk

No one can be sure who first thought of sitting a monkey at a typewriter, or who would have the patience to sift through the monkey's output looking for meaningful text. Two things are for sure, however: (1) references to immortal monkeys abound in novels, plays, short stories, and television; and (2) the Borel-Cantelli Lemma implies that if you're immortal, you can be a monkey and still get published. In this talk, we'll investigate both these two items that are for sure, and some that are not.

Pedro Teixeira, Union College

### Googling with Math

The success of web search engine Google can be attributed to a system devised to rank the importance of websites,  where the importance of a website is related to the importance of the websites that link to it.  In this talk I'll discuss the system used by Google and the mathematics behind it, and in particular how it relates to topics one typically learns in undergraduate courses. We'll see how Google's ranking system  leads to what has been called "the world's largest matrix calculation," and examine the techniques used to handle such a monstrous computation.

Joanna Masingila, Syracuse University

### Teachersâ€™ Evolving Practices in Supporting Studentsâ€™ Mathematics and Literacy Development

Sociocultural research on mathematics and literacy frames this interdisciplinary investigation of the evolving practices of secondary mathematics teachers as they seek to understand and support their studentsâ€™ mathematics and literacy development.  Teachersâ€™ evolving practices included (a) their use of the â€œProblem of the Dayâ€ to engage students in thinking and communicating mathematically, (b) their development of â€œtemplatesâ€ as scaffolding tools for mediating the literacy demands of the textbook, and (c) their choice to explore student engagement and mathematical communication in connection with their classroom practices.

Ryan Grover & Matthias Youngs, SUNY Geneseo

### Infinite Levels of Infinity

Do you accept the idea of infinity?  If so, in what ways?  Together we will explore the concepts of infinity to the mathematician.  For instance, if there is somethign of infinite size, is there something else of even more infinite size?  If so what does it mean to be more infinite?  What does it mean to be infinite?  How big is infinity?

Joel Foisy, SUNY Potsdam

### Knots and Links in Spatially Embedded Graphs:  Tangled-Up Mathematics.

This talk will be about graphs that have knotted and/or linked cycles in every spatial embedding.  Informally, a graph is a set of vertices and a set of edges.  A graph is defined by the number of vertices is has, and by which vertices are connected by edges.  A particular way to place a graph in space is called an embedding of a graph.  A cycle in a graph is a sequence of distinct edges in the graph such that any two consecutive edges share exactly one vertex, and the last edge and the first edge share exactly one vertex.  We will discuss what is known about graphs that have a pair of linked cycles in every embedding.  (Intuitively speaking, cycles are linked if they can't be pulled apart, like two looped pieces of string).  We will also discuss what is known about graphs that have two disjoint pairs of linked cycles in every embedding, as well as graphs that have a knotted cycle in every spatial embedding.

Katia Noyes, University of Rochester School of Medicine

### Certainty Uncertain:  performing and interpreting multivariate cost-effectiveness sensitivity analysis

A healthcare system comprises complex relationships across many levels of organization focused on providing healthcare services to individuals and populations.  Health Services Research is the multi-disciplinary field of inquiry that combines approaches of health economics, mathematics, epidemiology, biostatistics, anthropology and other disciplines to study healthcare system and examine the use, costs, quality, accessibility, delivery, organization, financing and outcomes of health care service.

Cost-effectiveness research is one of many areas that constitute health services research. Cost-effectiveness analysis is based on the incremental cost-effectiveness ratio (ICER), i.e., the ratio of difference in costs to the difference in health effects of two competing interventions. By its nature, cost-effectiveness research is applied: the end result of a cost-effectiveness evaluation should be a black-and-white decision whether to fund or not to fund. In reality, two main problems make this happy end problematic. One is the lack of an analytical solution for variance of a ratio which makes formal hypothesis testing (e.g., ICER < l) impossible. The other problem is that nobody really knows what the threshold value l should be. In her presentation, Dr. Noyes takes standard cost-effectiveness methods a little further and describes approaches to make sense out of analytically uncertain cost-effectiveness results.

This presentation is a part the Information and Student Recruitment Efforts of the Division of Health Services Research at the University of Rochester School of Medicine. For more information please contact Dr. Katia Noyes at 585-275-8467 or katia_noyes@urmc.rochester.edu or visit http://www.urmc.rochester.edu/cpm/education/phd_hsr/index.html.

