Mathematics, Computer Science and Statistics Department Seminar

 
 

 

Fall 2014

Friday, 5 September, 3:00 pm, Fitzelle 205

Fish, Forests, and Functions

Karl Seeley, Economics, Hartwick College

Abstract:

The workhorse models in environmental economics are some elegant applications of a simple logistic growth function, tweaked in different ways to reflect stylized facts about populations of fish, stands of trees, or some catch-all of renewable resources. This seminar will demonstrate these uses and what can be learned from them, as well as discussing some important parts of reality that get left out.

Friday, 3 October, 3:00 pm, Fitzelle 205

TBA

Jens Christensen, Mathematics, Colgate University

Friday, 7 November, 3:00 pm, Fitzelle 205

Bhāskarācārya: 900th Birth Anniversary

Keith Jones and Toke Knudsen, Mathematics, SUNY Oneonta

Friday, 5 December, 3:00 pm, Fitzelle 205

My Favorite Groups

Rachel Skipper, Mathematics, Binghamton University

Spring 2014

Friday, 31 January, 3:00 pm, Human Ecology 105

Introduction to Lie Algebras

Jonathan Brown, Mathematics, SUNY Oneonta

Abstract:

Algebraic Lie theory is an active and exciting area of mathematics research with many applications to other areas of mathematics and physics. At the heart of algebraic Lie theory are Lie algebras. They arise naturally as tangent spaces of Lie groups. In this talk I will give the basic axioms for Lie algebras as well as some examples. I will also explain their relationship with Lie groups, and I will talk about one of the connections between Lie theory and physics. The talk is intended for math undergraduates, and it will be accessible to anyone who has taken Calculus, though knowledge of elementary linear algebra will help.

Friday, 28 February, 3:00 pm, Morris 104

Non-transitive dice and directed graphs

Alex Schaefer, Mathematics, Binghamton University

Abstract:

A set of three dice A, B, and C are said to be non-transitive if the three probabilities (A beats B), (B beats C) and (C beats A) all exceed 1/2. I will prove that such dice can be constructed with any number of (at least 3) sides, and the number of dice can also in fact be arbitrary. Then I will discuss the connections to directed graphs and show that any directed graph has a set of dice that can be associated to it.

Friday, 21 March, 3:00 pm, Morris 104

POJOS, POGOS, and beans: Are these MREs or your employment future?

Dennis Higgins, Computer Science, SUNY Oneonta

Abstract:

POGOs and POJOs are "plain old" Groovy or Java objects which would typically follow the bean convention -- a standard way to build them and access their fields or properties. Beans provide a simple mechanism that is leveraged in Java and Groovy to facilitate object manipulation. In this talk I will develop a few beans and show how they can be used in a variety of ways, like application programs involving a database connection, web services and web applications. Time permitting, I'll show how Grails uses beans and talk about the vacation diary my students will build as one of their projects this semester. No programming background is needed.

Friday, 18 April, 3:00 pm, Morris 104

Making a Really Cheap Quantum Graph

Kevin Schultz, Physics, Hartwick College

Abstract:

The study of quantum systems, whose classical counterparts are chaotic, is called Quantum Chaology. In my talk I will describe quantum chaos and how we can measure its effects as well as describing the experimental realization of quantum graphs, which are an ideal test bed for investigating quantum chaos. In particular how we can use acoustic analogs of quantum graphs, which allows for low cost experiments (we use PVC from Home Depot and a sound card from a computer), and a smaller learning curve for undergraduates.

Fall 2013

Friday, 20 September, 3:00 pm, Morris 104

Suspect Something Fishy? How Statistics Can Help Detect It, Quickly

Aleksey Polunchenko, Mathematical Sciences, Binghamton University

Abstract:

Suppose you are gambling at a casino in a game where you and a dealer take turns rolling a die. Suppose next that the die is initially fair, that is, each of its six faces has the same probability of showing up. However, at some point during the course of the game the evil dealer - without you seeing - replaces the die with an unbalanced one, and so from that point on the die's faces are no longer equally probable. Yet as the new die looks exactly the same as the old fair one, you continue to gamble without suspecting anything. The natural question is: as the game progresses, can you somehow "detect" that the die has been tampered with?

