Newton 202

Alden Edson, Western Michigan University

Digital instructional resources are rapidly replacing print
materials offering a promising direction for education at all
levels. In this interactive session, we will examine pedagogical and
tool features of a highly interactive digital instructional unit
focusing on binomial distributions and statistical inference. This
experience will be connected to a summary of how tasks originating
in print curriculum materials can be adapted for highly interactive
digital instructional materials and their affordances for student
learning. We will next examine more closely the additional
affordances of the tool features of the digital materials.

Newton 202

Michael Pawlikowski, Depew High School and University at Buffalo

Wednesday, April 16 2:30-3:20p

Newton 203

Kristin Camenga, Houghton College

The numerical range of a matrix is a set of numbers defined by how the matrix acts on the unit sphere. Most frequently, we work with matrices with complex numbers as entries and the resulting numerical range is a set of complex numbers, which can be visually represented on the complex plane. The resulting numerical range is convex and contains the eigenvalues of the matrix. Based on this foundation, we can ask many different questions. Given a specific type of matrix, what shapes of numerical range result? Which transformations of the matrix leave the numerical range the same? What happens if we use different fields for the matrix entries? What if we associate a set of numbers to a pair of matrices instead of just one? In this talk we will define numerical range precisely and look at some of the different questions that can be explored. Previous experience with linear algebra is recommended, but all definitions will be reviewed.

The past two decades have been an exciting time in biomedicine, with the completion of the human genome project and the beginning of the "big data" era. This has made collaborations between medical professionals, biologists, computer scientists and mathematicians both commonplace and necessary. These interdisciplinary efforts have resulted in scientific breakthroughs in our understanding of fundamental biological principles, such as the role of DNA and how it is regulated, as well as increased our understanding of disease mechanisms. A wide variety of mathematics and other computational techniques have played important roles in driving these biomedical advancements, ranging from differential equation models to data processing and compression methods (such as wavelet transforms) to optimization methods and statistical analysis. I will give a broad overview of the exciting and active areas of computational biomedical research, as well as go more in depth into my particular area of research in cancer genomics.

Nathan Reff, Alfred University

Kevin Palmowski, Iowa State University, Geneseo Class of 2011## Polytopes, Pick's Theorem and Ehrhart Theorem

Suppose P is a convex polygon with integer points. Pick's Theorem says that the area of P can be computed by counting integer points on the interior and the boundary of P. I will discuss Pick's theorem as well as a generalization to higher dimensions via Ehrhart polynomials. If time permits, I will also discuss some applications.

**R**n,
x
∈
**R**m,
and n
< m.
The problem of reconstructing x
given
y
and
A
is
generally ill-posed. In the case where x
has
sparse support (|supp(x)|
≪ m),
compressed sensing literature provides algorithms, many of
which use l1
minimization,
by which we can accurately reconstruct x
under
certain sufficient conditions. In the case where supp(x)
is partially known, the Modified-CS algorithm can be used to
reconstruct x,
and the sufficient conditions for reconstruction can be
relaxed. Weighted-l1
minimization
uses a combination of standard CS and Modified-CS. We
present new sufficient conditions for exact reconstruction
by weighted-l1
minimization,
compare these to known bounds, and report results of
simulations. An application to image processing is included
as a motivating example.

Dangerous heart conditions can arise from disturbances in the electrical signals that trigger the heart to contract. Substantial experimental evidence links these serious electrical disruptions, called arrhythmias, with spiral waves of electrical activity in cardiac tissue. After a single spiral wave has formed, a number of known mechanisms can destabilize it and generate additional spirals that complicate restoration of cardiac function. These waves repetitively excite the tissue at fast rates and alter the sequence of activation, both of which compromise the hearts ability to pump effectively. This talk will describe the current understanding of the spatiotemporal organization of electrical waves in the heart during normal rhythm and arrhythmias using state-of-the-art experiments, theory, and simulations. The electrophysiology of cardiac cells and tissue, mathematical modeling approaches, physical principles, numerical methods, and high-performance computing issues will be discussed.