Daniel Birmajer, Nazareth College

### A Gentle Introduction to the Polynomial Identities of Matrices

Abstract:  Finding unexpected relations between apparently unrelated quantities is one of those things that make us, mathematicians, love our profession.  We call these mathematical relations identities.  Some identities are well for their beauty, simplicity and usefulness:
• 1 + 2 + . . .  + n = n(n + 1)/2 for all natural numbers n;
• sin2(x) + cos2(x) = 1 for all real numbers x;
• xy - yx = 0 for all complex numbers x and y.
Other identities are not so popular and they require a lot of work to be understood.  Of course, many identities still wait to be discovered by new generations of mathematicians.  In this talk we will discuss some of the relations that have been found among matrices, and the many questions that are open to explore.

Sergio Fratarcangeli, McMaster University

### Model Theory and Real Geometry

Abstract:  Model theory was born in the effort to shore up the foundations of mathematics.  Gradually, the field shifted away from its syntactic origins, toward a more geometric approach.  Consequently, model theory has found applications in diverse branches of mathematics.  The focus of this talk will be on the interaction between model theory and (generalizations of) real algebraic geometry.  We will see how the use of so-called o-minimal structures can simplify some otherwise very messy mathematics.

Markus Reitenbach, Syracuse University

### Configurations of Subspaces of Euclidean Space

Abstract: The (proper) subspaces of 3-dimensional Euclidean space are the lines and planes through the origin, and the origin itself. I will explain what is meant by a configuration of subspaces, and will give a classification of configurations, including the ones in higher dimensions.

Eric Gaze, Alfred University

### To Infinity and Beyond: An Irrational Tale

Abstract:  This talk will explore the history of infinity, going back to ancient Greece and the deep philosophical problems that arose from considering the infinite.  Aristotle was effectively able to put to rest the paradoxes of infinity with an argument later championed by the Church, but the development of the Calculus brought infinity back from the depths of human consciousness and forced mathematicians to study it with rigor.  Is it possible to go beyond infinity?  And if so just what would that mean?

Gregg Hartvigsen, SUNY Geneseo

### Modeling the spread of influenza through a spatially-structured host population.

Dr. Hartvigsen is going to present the results from a model that investigates the dynamics and prevention of influenza in realistically-structured human populations. This should be of interest to a broad range of folks interested in how computer and mathematical modeling can be used to better understand a biological system.

### An Odyssey of Discovery:  Vertical Development of Geometric Thinking in the Secondary School Curriculum

Dynamic geometry systems like Cabri Jr. and Cabri Geometry II Plus provide all students and teachers with a unique opportunity to experience the true sense of mathematical discovery usually reserved for a few elite mathematicians.  Through the guidance of a skilled teacher, students can begin with some basic ideas, investigate and explore these notions in traditional and non-traditional ways, and eventually reach levels of discovery never before available to students.  We will begin with a simple triangle and its four basic centers, stretching students' thinking past this rudimentary level with historic constructions that lead to the generalizations of geometric relationships and eventually to new representations of geometric ideas.  Yet, all of this vertical development is accessible to secondary school students because of the power of visualization provided by dynamic geometry software.

Rachel Schwell, University of Connecticut

### Knot Theoryâ€”What They Didnâ€™t Teach You in Boy Scouts

Knot theory is a relatively â€œnewâ€ field of mathematics; new in that it has only begun to be explored in the past one hundred or so years.  We will examine knots from a more mathematical angle, including the accepted mathematical procedure of â€œuntanglingâ€ a knot, if it can be so done, and determining whether two different-looking knots are actually the same.  We will then consider a way to â€œadd two knots together,â€ and compare this algebraic operation to addition and multiplication of natural numbers.  The only knowledge that is required is to know what a knot is and how to multiply integers!

Immediately following the talk there will be extensive time to talk with Rachel about graduate school in mathematics . . . why to consider it and what itâ€™s like.

Olympia Nicodemi, SUNY Geneseo

### An Invitation to Galileo's World

In this talk, we take an informal tour of the life and work of one of history's most important scientists. Galileo was a natural philosopher who thrust us into the mix of mathematics and experimentation that physics had become today.  He was also a musician, a talented writer, a wine maker, and a dedicated father. We will look at how many these facets come together in his work.  The invitation is extended for you to come explore this further.

