Arthur Benjamin, Harvey Mudd College
In this course, you will learn about the mathematics that underlies many of the great games and puzzles that people enjoy today. Imagine impressing your friends, students, or fellow classmates with your ability to solve the Rubik's cube or almost any Sudoku. Learn the optimal basic strategy for playing blackjack, along with some simple card counting techniques. Learn the mathematics needed to play great poker or expert backgammon. Since you're a mathematician, most people assume that you're already good at these things. This course will teach you those skills and you'll learn some fun mathematics along the way. The game plan includes:
Great Expectations and Winning Wagers
Optimal Blackjack and Simple Card Counting
Games You Can't Lose and Impossible Puzzles
How to Solve and Understand Rubik's Cube
Zero Sum Games and Practical Poker Probabilities
Solving Sudoku and KenKen
Chess and Games of Pure Strategy
Anna Barry, Institute for Mathematics and Its Applications, Hans Kaper, Georgetown University, Richard McGehee, University of Minnesota, Samantha Oestreicher, University of Minnsota, James Walsh, Oberlin College, Esther Widiasih, University of Arizona, and Mary Lou Zeeman, Bowdoin College
In this two-day short course, the presenters will introduce various conceptual models of the Earth's climate system. The first day will be devoted to Energy Balance Models (EBMs)—differential equations which express the physical law of energy conservation in mathematical terms. It will be shown how the models can be modified to include the effects of greenhouse gases and the ice-albedo feedback mechanism. The second day will be devoted to paleoclimate studies. It will be shown how observational data from the paleoclimate record and computational data from simulations of the Earth's orbit during the Pliocene and Pleistocene can be incorporated into EBMs.
During the two-day course, participants will have the opportunity to conduct hands-on simulations with models to explore the interplay between energy balance, ice-albedo feedback, Milankovitch cycles in Earth's orbit, and other feedback mechanisms. This will build insight into which features of the paleoclimate record can be explained by the dynamics of low-dimensional conceptual models. Modules for bringing the material into a range of core undergraduate mathematics classes will be provided.
Zero-dimensional Energy Balance Models, Hans Kaper, Georgetown University
Hands-On: Dynamics of Energy Balance Models, Anna Barry, Institute for Mathematics and its Applications, and Samantha Oestreicher, University of Minnesota
One-dimensional Energy Balance Models, Hans Kaper, Georgetown University
Hands-On: Dynamics of Energy Balance Models, Anna Barry, Institute for Mathematics and its Applications, and Samantha Oestreicher, University of Minnesota
PaleoClimate Data, Milankovitch Cycles, and Extending Energy Balance Models, Richard McGehee, University of Minnesota
Hands-On: Comparing Energy Balance Models with the PaleoClimate Record, Richard McGehee, University of Minnesota, and Esther Widiasih, University of Arizona
The Greenhouse Effect in Energy Balance Models, Jim Walsh, Oberlin College
Hands-On: Green House Gas Effect Explorations, Anna Barry, Institute for Mathematics and its Applications, and Esther Widiasih, University of Minnesota
Satyan L. Devadoss, Williams College, and Joseph O'Rourke, Smith College
Although geometry is as old as mathematics itself, discrete geometry only fully emerged in the 20th century, and computational geometry was only christened in the late 1970s. The terms “discrete” and “computational” fit well together as the geometry must be discretized in preparation for computations. “Discrete” here means concentration on finite sets of points, lines, triangles, and other geometric objects, and is used to contrast with “continuous” geometry, for example, smooth manifolds. Although the two endeavors were growing naturally on their own, it has been the interaction between discrete and computational geometry that has generated the most excitement, with each advance in one field spurring an advance in the other. The interaction also draws upon two traditions: theoretical pursuits in pure mathematics and applications-driven directions often arising in computer science. The confluence has made the topic an ideal bridge between mathematics and computer science. It is precisely to bridge that gap that we hope to accomplish with this short course.
The material covered is accessible to faculty and scholars at several different levels, whether they are interested in teaching or research: whether teaching students at an advanced high school level, a collegiate setting, or at the graduate level, and research specifically on the topics covered or in allied fields. The reason this course allows for such breadth is due to the subject material. A solid understanding of proofs is all that is needed to tackle some of the most beautiful and intriguing questions in this field. Moreover, a strong intuition of this subject can be obtained and developed through visualization. Due to the relative youth of the field, there are many accessible unsolved problems, which we highlight throughout the course. Although some have resisted the assaults of many talented researchers and might be awaiting a theoretical breakthrough, others may be accessible with current techniques and only await significant attention by an enterprising researcher. The field has expanded greatly since its origins and now the new connections to areas of mathematics and new application areas seems only to be accelerating. We hope this course can serve to open the door on this rich and fascinating subject. (link)
Holly Gaff, St John's University/College of St. Benedict and Jennifer Gallovich (Old Dominion University)
The purpose of this short course is to introduce participants to a range of curren topics in mathematical biology. Moreover, mathematical biology has exploded in recent years, developing new perspectives on parent disciplines by combining biological and mathematical ideas and tools in sometimes unexpected ways. So we also hope that this short course will begin a continuing conversation on how we might integrate such modern applications in the undergraduate mathematics program.