Ernest Fokoue, RIT

The technological advances of the recent decades have made it possible for huge amounts of data to be collected in all walks of life. In fact, its quite common these days to hear people say that we live in an era of the so-called Big Data. Interestingly, large complex data come in various shapes, sizes and characteristics. In this lecture, I will present a simple taxonomy of large/massive data sets, and I will briefly highlight some of the mathematical and statistical tools that are commonly used in data mining and machine learning to extract meaningful information from different forms of complex data. I intend to touch on data from fields such as image processing, speech/audio recognition, DNA micro-array analysis, classification of text documents, intrusion detection and consumer statistics, just to name a few. My focus throughout the lecture will be to show and possibly demonstrate computationally that while some of the mathematical/statistical tools needed to tackle these complex data structures are very novel and cutting edge, some are just straightforward applications or gentle extensions of the traditional statistical arsenal.

Jeff Johannes & Gary Towsley, SUNY Geneseo

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.

Yusuf Bilgic, SUNY Geneseo

ginormous? Lets make a decision at 5% level using simulation studies.

Students are encouraged to bring devices with internet access (laptop, smart phone etc).

This talk discusses my study in progress, which centers on Hispanic students math achievement using data from the Grade 12 National Assessment of Educational Progress (NAEP). I explain the rationale for focusing on functions as a representation of math achievement and highlight methods I use to create a profile of achievement for first-generation college-bound Hispanic students using their performance on function items. In addition, I discuss the implications my findings have for policymakers and others interested in addressing achievement gaps in mathematics.

Dana Dodson, Indiana University Northwest

Students recall and model the square root as the length of one side of a square. But what if the square is not a perfect square? This collection of activities explores the square roots of not-so-perfect squares and develops an algorithm to express the not-so-perfect square root as a rational value.

Christian Northrup, East Henderson High School

In an era of technology, students benefit from instant feedback enabling them to focus more on analysis and drawing conclusions. When utilized properly, technology can be a useful tool for helping students learn new statistical concepts. What are other methods that teachers can employ in a classroom that will aid and support students as they learn statistics? How can statistics teachers be confident that students are truly grasping concepts? Writing is one answer to these questions. Join me as we examine the benefits of writing. In addition, we will discuss the hurdles that will need to be overcome by students and teachers to successfully implement writing in a statistics classroom. Please bring your Smartphone, iPad, or an equivalent device.

Salam Khan, Alabama A&M University

Probability theory is a very important tool for mathematical and statistical analysis. The aim of this talk is to discuss basic concepts of probability and a probabilistic social conflict model for non-annihilating multi-opponent. In this probabilistic model opponents have no strategic priority with respect to each other. The conflict interaction among the opponents only produces a certain redistribution of common areas of interests. The limiting distribution of the conflicting areas, as a result of infinite conflict interaction for existence space is investigated.

Ranking is a commonly used procedure to evaluate daily life
situations, for example in medicine, business, sports, and many
other fields. This method is also used in nonparametric statistics,
the basic idea here being the ordering of the observations on a more
abstract level. The talk will explain the usefulness of this concept
in testing procedures, the famous two-sample problem being an
important example. A more advanced idea of ordering is used to
analyze the ``shoulder tip pain data set that appears in Brunner,
Domhof and Langer, 2002. This is a clinical study with 41 patients
who had undergone a laparoscopic cholecystectomy and developed
shoulder pain after the surgery. The main question here is to test
the effectiveness of the treatment for shoulder pain.

To get a better understanding of the weather systems, atmospheric scientists and meteorologists are generally interested in having available data spatially interpolated to much finer spatial and temporal grids. Many physical or deterministic models that are used to generate regional or global climate models are able to provide accurate point predictions, but it is very difficult for them to give realistic uncertainty estimates. This calls for the development of statistical models that can produce uncertainty estimates through conditional simulations. In this project, we look at the minute-by-minute atmospheric pressure space-time data obtained from the Atmospheric Radiation Measurement program. We explain how spatial statistics can be used to model such data that are sparse in space but high frequency in time. Due to the interesting local features of the data, we also take advantage of the localization properties of wavelets to capture the local dynamics of the high-frequency data. This method of modeling space-time processes using wavelets produces accurate point predictions with high precision, allows for fast computation, and eases the production of meteorological maps on large spatial and temporal scales.