Brigitte Servatius, Worchester Polytechnic Institute

### Student Workshop:  Bracing of Grids

A grid of rectangles is only useful as a rigid supporting structure if it has sufficient diagonal bracing.  How to brace the grid is both a geometric and a combinatorial problem.   We will examine this problem both theoretically, and practically with the help of models.

Brigitte Servatius, Worchester Polytechnic Institute

### Firing Cannons

Euler was the first to prove that the path of a cannon ball is a parabola, provided that the only acting force is the force of gravity.  It is well known that firing a cannon on a horizontal plane at an angle of 45 degrees yields the trajectory having the cannon ball landing at maximum horizontal distance from the cannon.

We will present what Halley knew about twists to this problem, as well as some new student thoughts on this old result.

Bonnie Gold, Monmouth University

### What IS Mathematics?

OK, you've studied calculus, and (at least in high school) geometry and algebra; and maybe linear algebra or number theory or differential equations.  So you've seen some examples of mathematics.  But just what IS mathematics?  Why do we call all of these topics mathematics?  Is computer science mathematics?  What about economics?   Why not?  In this talk, we'll look at some answers people have given, and what a good answer might look like.  In the process, we'll also introduce the more general topic of the philosophy of mathematics, and some of the questions it considers.

Rudy Rucker, San Jose State University

### Seek the Gnarl: Adventures in Computer Science

In 1986 former Geneseo mathematics professor Rudy Rucker retooled and became a computer science professor and professional programmer.  But he never stopped thinking like a mathematician.  This talk describes some of the work he's been involved in over the last twenty years in Silicon Valley.  This will include brief discussions of both the computer culture and of particular topics in chaos, fractals, artificial life and computer games.  In addition, he'll give a more detailed discussion of his specialty, cellular automata.  The theme of "gnarl" relates to the class four computations described in Stephen Wolfram's A New Kind of Science and elaborated upon in Rucker's forthcoming nonfiction book, The Lifebox, the Seashell and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning of Life, and How to be Happy.

Heather Lewis, Nazareth College

### The Mathematics of Time

Is it true that there was no October 10 in 1582? Was George Washington born on February 11, 1731 or February 22, 1732, or both?  Why was an hour equal to 144 minutes after the French Revolution?  Does the 13th of the month really fall on a Friday more than any other day of the week?

These questions will be answered and other tidbits of calendar trivia revealed as we look at the mathematics of time.  And there will indeed be some mathematics (modular arithmetic plays a natural role), but the majority of the talk will be accessible to people of all backgrounds.

Tom Head, Binghamton University

### Splicing Systems:  The formal generativer power of enzyme systems

The application of computational mathematics has aided the understanding of biological systems. A new scientific frontier is emerging where biology can aid computational mathematics. The  interdisciplinary field of biomolecular computing explores new biological paradigms to perform calculations and to use biomaterials in the fabrication and design of computing architectures at the molecular level.  This talk describes one such approach.

The splicing concept models the 'wet' cut & paste operations performed  by genetic engineers on DNA.  The abstract 'dry' version of splicing has contributed a new generative scheme that has been studied extensively in the theories of formal languages and computation.  The  wet (motivational) aspect will be discussed in detail.  The deepest  results in the dry aspect will be discussed briefly without  proofs.  Suggestions for further work will be made.

Olympia Nicodemi & Melissa Sutherland, SUNY Geneseo

### The Art and Math of Friezes

A frieze is a horizontal decorative strip. Often we see them as a strip of wallpaper near the ceiling of a room. In this talk, we look for the mathematics hidden in these designs. We are looking for your input too. The math club PRISM hopes to present this topic as a workshop for high school girls. We hope to spark ideas as to related math activities and related art activities.

Nancy Boynton, SUNY Fredonia

### Modeling a Birth and Death Process and What Does That Have to Do With Waiting in Line?

We will start with certain assumptions about births in a system, what we call a pure birth process. We will see what equations and solutions this leads us to. Next we will add the possibility of deaths to the system and see what equations this gives us. These are more complex and so we can simplify the model by considering the long run behavior of the system. Finally we will model a waiting line (like at the bank) as a birth and death process. We can view the births as customers arriving and joining a waiting line. When a customer completes service and leaves the system we will interpret that as a death and look at the probabilities for various numbers of customers in the system.