Planting Seeds by Mining the Mind: Examples from Mathematical Neuroscience Get Students Thinking by Janet Best, Ohio State University and Mathematical Biosciences Institute.
Some Applications of Game Theory in Evolutionary Ecology by Philip Crowley, University of Kentucky.
Can Math Cure Cancer? by Renee Fister, Murray State University
Ticks Can Give You More Than the Creeps—Mathematical Modeling of Tick-Borne Diseases by Holly Gaff, Old Dominion University
Drug, Sex, and Rock 'n' Roll: Biology Examples to Motivate Undergraduate Math Classes by Lou Gross, Director, NIMBioS, and University of Tennessee Knoxville
Tree Tapping and Network Mapping: Adventures in, and Applications of, Modern Discrete Mathematics by Terrell Hodge, Western Michigan University
Modeling the Dynamics of Biological Networks by Winfried Just, Ohio University
Nancy Ann Neudauer, Pacific University
Gian-Carlo Rota said that "Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day."
Hassler Whitney introduced the theory of matroids in 1935 and developed a striking number of their basic properties as well as different ways to formulate the notion of a matroid. As more and more connections between matroid theory and other fields have been discovered in the ensuing decades, it has been realized that the concept of a matroid is one of the most fundamental and powerful in mathematics. Examples of matroids arise from networks, matrices, configurations of points, arrangements of hyperplanes, and geometric lattices; matroids play an essential role in combinatorial optimization. We all know some matroids, but not always by name.
In mathematics, notions of independence akin to linear independence arise in various contexts; matroids surface naturally in these situations. We provide a brief, accessible introduction so that those interested in matroids have a place to start. We look at connections between seemingly unrelated mathematical objects, and show how matroids have unified and simplified diverse areas. (link)
Neal Calkin, Clemson University and Dan Warner, Clemson University
In recent years, a new piece of mathematical software has appeared on the scene: Sage (www.sagemath.org) is an open source package capable of doing high-powered symbolic and numerical computations. It features a web-based notebook interface, local or remote operation, and can interact with other packages, both open source and commercial (if available). In this short course we will introduce the package, giving multiple examples of how to use it for mathematical explorations, both elementary and advanced. We will focus on algebraic and combinatorial investigations.
Amy Shell-Gellasch, Beloit College, and Glen Van Brummelen, Quest University
Every intellectual endeavor has key moments when some new monumental work shakes its foundations and builds new ones. How these great books affect the future might be clear within months of publication, or may take centuries to develop; in ways we may not even fully recognize, they shape our thoughts. We shall concentrate on only four great books in mathematics, spending half a day on each. We shall delve deeply into the texts, translations, and commentaries, do some reading in each of the original texts, and consider their influences on later generations. (link)
Alex Jones, New York University, Ptolemy's Almagest: Greek mathematics and the heavenly bodies
George Smith, Tufts University, Newton's Principia
Ivor Grattan-Guinness, Middlesex University Business School, Tracking the great writings of mathematics
Robert E. Bradley, Adelphi University, and Ed Sandifer, Western Connecticut State College, Cauchy and the Cours d'analyse
Fernando Q. Gouvêa, Colby College, How algebra became modern
Steven Shreve, Carnegie Mellon University
Over the past 20 years, mathematical methods have permeated the finance and insurance industries. Universities have responded by offering undergraduate courses or degree programs in mathematics related to finance. This short course is based on the core of such a program at Carnegie Mellon. The purpose of this short course is to acquaint potential undergraduate instructors of financial mathematics with the main financial concepts and mathematical methodology that one can include in an undergraduate curriculum on this subject.
The first part, Introduction to Mathematical Finance, requires only that students are familiar with differential calculus. It presents calculations related to loans, annuities and bonds, no-arbitrage pricing of derivative securities, and mean-variance analysis.
The second, Discrete-Time Finance, requires students to understand probability on finite event spaces. It covers dynamic models for financial markets within that context and a derivation of the Nobel-Prize-winning Black- Scholes formula as a limit of a discrete model.