This question is a gamble on its own. On one hand, it would be desirable to find out that the die is no longer fair as fast as possible, so as to quit the game to prevent further losses and subsequently file a lawsuit against the casino. On the other hand, if you are too trigger-happy there is a risk of stopping the game too quickly, i.e., stopping the game before the fair die was replaced with the unbalanced one, which is not desirable. How does one go about solving this problem? Turn to statistics!

Statistics is a branch of mathematics concerned with rational decision-making among uncertainty. This is essential in real life, as only a well-thought-out decision can enable one to take the best action available given the circumstances. This talk's aim is to provide an introduction to the nook of statistics that deals with cases when a solution has to be worked out "on-the-go", i.e., when time is a factor as well. Specifically, the talk will focus on the so-called quickest change-point detection problem. Also known as sequential change-point detection, the subject is about designing fastest ways to detect sudden anomalies (changes) in ongoing phenomena. One example would be the above biased die detection problem. However, there are many more, arising in a variety of domains: military, finance, quality control, communications, environment - to name a few. We will consider some and touch upon the subject's basic ideas.

Friday, 11 October, 3:00 pm, Morris 104

Decision Making using Analytical Hierarchies

Ronald Brzenk, Mathematics, Hartwick College

Abstract:

This presentation will discuss the decision making process, Analytical Hierarchies. I will describe how I have used it in my teaching. It has been the basis for students in my course Mathematical Modeling to actually do some "math modeling". I have also used it in some lower level courses. Finally, I will describe how it has also been the basis for senior projects in mathematics.

Friday, 8 November, 3:00 pm, Morris 104

Economics, Mathematics, and Job Market Signalling

Kristen Jones, Economics, Hartwick College

Abstract:

Economists use mathematical tools extensively in the analysis of consumer and producer behavior. This talk will cover the two core optimization problem used in microeconomic theory - utility maximization and profit maximization - focusing both on the mathematical tools used as well as the economic theory. The lecture will then turn to a more sophisticated application of mathematics, Gibbon's (1992) game theory model of the job-market dynamics introduced in Spence (1973). Spence's model of job-market signaling involves decisions of two parties (an employer and an employee) under uncertainty and we will discuss the structure and solution methodology of Spence's models as well as the interesting (and unexpected) results of the model.

Friday, 6 December, 3:00 pm, Morris 104

Generalizing Mundici's Gamma Functor

Joshua Palmatier, Mathematics, SUNY Oneonta

Abstract:

In this talk, we will discuss Chang's Gamma functor, a natural equivalence between the category of MV-Algebras and the category of abelian l-groups with strong unit, as refined by Cignoli & Mundici, and the possibilities of extending this functor to a natural equivalence between the category of m-zeroids and some category containing the category of abelian l-groups with strong unit.

Spring 2013

Friday, 25 January, 3:00 pm, Craven Lounge

Bumping and Sliding for Beginners: An Introduction to Young Tableaux

James Ruffo, Mathematics, SUNY Oneonta

Abstract:

The primary goal of this talk will be to present some of the basic properties on an interesting class of combinatorial objects called Young tableaux. After discussing the basic definitions and constructions, such as the Jeu de Taquin and the Robinson-Schensted algorithm, we will discuss an application to some problems in enumerative geometry.

Friday, 8 March, 3:00 pm, Craven Lounge

The Little Theorem that Could: Finding Pseudoprimes by Matching Terms of Polynomial-Value Sequences

Robert Sulman, Mathematics, SUNY Oneonta

Abstract:

Pseudoprimes are counter-examples to the converse of Fermat's Little Theorem: If p is prime and p does not divide a, then p divides ap-1-1. Thus, a pseudoprime (with respect to a∈Z+) is any composite n satisfying: n divides an-1-1. In this talk, a modest set of divisibility relations are shown to generate two polynomial-valued sequences. A match between them produces a pseudoprime (with an additional non-divisibility condition). A generalization of this construction is then given, which yields sets of pseudoprimes with respect to a given base, and a variety of modifications are seen. Finally, we describe primes q that are a divisor of a pseudoprime with respect to a for all a=2,3,4,...,q-1.

Friday, 22 March, 3:00 pm, Craven Lounge

Why Study Finite p-Groups

Joseph Brennan, Mathematics, SUNY Binghamton

Abstract:

Given a prime p, a finite p-group is a group whose order is a power of p. Though major structures of a p-group lie outside the scope of a typical undergraduate abstract algebra course; I would like to spark interest in finite p-groups by defining their basics and outlining their role in the study of groups. If time permits, I will outline some major developments in the field.