Cheryl C. Miller, SUNY Potsdam

### Logic and the Natural Numbers

How unique are the Natural Numbers?  Some of the simplest properties to express (without naming specific numbers) include the fact that the set is infinite, there is always a successor or next natural number, and that only one of them has no previous element.  Does this completely describe the set?  Come see how some logic formulas and theorems can help us find out more about the natural numbers, and the possibility of other sets that can also satisfy these statements.

The talk requires only a basic knowledge of logical formulas, the symbols used will be explained as needed.
Uma Iyer, SUNY Potsdam

### An Introduction to Noncommutative Algebras

Polynomials can be added and multiplied; at the same time, real numbers can be thought of as polynomials in the form of constant polynomials.  Hence, all the polynomials in variable x with real number coefficients form an "algebra".  Suppose we look at objects which can be added and multiplied, but the multiplication is not commutative.  Then we get a noncommutative algebra.  Matrices are one example of a noncommutative algebra.

The study of noncommutative algebras has been of interest for more than a century because of the study of matrices.  In recent decades, as the interest in noncommutative geometry grew, noncommutative algebras have become quite important.  In the 90s quantum groups were widely studied, which had relevance to diverse areas like knot theory and Lie theory.

In this talk, I will introduce noncommutative algebras through examples.

Julia Wilson, SUNY Fredonia

### Eine Kleine Mathmusik

Abstract:  Math and music have been linked in curious ways for thousands  of years.  In fact, in the Middle Ages music was considered a mathematical subject.  In Ancient Greece, the Pythagoreans built their theory of the universe on some basic observations about the role of number in music.  We will look at ways in which people have used mathematical ideas to describe and understand music over the millennia.
Dave Bock, Ithaca High School

### ESP and Derangements (a journey into probability with a couple of surprising punch lines)

Robert Kantrowitz, Hamilton College

### Matrices and Their Square Roots

Abstract:  If A and B are square matrices, and B2 = A, then B is called a square root of A.  In this talk, we shall look at several examples of matrices and their square roots.  The examples will serve also to motivate discussion of some general facts about square roots. Only knowledge of matrix multiplication is required.

Bob Rogers, SUNY Fredonia

### Calculus before Calculus

Abstract:  A number of mathematicians used their ingenuity to solve calculus problems before its invention by Newton and Leibniz.  This talk explores some of these accomplishments and discusses their place in the invention of the Calculus.

Brad Emmons, Utica College

### Rational Points on Curves

The Pythagorean theorem tells us that the sides of a right triangle are related by the equation a2 + b2 = c2.  One of the main goals in classical number theory is finding all integral solutions to equations, like the Pythagorean equation.  Many of these problems have rather elegant solutions when viewed graphically.  In this talk we will investigate a few problems related to the Pythagorean theorem, and the graphical approaches to the problems.  This will lead to a discussion on elliptic curves and how you can earn an easy million dollars.
Darwyn Cook, Alfred University

### Is It Serendipity?

We will look at some results in mathematics that have had a large impact in other areas of science. In particular we will be looking at how closely the result in mathematics was followed by the applications in other fields.  The goal is to discuss these results - please come prepared to participate.
Tom Pfaff, Ithaca College

### Mathematical Ideas in Everday Life

Abstract:  As the title suggests, this talk will take a look at mathematical ideas in everyday life.  In no particular order some of the topics will be, measuring spoons and how not to give advice to a cook; least common multiples; counting numbers, letters and rocks; Kevin Garnett; geometric and arithmetic means; cookies and chocolate.

Paul Loomis, Bloomsburg University

### Perfect numbers, unpredictable sequences, and other number theoretic nuggets

Abstract:  The concept of a perfect number - a number that is the sum of its proper divisors - has been around since Euclid, 2300 years ago, yet there are still open questions and active research about perfect numbers and their relatives. I'll talk about perfect numbers, the unpredictable sequences that result when we iterate the function s(n) = the sum of the proper divisors of n, and many close relatives of these ideas.  If you are comfortable with functions and basic arithmetic, none of the main ideas in this talk will be over your head.

Mark McKinzie, Monroe Community College

### Eighteenth Century Precalculus

Abstract:  "Precalculus" is an odd topic for a course of study.  The point of a precalculus class isn't to learn any specific, coherent,
self-contained body of knowledge, but rather to build upon prior algebraic and geometric ideas, acquiring the prerequisite tools for
understanding the calculus.  As the teaching of calculus has evolved over time, so too has the content of the precalculus curriculum.  Thus one can gain insight into how people conceive of the calculus by examining what they teach in their precalculus classes.  Leonhard Euler's "Introductio in analysin infinitorum" ("Introduction to the analysis of infinities", 1748) was explicitly presented as a precalculus text, and has been described as the most influential textbook of the modern era.  In this talk, we will examine the content of the "Introductio...", and discuss its relation to the notions of the calculus prevalent in the 18th Century.