The third part, Continuous-Time Finance, expects students to know calculus-based probability and have the facility to handle analysis arguments at an undergraduate level. It introduces Brownian motion and stochastic calculus, and then derives the Black-Scholes formula within this context. We conclude with an introduction to problems in optimal consumption and investment, which provide opportunities for student projects in financial mathematics.
Richard D. De Veaux
There are two main themes. It will serve as a practical introduction to and an overview of data mining. It will also highlight some of the ways that technology has changed the way we practice and teach statistics. Forty years ago the emphasis in introductory statistics was on formulas and their calculation. For example students were taught the formula for standard deviation and learned alternatives for avoiding rounding errors and short cuts for grouped data. Technology has made much of that subject matter irrelevant and obsolete. Today, we have been freed by technology to focus on the concepts of data analysis and inference. Where is this trend taking us? Computational methods in statistics are rendering some of our methods obsolete as well. How much should be introduced in the introductory statistics course? Data mining is the exploration and analysis of large data sets by automatic or semiautomatic means with the purpose of discovering meaningful patterns. The knowledge learned from theses patterns can then be used for decision making via a process known as ’knowledge discovery.â? Much of exploratory data analysis and inferential statistics concern the same type of problems, so what is different about data mining? What is similar? In the course I will attempt to answer these questions by providing a broad survey of the problems that motivate data mining and the approaches that are used to solve them. The course will start with an overview of how the introductory statistics course is taught today and what the main concepts are. Examples of how technology enables us to get to the heart of the subject early will be given. Some elementary modeling concepts will be reviewed before we embark on an introduction to data mining. Then, we will use case studies and real data sets to illustrate many of the algorithms used in data mining. The applications will come from a wide variety of industries and include applications from my personal experiences as a consultant for companies that deal with such topics as financial services, chemical processing, pharmaceuticals, and insurance.
Michael A. Jones, Montclair State University
The object of this short course is to learn about both the mathematical techniques that collectively can be called game theory and the range of applications that can be modeled using these techniques. Techniques will include simultaneous and sequential move games under different information assumptions, cooperative games, mechanism design, theory of moves ’ a dynamic extension of game theory, and a qualitative approach to evolutionary game theory. Applications will be drawn from biology, economics, environmental science, literature, political science, and popular culture.
Michael A. Jones, Montclair State University
Anytime one person’s decision can affect another person’s outcome is a situation that can be modeled by game theory. This cocktail party description hints at how game theory can rightfully be considered a collection of tools and techniques for modeling diverse applications. For two-person, simultaneous move games, I will discuss how Nash’s equilibrium solution generalizes optimization in decision theory and Von Neumann’s Minimax Theorem for zero-sum games. I will conclude with an overview of the short course to demonstrate how game theory has evolved from these historic roots. non-cooperative
Paul Coe, Dominican University
Besides sharing an adjective, what do game shows have to do with game theory? I will introduce concepts and well-known games (e.g., the Prisoner’s Dilemma) from non-cooperative game theory by using actual games from television game shows including The Price is Right and Friend or Foe, among other sources. Optimal behavior for these games will demonstrate different solution concepts for both simultaneous and sequential move games.
D. Marc Kilgour, Wilfrid Laurier University
I will highlight the difference between Nash and subgame-perfect equilibria for games in which players move sequentially and explain how subgame-perfect equilibria use a stronger criterion of rationality to refine Nash equilibria to a more compelling (or demanding) solution. I will extend subgame-perfect equilibria to games of imperfect information and incomplete information. Applications will include models of deterrence and truels (3- person duels).
Jennifer Wilson, New School University
Cooperative game theory models situations in which players form coalitions whose value is greater than the sum of their parts. In this talk, I will discuss several well-known methods, including the core and Shapley value, which assign player’s values based on the coalitions that they can join. Applications include sharing the cost of building an airport runway and cleaning up a polluted river, as well as determining power in voting games. I will discuss recent extensions of these ideas to multi-choice and fuzzy games.
Michael Rothkopf, Penn State University
Auctions are a particularly structured form of competition that invites formal analysis. This talk will review briefly the results from the game theoretic literature on single, isolated auctions. It will then raise issues related to modeling auctions and argue that improved models produce significantly, and sometimes radically, different results. Some of these results can be obtained using game theory, but some come from disciplines that are less demanding mathematically.