Friday, 26 April, 3:00 pm, Human Ecology 106

On Sums of Finitely Many Distinct Reciprocals

Donald Silberger, Mathematics, SUNY New Paltz

Abstract:

Let F denote the family of all nonempty finite subsets of N := {1, 2, 3, ...}, and let I ⊆ F be the family of all intervals [m, n] := {m, m+1,..., n-1, n}. We define the function σ : F→Q+ by

σ : X → σ X := Σk∈ X 1⁄k.

Since the harmonic series 1 + 1⁄2 + 1⁄3 + ... diverges to ∞, and since its terms are positive, with limk→∞ 1⁄k = 0, it is easy to see that the set H := {σ[m, n] : m ≤ n ∧ {m, n} ⊂ N} of harmonic rationals is dense in ℜ}+ . So it is natural to ask: Is H = Q+? In 1918 J. Kürschák answered this question in the negative by proving that σ[m, n] ∈ N only if m = n = 1. We showed last year that in fact σ[m, n] = 1⁄k for k ∈ N only if m = n = k. This suggested a generalization, which we finally managed to establish:

Theorem: σ|I is injective.

As of the present writing, my brother Allan, my daughter Sylvia and I are attempting to pick the final burr out of a proof of the following

Conjecture: The function σ is a surjection from F onto Q+.

Indeed, if the approach we are using establishes this conjecture then it will moreover, for each a ∈ N, facilitate the construction of a partition Sa of N into infinitely many finite sets such that σ X = a for every X ∈ Sa. Furthermore, it will give us that, for each r ∈ Q+, there are infinitely many distinct Y ∈ F such that σ Y = r.

Fall 2012

Friday, 7 September, 3:00 pm, Room: Human Ecology 216

Lattice, You Have Seen One Don't Even Know It

Martha Kilpack, Mathematics, SUNY Oneonta

Abstract:

A lattice is a partially ordered set with some extra conditions. We will look at these conditions and some examples of lattices. We will then look at how these lattice are actually algebraic structures and what questions then arise.

Friday, 5 October, 3:00 pm, Room: Human Ecology 106

Trying to understand evolution? Get help from the Clergy (Three Cheers for the Good Reverend Bayes)

Jeffrey Heilveil, Biology, SUNY Oneonta

Abstract:

The study of evolutionary relationships, whether between or within species, often requires a "backward view", as we are unable to observe evolution in action for many species. Answering evolutionary questions therefore requires heavy reliance on probability. One of the most helpful analytical paradigms involves the application of Bayesian statistics to evolutionary datasets. After briefly discussing the nature of evolutionary research, Bayesian statistics will be explained at a basic level, including showing how conditional probability impacts our understanding of topics such as breast cancer. We will then look at an example of how one can use Bayesian statistics to evaluate the recolonization of the northern US following the retreat of the Wisconsinan Glaciation.

Friday, 2 November, 3:00 pm, Room: Human Ecology 106

Hyperbolic Geometry: Logic Takes Us to a Strange Place

Charles Scheim, Mathematics, Hartwick College

Abstract:

The geometry that most people are familiar with from their high school math days, Euclidean Geometry, has a long history and great importance from both practical and intellectual viewpoints. But the discovery in the 1800's of non-Euclidean geometries caused a revolution in the fundamental assumptions about the relationship of mathematics with the world around us. This presentation will use the software Geometer's Sketchpad to re-awaken the basic principles of Euclidean geometry for the listeners and to help them visualize a model of a non-Euclidean geometry. We'll explore this new geometry and examine some of the philosophical questions that arise because of its existence.

Friday, 30 November, 3:00 pm, Room: Human Ecology 106

Games, Fractals, and Groups: From Hanoi to Sierpinski

Keith Jones, Mathematics, SUNY Oneonta

Abstract:

The long-studied game The Tower of Hanoi has a fascinating connection to the famous Sierpinski Gasket fractal, which provides a clear visualization of the recursive nature of its solution. This connection is strengthened by a group structure, which exhibits the same self-similarity. The fractal nature of the group lends itself to a natural description in terms of a finite state automaton a theoretical model for computing. In this expository talk, I will introduce the various concepts involved, and illustrate how these seemingly disparate concepts are tightly intertwined.