Huaien Li, Los Alamos National Lab

### A complete system of orthogonal step functions

Abstract: We deduce a complete orthogonal system of step functions for the interval [0,1]. Its step functions are expressed in closed form using the MÃ¶bius function. Each step function exhibits only one step length; two functions of the system have length equal to 1/2n for each natural number n. Hence number theory is involved. Furthermore, all the step heights are rational. This talk is designed with undergraduates in mind.  During the talk we will discuss the following topics:  Gram-Schmidt orthogonalization, method of least squares, Fourier series, linear spaces and the MÃ¶bius function.

Blair Madore, SUNY Potsdam

### Why Study Dynamics?

Abstract:  What is the field of Dynamical Systems? Why would anyone be interested in it?  In this presentation weâ€™ll see how the dynamical systems point of view is useful in solving some interesting number theoretic problems. Please bring pen, paper and a calculator â€“ the audience will have an opportunity to participate in solving these problems. Additionally we hope to introduce some popular concepts from dynamical systems theory including orbits, fixed points, periodicity, fractals and chaos.  No mathematical prerequisites. All are welcome.

Bio:  Blair Madore has a Ph. D. in Ergodic Theory (a field of measure theoretical dynamical systems) from the University of Toronto. He has a BMath from University of Waterloo where he had the opportunity to work in the Computer Science research lab that created Maple. A native of Newfoundland, he is currently enjoying the fun of teaching math at SUNY Potsdam and all the outdoor excitement that the North Country has to offer including skiing, snow shoeing, hiking, canoeing, and fishing.
Chris Leary, SUNY Geneseo

### On Number

We briefly consider the concept of number and some of the ways that mathematicians have tried to make the idea of number precise.  The talk will be of a mixture of some mathematics, some history, a couple of half-truths, and at least one outright lie.

Prerequisite:  A nodding acquaintance with the numbers 0, 1 and 2.
Michael Knapp, University of Rochester

### A Trip to the Fun House: the World of p-Adic numbers

Have you ever stood in front of one of those fun house mirrors which distort distances and perspectives? Imagine standing on a number line and looking at one of those mirrors. You're standing on the number 0, and the number 3125 appears to be very close to you. But the numbers 1, 3124 and 3126 all appear to be much farther away from you, and all are the same distance away. The number 1/3125 is even farther away!

This is the way distances can look in the world of p-adic numbers. Despite this strange notion of distance, p-adic numbers can be used to help answer questions about the "normal" world of numbers. For example, they can be used to help determine whether some equations have solutions in which the variables are all integers.

This talk will be a brief introduction to this brave new world of p-adic numbers. First, I will talk about trying to determine whether an equation has any integer solutions, and this will lead to a very informal definition of the p-adics. Then I will show a more formal way in which they can be defined, which will explain the strange notion of distance mentioned above. Finally, if we have time, I will talk a little more about how the p-adics and "normal" numbers relate to each other, and also mention a few interesting theorems about solving equations where the variables are p-adic numbers.

Frank Vafier (Geneseo B.A. Mathematics '74)

### Getting your first job and accelerating you career.

Having interviewed and hired hundreds of applicants, I know what to look for and how to separate the doers from the talkers.  I've also been deeply imbedded in great companies (like IBM) and horrible ones (to remain nameless) and I feel qualified to talk about recognizing great companies, getting a job with them and moving through the hierarchy.

Bio:  Frank Vafier is cofounder and CEO of Prolifics, provider of Enterprise Business Solutions to Fortune 2000 companies around the globe by leveraging a 24-year wealth of technical expertise and business acumen.  Mr. Vafier has a Bachelors Degree in Science in Mathematics and Physics from SUNY Geneseo, where he graduated Magna Cum Laude, and a Masters Degree in Computer Science from NYU.