Steven J. Brams, New York University
Emotions such as anger, jealousy, and love would seem to be spontaneous feelings that overtake us suddenly and hence not the product of careful means-ends analysis that we normally associate with rational choice. On the contrary, I argue that the passionate pursuit of certain ends may be eminently rational in expressing strong commitment, extreme frustration, and the like, which in turn affect the responses of others in gamelike situations. I will use ’theory of moves,â? a dynamic extension of game theory, to illustrate this thesis, focusing on frustration and its most common manifestation in anger. My principal sources will be literary, from the Bible to Shakespeare to such modern authors as William Faulkner and Joseph Heller.
Donald G. Saari, University of California, Irvine
Evolutionary game theory has proved to be popular in explaining different social and biological behavior. Unfortunately the approach is too difficult for most to use and it is very difficult to accept the ’behavioral dynamics.â? A new, easily understood approach is introduced to avoid these problems. Panel discussion: Game Theory In and Out of the Classroom Not only has game theory been successfully taught in economics and political science departments, game theory has been an integral part of non-major, general education math courses and has been a popular, yet infrequent math major elective. We will discuss how game theory can also be introduced in math major courses like calculus, combinatorics, probability, and differential equations. Further, we will discuss areas of open research that would be suitable for faculty and for faculty/student collaborations.
Attend this short course to learn more about the development of a wide range of combinatorial topics, from earliest times up to the present day and beyond. The topics presented will include early combinatorics from non-Western traditions, European combinatorics during the Renaissance, the combinatiorial work of Leonhard Euler, and various topics from the modern era. Early Combinatorics (up to the 17th century): Andrea Breard, China Victor Katz, Combinatorics in the Islamic and Hebrew traditions Europe Before and After Euler: Eberhard Knobloch, European combinatorics, 1200’1700 Robin Wilson, Early graph theory and Cayley’s work on trees, to the early attempts to solve map-coloring problems George Andrews, Euler’s ’De Partitio Numerorumâ? Lars Andersen, Latin squares Robin Wilson, Triple systems, schoolgirls, and designs Combinatorics Comes of Age: Lowell Beineke, 20th-century graph theory Herb Wilf and Lily Yen, Sister Celine as I knew her Bjarne Toft, The game of Hex: History, results and problems Toward the Future: Ronald L. Graham, Combinatorics: The future and beyond
(in cooperation with the joint meetings of the Society for Mathematical Biology the Japanese Society for Mathematical Biology)
John R. Jungck
Many mathematics educators are faced with the challenge that the majority of students enrolled in their classes are from the broader life sciences (e.g., biology, allied health, environmental sciences, agriculture, etc.), while most mathematicians have very little background in the life sciences themselves. Therefore, the MAA has chosen to meet this year in combination with the joint meeting of the Society for Mathematical Biology and the Japanese Society for Mathematical Biology. This short course, while preceding MathFest, is concurrent with those joint meetings and has the advantage that participants will not only be able to be involved in the short course, but will also be able to attend the plenary lectures of those societies as guests of the societies at no additional cost. Besides the Society for Mathematical Biology and the SIGMAA on Mathematical Biology, the individual lecturers in the short course also represent several organizations committed to the inclusion of much more mathematics in biology education and much more biology in mathematics education: the BioQUEST Curriculum Consortium (in particular, several of its projects: NUMBERS COUNT! [Numerical Undergraduate Mathematical Biology Education: exploRing with Statistics, Computation, mOdeling, and qUaNtitative daTa]; the Biological ESTEEM Project [Excel® Simulations and Tools for Exploratory, Experiential Mathematics]; the BEDROCK Project [Bioinformatics Education Dissemination: Reaching Out, Connecting, and Knittingtogether] http://www.bioquest.org); and CoMBiNe: [the Computational and Mathematical Biology Network] http://muweb. marymount.edu/~eschaefe/combine/welcome.htm). Biological subjects will include evolution, ecology, epidemiology, biometrics, genetics, bioinformatics, microbiology, and biochemistry. Mathematical subjects will include probability and statistics, linear algebra, differential equations, combinatorics, number theory, graph theory, and geometry. The examples employed will be appropriate for inclusion in courses aimed at the first two years of the undergraduate curriculum and will serve to introduce mathematicians to many current avenues of research in mathematical biology, as well.
Raina Robeva, Sweet Briar College
This part of the course will focus on biological and medical models that utilize methods from the fields of probability and statistics. We will begin with examples from genetics to illustrate the binomial, normal, and Poisson distributions and discuss the underlying biological mechanisms and mathematical connections. More specifically, we will outline the experiments of Nilsson ’ Ehle and discuss the emergence of quantitative traits based on the Central Limit Theorem. We will examine the Luria-DelbrÃ¼ck experiments and show how using a Poisson distribution to describe the count of resistant bacterial variants allows for statistically distinguishing between the hypothesis of mutation to immunity and the hypothesis of acquired immunity. Next, we will examine some medical models for risk assessment, such as assessing the risk for hypoglycemia in diabetes, quantified from self-monitoring blood glucose data, and the risk for neonatal sepsis, quantified from electrocardiographic (EKG) data.