Spring 2012

Friday, 3 February, 3:00 pm, Fitzelle 206

From the Four Color Theorem to Thompson's Group F

Garry Bowlin, Mathematics, SUNY Oneonta

Abstract:

The Four Color Theorem states that given any map on the sphere (or plane) one can color the map with four colors so that no two regions that share an edge are the same color. The question was first posed by Francis Guthrie in the early 1850's and likely publicized by him in The Athenæum in 1854. We will discuss various reformulations of the Four Color Theorem, which will take us from the world of graphs and topology into the realm of group theory. No prior knowledge of group theory or graph theory is necessary.

Friday, 2 March, 3:00 pm, Fitzelle 206

Statistics: The Good, the Bad, and the Ugly

Grazyna Kamburowska, Statistics, SUNY Oneonta

Abstract:

``There are three kinds of lies: lies, damned lies, and statistics.'' The statement, attributed to Benjamin Disraeli, refers to the persuasive power of numbers, the use of statistics to bolster weak arguments, and the tendency of people to disparage statistics that do not support their positions. There are many researchers who are passionate about exposing poor studies but, unfortunately, the incorrect use of statistics is still common. Many researchers still shy away from the rigorous application of statistical methods or, worse, use them incorrectly. We will discuss various examples of the misuse and abuse of statistics.

Friday, 30 March, 3:00 pm, Fitzelle 206

Perturbations of Fourier Bases and the Haar Wavelet

Min Chung, Mathematics, Hartwick College

Abstract:

A Riesz basis is the image of an orthonormal basis under an invertible continuous linear mapping. Both orthononal basis and Riesz basis provide us with a simple representation of an element in Hilbert space. Since perturbing an orthonormal basis in a controlled manner yield a Riesz basis, this is an important subject of study which goes back to Paley and Wiener who were interested in the question of which perturbations {1/(2π)eλnx} of the orthonormal basis {1/(2π)enx} are still a Riesz basis for L2[-π,π]. In this context, it is then natural to consider when the sequence {&lambdan-n} is small, so that the perturbations of local Fourier bases are still a Riesz basis. In this talk, first we find Riesz basis for L2[0,1] of the form {sin(λkx)}, by perturbing the local sine and cosine orthonormal bases of Coifman and Meyer. Second, the Haar function, the simplest compactly supported but discontinuous wavelet. By using the properties of Bessel sequences, we provide an explicit and more practical way of constructing locally continuous perturbations of the Haar wavelet. In this way, we can generate the continuous or even smooth perturbation of the Haar wavelet which then will lead to an alternative way to the approach given by Aimer, Bernardis, and Gorosito, but we provide better conditions for perturbations and frame bounds.

Friday, 27 April, 3:00 pm, Fitzelle 206

Ruler Constructions

Marius Munteanu, Mathematics, SUNY Oneonta

Abstract:

Ruler and compass constructions have a rich and fascinating history going back more than two thousand years. We will see that many of these constructions can be carried out with a ruler alone, as long as an appropriate ``starter set'' is given.

Fall 2011

Thursday, 1 September, 3:20 pm, Fitz 308

From Galaxy Clusters to Cosmology: Using Mathematics to Solve Some of Astronomy's Biggest Problems

Parker Troischt, Physics, Hartwick College

Abstract:

In the last decade or so, astronomy has advanced extremely rapidly due to technological advances used in telescopes, satellites and large sky surveys. We now have quantitative data on hundreds of extra-solar planets, countless stars and over a million galaxies. Future telescopes like the Large Synoptic Survey Telescope (LSST) promise to extend this much further and will survey the entire visible night sky every three days. Here, we discuss how mathematics is used to study the properties of everything from nearby stars, to galaxies, to clusters of galaxies located over 100 million light years away. In addition, we will show how solutions to Einstein's equations can be used to study some of the most interesting questions about the universe starting with the Big Bang.

