Anna Mummert (Geneseo class of 2000), Alfred University ### Fractals and Hausdorff Dimension

Patrick Rault, University of Wisconsin

### Mathematical game theory

Sharon McCathern, University of Illinois at Chicago

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

Palalanivel Manoharan, Penn State University

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

Jeff Johannes & Gary Towsley, SUNY Geneseo

### A Concise History of Calculus

A lively overview of over two thousand years of calculus history. Not only who-did-what along the way, but the cultural and sociological causes and effects of the calculus. Strongly recommended for anyone who has taken or is taking calculus.

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

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 ?best? 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.

### 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.

A fractal is an object with complicated repetition of structure at many size scales. Fractals appear in nature as ferns, coastlines, and clouds, to name a few. Mathematically, a fractal has a Hausdorff dimension that is not an integer. In this talk, I will explore the idea of fractals in nature and in math. The Hausdorff dimension will be defined and computed for the middle-third Cantor set, the Koch curve, and the Sierpinski triangle.

Patrick Rault, University of Wisconsin

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

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

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.

Jeff Johannes & Gary Towsley, SUNY Geneseo

Patrick van Fleet, University of St. Thomas

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

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!

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

Lingji Kong, Union College (Kentucky)

Shubiao Li, Central Michigan University

Amy Stornello, Rochester Institute of Technology

Matthew Rashford, SUNY Geneseo

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

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.

### 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 ?

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.

### 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

### Teachers? Evolving Practices in Supporting Students? 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 students? mathematics and literacy development. Teachers? evolving practices included (a) their use of the ?Problem of the Day? to engage students in thinking and communicating mathematically, (b) their development of ?templates? 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?

Jim Conklin, Ithaca College

Christopher Andrews, University at Buffalo

Michael Fisher, CSU Fresno

Alison Setyadi, Dartmouth College

Aaron Heap, University of Rochester

Sharon Garthwaite, University of Wisconsin

David Perkins, Houghton College

Pedro Teixeira, Union College

Joanna Masingila, Syracuse University

Ryan Grover & Matthias Youngs, SUNY Geneseo

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:

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.

Eugene Olmstead, Elmira Free Academy

### 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

Katia Noyes, University of Rochester School of Medicine

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

- 1 + 2 + . . . + n = n(n + 1)/2 for all natural numbers n;
- sin
^{2}(x) + cos^{2}(x) = 1 for all real numbers x; - xy - yx = 0 for all complex numbers x and y.

Sergio Fratarcangeli, McMaster University

Markus Reitenbach, Syracuse University

Eric Gaze, Alfred University

Gregg Hartvigsen, SUNY Geneseo

Eugene Olmstead, Elmira Free Academy

Rachel Schwell, University of Connecticut

Knot theory is a relatively ?new? 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 ?untangling? 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 ?add two knots together,? 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 it?s like.

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

Olympia Nicodemi, SUNY Geneseo

Brigitte Servatius, Worchester Polytechnic Institute

Brigitte Servatius, Worchester Polytechnic Institute

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

Rudy Rucker, San Jose State University

Heather Lewis, Nazareth College

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

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.

Jeff Johannes & Gary Towsley, SUNY Geneseo

Abstract: A lively overview of over two thousand years of calculus history. Not only who-did-what along the way, but the cultural and sociological causes and effects of the calculus. Strongly recommended for anyone who has taken or is taking calculus.

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 variablexwith 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: IfAandBare square matrices, andB, then^{2}= ABis called a square root ofA. 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 a^{2}+ b^{2}= c^{2}. 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 M?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 M?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 we?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 ? 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 denominationsa_{ 1}, . . . ,a. Can one find a formula for the number_{d}c(n)of ways to changencents? A seemingly easier question is:canyou changencents, using only our coins?

We will see that ifa_{1}, . . . ,aare relatively prime then we can be certain that we can change_{d}n, providednis large enough. A natural task then is to find the largest integer that cannot be changed. This problem, often called thelinear Diophantine problem of Frobenius, is solved ford= 2 but wide open ford> 2.

We will use the above counting functionc(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 toc(n).