Anton E. Weisstein, Truman State University
Population geneticists apply a wide range of mathematical techniques in seeking to understand and predict changes in the genetic makeup of real-world populations. In this session, we will: (1) review the recursion equations for calculating allele frequencies under the assumptions of Hardy-Weinberg Equilibrium, (2) mathematically model the effects of specific evolutionary forces, such as selection and migration, and (3) apply linear algebra to understand why natural selection disfavors a specific genetic variant that provides the best-known resistance to malarial infection. These investigations will introduce some of the Excel tools from the BioQUEST Consortium’s Biological ESTEEM collection.
Jennifer R. Galovich, St. John’s University and the College of St. Benedict
RNA folding, Smith-Waterman Sequence Alignment, and other topics will be presented in the context of a new bioinformatics course taught in an undergraduate institution’s mathematics department by an applied combinatorist who spent her sabbatical last year at the Mathematical Biosciences Institute at Ohio State University and with the BEDROCK Project (Bioinformatics Education Dissemination: the Reaching Out, Connecting With, and Knitting Together BioQUEST Curriculum Consortium at Beloit College).
Holly D. Gaff, University of Maryland School of Medicine
A wide variety of mathematical models have been used to study an equally wide variety of infectious diseases. We will discuss the basics of infectious disease epidemiology, the building blocks for models, the types of mathematical approaches, and the history of epidemiology models. We will walk the examples of disease models, including measles and tick-borne diseases.
Gretchen A. Koch, Goucher College
When using mathematics to model biology, one must decide the level at which to present the material. In this session, I will present several modules from the BioQUEST Consortium’s Biological ESTEEM collection and demonstrate to the audience how each module can be used at varying levels of mathematical and biological ability. The modules will include a logistic growth model, a competing species model, and an SIR epidemiological model. Time permitting, an additional application based in MATLAB will be demonstrated to compare and contrast the ESTEEM competing species model.
John R. Jungck, Beloit College
Graph theory is generally applicable to many areas of biology, including pedigrees and multiple allele genetic graphs in genetics, fate maps in developmental biology, phylogenetic trees in evolution and systematics, metabolic pathways and RNA folding in biochemistry, interactomes in genomics-molecular biology, restriction maps in biotechnology, food webs in ecology, infection contact maps in epidemiology, and Delaunay triangulations in image analysis. Despite this breadth of utility, there has been a lack of easy-to-use tools for entering biological data into graph visualization packages with tools for graph theoretical analysis. BioGrapher is an Excel® and open source graph visualization package for importing, illustrating, and analyzing biological data. Interval graphs, planar graphs, trees, de Bruijn graphs (Euler paths), n-cubes (Hamiltonian paths), and Voronoi tessellations-Delaunay triangulations will be illustrated through biological examples.
Julius H. Jackson, Michigan State University
Number theory is used in a study of bacterial and archaeal genomes as information systems that determine the physiological states of an organism. The larger goal is to model the dynamics of information evolution and exchange in prokaryotes and to derive the theory base to explain the origin, evolution, and function of genes and chromosomes. Our goal is to discover and model gene-specific and genome-specific information that defines metabolic properties and physiological behavior of prokaryotes in adaptive response to their environment(s). The limits of coding space, protein mobility, and variation space are explored to understand the physiological consequences of such limits. This work utilizes experimental methods for genetic, molecular biological, biochemical, and microbiological studies in combination with mathematical and computational methods for modeling and simulating the function of natural systems. My teaching approach is to prepare students to view organisms and their environments as biological systems, to ask critical questions about how these systems work and interact, and to design experiments that yield quantitative assessments of systems behavior that will lead to construction of mathematical models for simulation.
Claudia Neuhauser, University of Minnesota
Today, most undergraduate biology majors take quite a bit of basic quantitative coursework early on, but then they never see it again. A few years later, when they’re graduate students, they encounter the new world of biology, full of massive amounts of data and analysis, and they’re not prepared. We’ve got to change that. Neuhauser will emphasize the need to train faculty in quantitative techniques and teaching. She envisions adding mathematically themed guest lectures to classes and possibly holding teaching workshops for faculty, as well as working with faculty one-on-one. She believes that this calls for a "logical step" in incorporating quantitative techniques across the curriculum. For several years, my goal has been to develop at least two solid years of undergraduate quantitative training for our biology majors. Now, we can do so much more.