Friday, 23 September, 3:00 pm, Fitz 221

Cantor's Diagonalization Revisited: Constructing Transcendental Numbers

Luise-Charlotte Kappe, Mathematics, Binghamton University

Abstract:

An evolving awareness of the deep nature of the real numbers began over 2,500 years ago, when the Pythagoreans were startled by their discovery that numbers such as the square root of 2 were not rational. A recurring theme in their history is that the set of real numbers is richer and much more complex than is generally assumed. The demonstration by Cantor, that the reals cannot be enumerated, is a well-known landmark of these developments. Knowing that the rationals can be enumerated, it follows from Cantor's diagonalization that there exist irrational numbers. Similarly, knowing that the algebraic numbers can be enumerated, it follows that there exist transcendental numbers. But can one use Cantor's diagonalization for the construction of such numbers? The topic of this talk is the explicit construction of a transcendental number using Cantor's diagonalization.

Friday, 21 October, 4 pm, Fitz 221

Computational Thinking in 24-Point Card Game

Sen Zhang, Computer Science, SUNY Oneonta

Abstract:

The 24-point game is a mathematical game in which the object is to construct an arithmetic expression using four integers (usually from 1 to 10) and three out of four possible elementary arithmetic operations (addition, subtraction, multiplication and division) so that the expression evaluates to 24. For example, given a hand of four integers 4,7,8,8, (7-8/8)*4 would be a possible solution. Notice that all four integers need but not every operation needs to be used in the expression. The game is relatively easy but entertaining enough to play for almost any people who have elementary arithmetic knowledge. Two questions that tend to captivate many people who have played the game are how to find out all the hands that have at least one possible solution and how to find out all possible solutions for each of such hands. Obviously these are the problems we don't want to solve manually. This talk will first examine a set of mathematical and computational thinking techniques that can help answer the above questions. After that, a straightforward software that implements the techniques will be presented as a typical computing solution where mathematical thinking and computational thinking go hand in hand perfectly.

Friday, 11 November, 3pm, Fitz 221

From Ordinary Differential Equations to Geometric Control Theory

Laura Munteanu, Mathematics, SUNY Oneonta

Abstract:

Many real life processes can be modeled by (systems of) ordinary differential equations. More complex processes involve a parameter/control that can be adjusted in order to affect their outcome. Due to the large number of variables or the nonlinear nature of these control systems, one seeks to find simpler systems whose solutions mimic (in a certain sense) the solutions of the original system. In this presentation, we look at how geometric objects such as differentiable manifolds and vector fields can be used to better understand the aforementioned problem.

Spring 2011

Friday, 4 February

Chess Endgame Composition: Proofs in Pictures

Robert Sulman, Mathematics, SUNY Oneonta

Abstract:

Although primarily viewed as a game to be played between two individuals, the very nature of chess leads to configurations of the pieces (called "positions") in which one player has a forced win, or a forced draw. Such positions have been composed as well, that is, they have not come about naturally from an actual game. The freedom to create such problems enables one to highlight a given theme in which the pieces interact on the 64 squares. Whether one tries to solve such a composition, or simply read the solution, the moves leading to the end result are intended to be surprising and beautiful. A brief review of the moves of the pieces will be followed by a series of endgame studies (as they are also called), beginning with relatively simple examples. The more subtle studies, including those by pioneer Alexey Troitsky will (I hope) draw you into this wonderful area of chess.

Friday, 4 March

Fibonacci's Not Just Counting Rabbits Anymore

Gary Stevens, Mathematics, Hartwick College

Abstract:

In 1202, Leonardo of Pisa, son of Bonacci, published his pioneering work, Liber Abaci, the book of calculation. In this book, Fibonacci presented a problem relating to the breeding of rabbits whose solution gave rise to a sequence of numbers which now bears his name. The Fibonacci Sequence has been studied and generalized for the last 800 years and is now one of the cornerstones of the area of mathematics known as combinatorics, the art of counting. The sequence shows up in some unexpected places and provides solutions to many counting problems. This lecture will look at some of the myriad uses of the sequence and will provide a gentle overview of and introduction to the subject of combinatorics.

Friday, 8 April

Techniques in Atmospheric Dynamics and Weather Forecasting

Melissa Godek, Meteorology, SUNY Oneonta

Abstract:

Within the field of meteorology, the study of atmospheric dynamics focuses on macro-scale motions and their ability to create the weather phenomena and climate patterns experienced at the surface. These circulations are described by fundamental and apparent Earth forces as well as the governing equations of motion. It is through calculus that atmospheric scientists learn how to apply these equations in dynamics to obtain a new set of equations referred to as Quasi-Geostrophic Theory. Just as important to meteorologists are Numerical Weather Prediction (NWP) models, which are used every day in short- and long-term weather predictions. These NWP models are essentially algorithms that represent the Quasi-Geostrophic Theory that describes macro-scale atmospheric circulations. This talk will describe how atmospheric scientists incorporate the science, mathematics and computer model predictions to produce multiple forecasts each day.