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

Amanda Beeson, University of Rochester

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.

May Mei, University of California, Irvine

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.

Emma Norbrothen, North Carolina State University

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.

Sue McMilllen, Buffalo State

President, Association of Mathematics Teachers of New York State
(AMTNYS)

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.

Arunima Ray, Rice University, SUNY Geneseo class of 2009

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

Xiao Xiao, Utica College

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

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.

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

Laurel Miller-Sims, Hobart & William Smith Colleges

Rebekah Yates, Houghton College

Prerequisite: Math 233 (Elementary Linear Algebra).

Elizabeth Wilcox, Colgate University

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.

This is a version of a search problem for succinct data structures that appeared in a paper written by Peter Bro Miltersen and Anna Gál 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

Douglas Haessig, University of Rochester

Jane Cushman, Buffalo State University

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

Jonathan Hoyle, Eastman Kodak

Ryan Gantner, St. John Fisher College

Bonnie Jacob, Rochester Institute of Technology

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 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

Candace Schenk, Binghamton University

Brandt Kronholm, St. Mary's College of Maryland

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. What’s the
formula? It’s less than 100 years old and you wouldn’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:

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!

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)≡0
(mod5)

p(7n+5)≡0 (mod7)

p(11n + 6) ≡ 0 (mod 11)

Fifty years later one more divisibility property modulo 17 was
discovered. Are there any others?p(7n+5)≡0 (mod7)

p(11n + 6) ≡ 0 (mod 11)

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

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

Bogdan Petrenko, SUNY Brockport

Aaron B. Luttman, Clarkson University

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

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

Tom Bleier, Syracuse University

Paul Seeburger, Monroe Community College

Phong Le, University of California at Irvine

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 x^{2}+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

Emilie Wiesner, Ithaca College

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 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.

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:
R^{n} -> R^{n}
whose inverse is continuous, such that f(Ax)=Bf(x)
for all x in R^{n}.

If f were the function induced by a matrix P, i.e., if f(x)=Px for all x in R^{n},
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 GeneseoIf f were the function induced by a matrix P, i.e., if f(x)=Px for all x in R

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.

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)

Michael Starbird, University of Texas

Vijay Sookdeo, University of Rochester

Keary Howard and friends, SUNY Fredonia

Scott Russell, SUNY Geneseo Computer Science

Chris Leary, SUNY Geneseo

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

Bob Rogers, SUNY Fredonia

Claudiu Mihai, Daemen College

Darwyn Cook, Alfred University

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?

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.

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

Patti Fraser-Lock, St. Lawrence University

### Marijuana Use, Goldfish, and Knee
Injuries

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.

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)

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.

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 …

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

### 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

Michael Fisher, CSU Fresno

Alison Setyadi, Dartmouth College

Aaron Heap, University of Rochester

Sharon Garthwaite, University of Wisconsin

David Perkins, Houghton 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

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.

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

Uma Iyer, 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.

## 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

Bob Rogers, SUNY Fredonia

Brad Emmons, Utica College

Robert Kantrowitz, Hamilton College## ESP and Derangements (a journey into probability with a couple of surprising punch lines)

## 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

Darwyn Cook, Alfred University## 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.

Tom Pfaff, Ithaca College## 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.

## 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.

## 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 Möbius 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 Möbius function.

Blair Madore, SUNY Potsdam

Chris Leary, SUNY Geneseo## 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.

Michael Knapp, University of Rochester## 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.

## 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)

Carl Pomerance, Bell Laboratories## 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.

## 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

Tony Macula, SUNY Geneseo## 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.

## 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

Matthais Beck, Binghamton University## 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.

## 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).