Edward Sandifer and Robert Bradley
In 2007 we celebrate the 300th anniversary of Euler’s birth. As the preparation and publication of more than 70 volumes of his works and correspondence in the Opera Omnia begins to wind down, this is a fitting occasion to take an in-depth look at what Euler did and how it fit in the context of his own times. Seven historians of mathematics will describe their recent work on Euler, his life, times, science and mathematics. The program includes the following lectures:
A mathematical life in the enlightenment, Ronald S. Calinger, Catholic University of America
Leonhard Euler and the function concept, Ruediger Thiele, University of Leipzig
D’Alembert, Clairaut and Lagrange: Euler and the French mathematical community, Robert E. Bradley
Enter, stage center: The early drama of hyperbolic functions in the age of Euler, Janet Barnett, Colorado State University at Pueblo
Euler and classical physics, Stacy G. Langton, University of San Diego
Elliptic integrals, mechanics and differential equations, Lawrence A. D’Antonio, Ramapo College
Euler’s great theorems, Edward Sandifer
Ben Fusaro, Florida State University
The goal of this two-day course is to introduce college teachers to a variety of topics in environmental mathematics and to the opportunities that this emerging field provides to interact with the larger society. Ben Fusaro has been active in lecturing, writing, and organizing activities in environmental mathematics since 1984. He will do the introduction and wrap-up.
Fred S. Roberts, DIMACS Center Rutgers University
Finding simple ways to measure the amount of pollution in the air we breathe, the water we drink, or the sounds we hear, has long been a goal of environmental scientists. We will discuss pollution indices in the context of a more general discussion of the theory of meaningful and meaningless statements and scales of measurement. A statement involving scales of measurement is called meaningless if its truth or falsity can depend on the particular versions of scales which are used in the statement. We will develop the theory and apply it to measurement of air, water, and noise pollution. We will discuss the possibility of averaging different measures of pollution in a meaningful way, or of combining different measures of pollution to get a consensus measure. We will also describe the use of expert judgments to assess pollution levels and describe ways to combine these judgments in the context of mathematical models of the level of air pollution and energy use in cities.
Suzanne Lenhart, University of Tennessee - Knoxville
This is an introduction to optimal control of systems of ordinary differential equations that model environmental processes. Examples will be taken from population, disease and the bacterial control of pollutants.
Catherine A. Roberts, College of the Holy Cross
The challenge of modeling oil supply and production is interdisciplinary, calling upon geology and environmental science, as well as mathematics. The issue is also laced with political and philosophical perspectives on the nature of our relationship with the planet. This talk will introduce this topic at a level suitable for a liberal arts course in mathematical modeling or environmental science. Models that provide insight into how oil production schemes impact this natural resource will be developed and discussed. As a specific example, the speaker will describe a model tied to oil drilling in the Arctic National Wildlife Refuge.
Roland H. Lamberson, Humboldt State University
We will explore some mathematical models in ecology with particular interest in the probability of extinction. We will look at measures of vulnerability, risky management strategies and how reliably models can predict the viability of a species. Species of interest will include blue whales, northern spotted owls and Pacific salmon.
Thomas O’Neil,California Polytechnic State University - San Luis Obispo
Since 1999, several Cal Poly students and I have been providing support to the Ventana Wilderness Society in their effort to establish a flock of California condors in the Big Sur area. A good recovery strategy requires an accurate population projection program. Unfortunately, there are several condor traits that make construction of such a program difficult. We will discuss these traits and how we have overcome many of the problems. Additionally, there is a lack of data. Critical to any population projection program is the survival rate data. There are estimates that can be used for first approximations but these data are based on observations of small populations of wild condors. Little was known of how the captive bred and reared birds will fare in the wild. To help in this area, we created a database of every California condor in captivity or in the wild, living or dead since 1987, the year the last wild condor was brought into captivity. We will discuss the problems encountered in creating this database and getting it into a format that has made it a useful tool for the biologists in the condor recovery project.
Charles Hadlock, Bentley College
Environmental consulting includes the use of modeling and encompasses a considerable range of activities depending on both the nature of the client organization and the objective of the investigation. For example, regulatory and legal cases might be conducted very differently from scientific and engineering investigations. The speaker will discuss his experience in a wide range of consulting assignments and will also suggest ways that mathematicians can involve themselves in this kind of work.