Friday, 29 April

Not So Boring Statistics

Jen-Ting Wang, Statistics, SUNY Oneonta

Abstract:

In this talk, we'll discuss three real court cases, which involved different types of statistical reasoning. In particular, we'll see the statistical evidence found in a serial killer case and how it was used in court. This presentation is accessible to everyone and no previous statistical knowledge is required.

Fall 2010

Friday, 10 September

Bounded Analytic Functions with Unbounded Parts

Angeliki Kazas-Postisakos, SUNY Oneonta

Abstract:

For any analytic function on the unit disc, f &isin Hp, there exists a factorization f(z)=g(z)F(z) where g is inner and F is outer. The properties of g and F are well known and in particular the function g is a bounded analytic function on the unit disc. More generally, for an inner function φ, there exists a factorization f(z)=h(z)F(φ(z)) where h is a φ-p inner function and F is outer. We will construct a function that is bounded and analytic on the unit disc with unbounded φ-p inner part. All the terms above will be defined and the talk should be accessible to students that have completed the calculus sequence.

Friday, 8 October

The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof

V. Frederick Rickey, US Military Academy at West Point

Abstract:

The Fundamental Theorem of Calculus (FTC) was a theorem with Newton and Leibniz, a triviality with Bernoulli and Euler, and took on the concept of "fundamental" when Cauchy and Riemann defined the integral. FTC became part of academic mathematics in the 19th century, but waited until the 20th century to take hold in classroom mathematics. We will discuss the transition from clear intuition to rigorous proof that occurred over three centuries.

Friday, 5 November

Mathematics and the Methods and Models of Morality

Michael Green, Philosophy, SUNY Oneonta

Abstract:

Mathematics and moral theorizing have had a long and tangled history. Philosophy has nurtured mathematical forms of thought that have, in turn, had a profound influence on ethical theorizing. The aim of this lecture is to reflect upon the relationship between mathematics and moral theorizing in Plato, Aristotle, Augustine, Hobbes, Spinoza, Bentham, and Rawls.

Friday, 3 December

On Symmetries, Supersymmetries, and Akan Symbolism

Michael Faux, Physics, SUNY Oneonta

Abstract:

Some of the intricate mathematics used by physicists to probe the laws of nature are rendered helpfully perspicuous by the use of symbolic representations. In some cases, such symbols exhibit a mathematical significance which transcends their original motivation. In this talk I will explain how a colorful graphical paradigm, reminiscent of West African tribal symbols, invented to represent supersymmetry algebras, which I will explain, have exposed an unexpected, fascinating, and rich connection between an emergent quantum theory of gravitation and the mathematics of coding theory.

Spring 2010

Friday, 5 February

Math Anxiety

Lynne Talbot, SUNY Oneonta

Abstract:

The presentation will outline the causes and symptoms of math phobia, as well as suggestions for instructors to alleviate the anxiety that some students feel when confronted with a math problem. Myths regarding math, such as math is not creative, will be dispelled. A pamphlet will be distributed for instructors to reference to assist in identifying math phobic students.

Friday, 5 March

Knot a Graph? Why Not?

Susan Beckhardt, SUNY Albany

Abstract:

Choose any seven points in space, and connect each pair of points with a curve in such a way that none of the curves intersect. No matter how you arrange your points and curves in space, I can always find a closed loop that is tied in a knot. The choice of points and curves is called a spatial embedding of K7, the complete graph on seven vertices, and a graph with the property that every spatial embedding has a knotted cycle is said to be intrinsically knotted. Although topological in nature, the remarkable fact that K7 has this property can be proved with little more than some basic combinatorics. We'll go over the proof, and then look at some further results in the field of intrinsically knotted graphs.