Carl Pomerance, Bell Laboratories

### A New Primal Screen

It is amazing that we are still finding new things about prime numbers, as old as Euclid, and new uses.  In the past few decades we've used prime number both to protect Internet message from eavesdroppers and to do the completely opposite task of making communication so transparent that we can potentially talkw ith aleins from other worlds.  Further, last August the mathematics and computer science communities were stunned with the announcement of a new and speedy screen for prime numbers.  What is perhaps more stunning is that two of the three researchers had just received their bachelor's degrees in June.  In this talk some of the principal ideas in the new test, and in the applications, will be described.

Bio:  Carl Pomerance received his B.A. from Brown University in 1966 and his Ph.D. from Harvard University in 1972 under the direction of John Tate.  During the period 1972-1999 he was a professor at the University of Georgia.  Currently, he is a Member of Technical Staff at Bell Laboratories and a Research Professor Emeritus at the University of Georgia.
Carl Pomerance, Bell Laboratories

### Recent Developments in Primality Testing

Last August, Agrawal, Kayal, and Saxena, all from the Indian Institute of Technology in Kanpur, announced a new algorithm to distinguish between prime numbers and composite numbers. Unlike earlier methods, theirs is completely rigorous, deterministic, and runs in polynomial time. Previous results, some of them quite deep, were close to this ideal in various ways, so, perhaps, it was not such a great surprise that such a result should exist. But the relatively easy algorithm and proof is stunning. In this talk, the new algorithm will be described as well as some more recent developments.

Jim Conant, Cornell University

### The Fractal Dimension of Art

Abstract:  Jackson Pollock is famous for his random-seeming drip paintings, and is often a target for anti-abstractionists. It has recently been discovered that Pollock's paintings have a consistent fractal dimension, one that rose over a period of time from 1.3 to about 1.7. It has also been discovered that fractal dimension functions as a sort of aesthetic barometer. A survey found that people typically like things of fractal dimension 1.4, comparable to dimensions found in nature. Thus the appeal of Pollock's paintings has been partially explained: they are not just random drips. In  this talk, we will explain what is meant by "fractal dimension" and we will participate in an experiment where audience members will judge their favorite pictures to see what fractal dimension they prefer.

Tony Macula, SUNY Geneseo

### Extraction of Complexes Using a DNA Computing Model

Abstract:  The (relative) specificity of hybridization between a DNA strand and its Watson-Crick complement can be used to perform mathematical computation.  In 1994, L. Adleman used DNA to "solve" a small Hamiltonian path problem. His experiment demonstrated that the existence and nature of a solution can be achieved by the formation and isolation of a certain DNA molecule.  Adleman's toy'' demonstration was the first indication that the massive (i.e, exponential) parallelism of DNA reactions could be exploited to overcome the exponential time complexity (via a silicon computer or Turning machine) of NP complete problems so that they could possibly be solved in linear \emph{real} time.  To achieve the potential of DNA computing, many bio-engineering hurdles need to be overcome.

In this general audience talk, we discuss an applied mathematical problem, a possible DNA approach to its solution and some of the bio-engineering problems that arise.

Look here for a more formal version of this abstract.

David Handron, Carnegie Mellon University

### An Introduction to Morse Theory

Abstract: Morse theory relates information about the critical points of a function (where the derivative is zero) to information about the shape (topology) of the domain.

In this talk I will discuss different types of critical points and describe some of the basic results of Morse theory.  I will also show how these ideas relate to the Energy function and equilibrium points of a physical system.

Tom Pfaff, Ithaca College

### Tiling the WWW

Abstract:  What makes for a good background on a webpage? What mathematics is involved?  This talk will address these questions.  In attempting to answer them we will come across Escher, the Pythagorean theorem,  puzzles of Sam Loyd, dissections, donuts and plenty of neat graphics.

Matthais Beck, Binghamton University

### The "Coin Exchange Problem" of Frobenius

Abstract: How many ways are there to change 42 cents? How many ways would there be if we did not have pennies? How about if nickels were worth four cents?

More generally, suppose we have coins of denominations a 1, . . . ,  ad .  Can one find a formula for the number c(n) of ways to change n cents?  A seemingly easier question is: can you change n cents, using only our coins?

We will see that if a1 , . . . ,  ad  are relatively prime then we can be certain that we can change n , provided n is large enough. A natural task then is to find the largest integer that cannot be changed. This problem, often called the linear Diophantine problem of Frobenius, is solved for d = 2 but wide open for d > 2.

We will use the above counting function c(n) to recover and extend some well-known results on this classical problem. En route we will discuss some basic Number Theory and Discrete Geometry connected to c(n) .