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.
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 2011Polytopes, 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.
Suppose that y = Ax, with A ∈ Rnm, y ∈ Rn, x ∈ Rm, 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.
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.
Yusuf Bilgic, SUNY Geneseo
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.
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.
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.
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.
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
Emma Norbrothen, North Carolina State University
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
Xiao Xiao, Utica College
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.
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.
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.
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.
Uma Iyer, SUNY PotsdamLogic 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 variable x with real number coefficients form an "algebra". Suppose we look at objects which can be added and multiplied, but the multiplication is not commutative. Then we get a noncommutative algebra. Matrices are one example of a noncommutative algebra.
The study of noncommutative algebras has been of interest for more than a century because of the study of matrices. In recent decades, as the interest in noncommutative geometry grew, noncommutative algebras have become quite important. In the 90s quantum groups were widely studied, which had relevance to diverse areas like knot theory and Lie theory.
In this talk, I will introduce noncommutative algebras through examples.
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.
Robert Kantrowitz, Hamilton CollegeESP and Derangements (a journey into probability with a couple of surprising punch lines)
Matrices and Their Square Roots
Abstract: If A and B are square matrices, and B2 = A, then B is called a square root of A. In this talk, we shall look at several examples of matrices and their square roots. The examples will serve also to motivate discussion of some general facts about square roots. Only knowledge of matrix multiplication is required.
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.
Darwyn Cook, Alfred UniversityRational Points on Curves
The Pythagorean theorem tells us that the sides of a right triangle are related by the equation a2 + b2 = c2. One of the main goals in classical number theory is finding all integral solutions to equations, like the Pythagorean equation. Many of these problems have rather elegant solutions when viewed graphically. In this talk we will investigate a few problems related to the Pythagorean theorem, and the graphical approaches to the problems. This will lead to a discussion on elliptic curves and how you can earn an easy million dollars.
Tom Pfaff, Ithaca CollegeIs 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.
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.
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.
Chris Leary, SUNY GeneseoWhy 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 RochesterOn 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.
Carl Pomerance, Bell LaboratoriesGetting 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 LaboratoriesA 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.
Tony Macula, SUNY GeneseoThe 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.
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.
Matthais Beck, Binghamton UniversityTiling 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 denominations a 1, . . . , ad . Can one find a formula for the number c(n) of ways to change n cents? A seemingly easier question is: can you change n cents, using only our coins?
We will see that if a1 , . . . , ad are relatively prime then we can be certain that we can change n , provided n is large enough. A natural task then is to find the largest integer that cannot be changed. This problem, often called the linear Diophantine problem of Frobenius, is solved for d = 2 but wide open for d > 2.
We will use the above counting function c(n) to recover and extend some well-known results on this classical problem. En route we will discuss some basic Number Theory and Discrete Geometry connected to c(n) .