Organized by Jonathan M. Borwein, Dalhousie University
The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computing power and accessibility, the view that ’real mathematicians don’t computeâ? no longer has any traction for a newer generation of mathematicians that can really take advantage of computer aided research, especially given the modern computational packages such as Maple, Mathematica, and Matlab. While a working knowledge of some mathematical computing package is an advantage, it is certainly not a prerequisite. Additionally, the course will be ’hands onâ? for those who wish to follow along using their laptops, via a wireless Internet connection. The goal of this course is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in doing so to highlight some of the key algorithms and to teach some of the key experimental approaches. The program includes the following lectures:
Despite his limited formal education, Franklin was dedicated to learning and to facilitating the learning of others. As he famously opined, mathematical exercises with no direct application could still be valuable simply because they hone one’s reasoning skills. This short course will focus on ways to use “fun” problems at all levels for the purpose of developing students’ mathematical abilities. Paul C. Pasles will begin the course with a few opening remarks.
Art Benjamin, Harvey Mudd College
The mathemagician invites a member of the audience to join him onstage and to give him any number (typically a number between 50 and 100). The mathemagician then draws a blank 4-by-4 grid, and asks the volunteer to point to the 16 cells in any order. As each cell is pointed to, the mathemagician immediately writes a number in the cell. When the grid is full, the rows, columns, diagonals, and many other groups of 4, will sum to the spectator’s number. This impressive feat of mathematical magic is very easy to do, as you will learn. Did you notice that my title is palindromic? All 3-by-3 magic squares have a beautiful, little-known property called “squarepalindromicity” To illustrate, using the 3-by-3 magic square 4 9 2 3 5 7 8 1 6 you can verify that the sum of the squares of the 3-digit numbers given by the rows satisfy (492)2 + (357)2 + (816)2 = (294)2 + (753)2 + (618)2. The same phenomenon occurs with the columns, and the (wrapped) diagonals. In fact, this property holds when the numbers are written in any base! Essentially this property holds for every 3- by-3 magic square (of any sum), and for a large class of n-by-n magic squares as well. These properties can be derived using elementary linear algebra. The proof was discovered with an undergraduate, Kan Yasuda, and was eventually published in the American Mathematical Monthly. Time permitting, I will also demonstrate and explain magical ways to “square” numbers.
Professor V. Frederick Rickey United States Military Academy
Recreational mathematics is as old as mathematics itself, so a survey of its history is out of the question. Instead we discuss a few neat things, setting each in its historical context and explaining their significance. As a benchmark for looking forward and back we shall take Charles Hutton’s Recreations in Mathematics, which in turn is based on works of Ozanam and Montucla on recreational mathematics.
Orin Chein, Temple University
My part of the program will be divided into two sessions. During the first session, at the end of Day 1, I will describe a course in recreational mathematics that we offer at Temple University, and introduce a variety of problems from the text. I will also perform some mathematical card and number tricks for participants to think about. On day two, we will discuss solutions to some of the problems as well as the mathematics behind some of the tricks.
Charles Ashbacher, Editor, Journal of Recreational Mathematics
The alphametic, an arithmetic problem where letters represent digits and the letters also create a message, has been a staple problem in the Journal of Recreational Mathematics since the first issue was published. The messages are simple, such as the classic SEND+ MORE = MONEY. Solving them is usually an exercise in algebra in combination with trial and error. Solving an alphametic also makes an excellent programming assignment in beginning programming classes, in that they can be solved in a brute force manner by creating a set of nested loops. This presentation will be a demonstration of the various forms of the alphametic and how they are solved. The messages of the alphametics that have been published in the JRM over the years have covered a wide area. Everything from political statements to congratulations and condolences has been published as math problems. Some of the more interesting examples of this area of mathematics will also be given.
Matthias Beck, San Francisco State University
How many ways are there to change 42 cents? How many ways will there be when all the pennies are gone? How about if nickels were worth four cents? More generally, suppose we have coins of denominations a1, a2, ..., 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? Depending on the culinary preference of the audience, we may state these questions in terms of bags of M&M’s or boxes of Chicken Nuggets (’Can you buy Chicken Nuggets so that our 34 friends get exactly one each?â?). We will see that if a1, a2, ..., ad do not have any common factors 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 in closed form for d = 2, in generatingfunction form for d=3, and wide open for d > 3. We will outline several elementary approaches to the d=2 case of this classical problem, including one that generalizes to d=3. These proofs are well suited for undergraduate classes in discrete mathematics, number theory, abstract algebra, combinatorics, or geometry. Going a step further, we will use the above counting function c(n) to recover and extend some well-known results on the Frobenius problem. En route we will discuss some basic number theory and discrete geometry connected to c(n). We will mention several open problems, some which are well suited for original undergraduate research projects.