Friday, 26 March

Embarrassing Moments in the History of Calculus

Kim Plofker, Union College

Abstract:

The development of modern calculus wasn't the smooth transition pictured in today's textbooks, but rather a long struggle between practicality and precision. Handling the tricky quantities known as infinitesimals, which might be zero or not as the situation required, led early modern mathematicians into some awkward contradictions and some vehement disputes. This talk surveys some of the peculiar innovations in calculus that you won't find in your calculus book, and the controversies, shock, and outrage that they provoked in their day.

Friday, 16 April

The Power Residue Problem and Arithmetic Geometry

John Cullinan, Bard College

Abstract:

If an integer is an nth power, then it is an nth power mod p for all primes p. However, the converse is not always true - there exist integers which are nth powers mod p for all primes p, yet are not the nth power of an integer. In general, given a polynomial with integral coefficients, it can be quite difficult to describe its integral (or rational) roots. One technique that has proved fruitful is instead to find roots of the polynomial mod p for prime numbers p, and from these roots deduce the existence (or non-existence) of a rational root. The success or failure of this approach often gives new insight into the original problem. This talk will focus on the example given above, and a similar problem phrased in terms of points on elliptic curves and abelian varieties.

Friday, 7 May

Let's Pack! An Introduction to Hyperspheres and Hypercubes

David Biddle, SUNY Oneonta

Abstract:

It is commonly said that beauty is in the eye of the beholder, but what if the eye is incapable of seeing something in its entirety? Higher dimensional geometry affords us the opportunity to explore beautiful objects and phenomena using a plethora of tools from all parts of mathematics. In this talk we compare the 'look and feel' of higher dimensional analogues of the circle and sphere (hyperspheres) and the square and cube (hypercubes) and come to the conclusion that exploring higher dimensions leaves plenty of room for the bizarre!

Fall 2009

Friday, 11 September

M-Zeroids: Structure and Its Effect on the Additive Operation

J. Palmatier, SUNY Oneonta

Abstract:

An m-zeroid is an algebraic structure with both operations and an inherent order on its elements. If we remove the order by making it totally ordered and finite, only the additive operation remains important. In this talk, we will discuss how the structure of the finite, totally-ordered m-zeroid, both algebraic and pictorial, restricts the additive operation table and allows you to generate such an m-zeroid with a minimum of fuss.

Friday, 9 October

The Mathematics of Origami

L. Bridgers, SUNY Oneonta

Abstract:

With paper folding, we can complete geometric constructions that are impossible using the classical geometry tools of a straight edge and compass. In this talk we will explore some origami constructions and the geometry proofs behind them. This will include both constructions that we could complete with a straight edge and compass, such as the construction of an equilateral triangle, and those that are impossible using a straight edge and compass, such as the trisection of an angle.

Friday, 6 November

Number Theory and Music

L. Alex, SUNY Oneonta

Abstract:

In this talk connections between musical intervals and Diophantine equations in number theory will be described. In particular the intervals will be viewed as "superparticular" ratios of the form (n+1)/n. In this form the octave would be viewed as the ratio 2/1. The ten superparticular ratios corresponding to the intervals preferred by the Western ear will be listed. Each of these ratios corresponds to a solution of a certain Diophantine equation in number theory. An elementary number theoretic method for solving the equation will be illustrated.
Diophantine equations in number theory will be described. In particular the intervals will be viewed as "superparticular" ratios of the form (n+1)/n. In this form the octave would be viewed as the ratio 2/1. The ten superparticular ratios corresponding to the intervals preferred by the Western ear will be listed. Each of these ratios corresponds to a solution of a certain Diophantine equation in number theory. An elementary number theoretic method for solving the equation will be illustrated.

Friday, 4 December

Disappearing Messages: Basic Steganography

J. Ryder, SUNY Oneonta

Abstract:

The practice of steganography involves surrepetitiosly hiding a secret within some object at a source location then allowing that object to be carried to some destination. At the destination, a receiver extracts the secret from the carrier. Steganography is secret communication in which only the sender and receiver of the secret know of its existence yet it travels through hostile territory unnoticed. In Ancient Greece, troops at one location would tattoo a secret onto a slave's shaved head. When the slave's hair had once again become sufficiently long, the slave would be sent to another camp some distance away. Upon arrival, the slave's head was shaved and the secret message was read. This brief talk will show a few ways that steganography is used today in the digital world. Examples will show methods of hiding secrets in images, in Internet web pages, in music, and even in plain text.