Maya Mohsin Ahmed, University of California – Davis
The problem of constructing magic squares is of classical interest. Enumerating magic squares is a relatively new problem. I will describe how to construct and enumerate magic squares as lattice points inside polyhedral cones using techniques from algebraic combinatorics. I will also look at the correspondence of magic labelings of graphs and symmetric magic squares.
Alan M. Frieze
The subject began properly with a sequence of seminal papers in the 1960’s by Paul ErdÃ¶s and Alfred RÃ©nyi. ErdÃ¶s had already used randomly generated graphs as a tool for showing the existence of various structures, but these papers began the study of random graphs as objects in their own right. Since that time there has been much research establishing the likely structure of various models of random graph and finding uses for this knowledge. In this course we provide some of the basic results and tools used in the area. Presenters include:
Random matrix theory: It is mathematics; it is statistics; it is physics; it is engineering. It is stochastic equation solving in its most glorious form. There have been applications for decades, yet the subject is not sufficiently well-known or wellunderstood for all of the applications to have been realized. One-by-one the word has to spread. This is a start. In this course we will review the theory from the finite to the infinite. We will consider Gaussian ensembles and Wishart matrices. There will be Riemann zeta and Painleve for the mathematicians. There will be zonal polynomials and hypergeometrics of matrix argument for the statisticians. We will emphasize applications to wireless communication not only for its own sake but as a case study for other applications to follow.
Ioana Dumitriu (UC Berkeley)
Many statistical, physical, and engineering problems require the use of random matrix theory, and with it, the computation of a certain messy multivariate integral (sometimes parameterdependent) over a subset of the real line. Using multivariate orthogonal polynomial theory, we have written and implemented in Maple a set of codes which provide a unified way of dealing with such computations, for certain classical types of random matrices. These codes are fast (in a relative sense, as the complexity of the problem is super-polynomial) and have the advantage of working both symbolically and numerically. We will discuss a few problems, the computations they involve, and demonstrate the performance of MOPs in each case.
Moe Win (MIT) and Marco Chiani (University of Bologna)
You are a medieval knight and you want to send an important message to the king in another country. Unfortunately, the messages that you send may become stained or soaked with water! Furthermore, some routes may be closed due to natural and unnatural events such as fire, floods, banditry, (and stray dragons). Your astrologer comes up with a brilliant idea: send multiple couriers along different routes in hopes that the received letters can be combined to make a coherent message. We consider a class of problems arising in wireless communications where we are given the statistical behavior of possible routes, and how we can use ’route diversityâ? to improve the reliability of a wireless communication system. Finally, we will show why it is better to have more antennas on your Wi- Fi, or in medieval terms, more horses in your stables. We will show how the beautiful theory of random matrices and eigenvalues can be used to model this class of problems and give insights into the design of future communication systems.
Raj Rao (MIT)
In traditional probability theory, independence of random variables allows one to easily compute the distribution of the sums or products of these variables. For random matrices, freeness is the analogous concept that allows us to compute the distribution of the eigenvalues of the sums or products or other functions of random matrices. Freeness turns out to be important and quite powerful because unlike scalar random variables, the order in which the random matrices are multiplied matters. Free probability is the theory that tells us what it means to be ’freeâ? and what we can or cannot do when the variables in question are ’freeâ?. We will discuss free probability and demonstrate how it can be used to compute the distribution of seemingly intractable functions of random matrices. This is joint work (more accurately, joint fun) with Alan Edelman (MIT)
Amy Shell-Gellasch and Glen Van Brummelen
This short course will explore the history, development, use, and significance of various mathematical devices throughout history. Devices investigated will include sun dials, linkages, navigational and surveying devices, early computing devices, and early computers. Presenters will bring in actual historical devices when possible. The sessions will be a mix of traditional presentations, followed by a hands-on demonstration and question period. Topics will cover calculations and Mensuration devices from various eras, from ancient to modern times. Our finale will be a presentation on mathematical devices at world’s fairs. Presenters include:
John R. Jungck
Bioinformatics is an emerging area that combines extensive mathematical and computer applications in molecular biology. This course will introduce how genetic sequences can be analyzed with coding and information theory, computational linguistics, dynamic programming of multiple sequence comparison, and phylogenetic trees. Other topics include: molecular surface calculations as well as X-ray crystallographic approaches to structure determination, knot theory for studying the topological changes involved in replication, recombination, and repair of DNA, RNA folding, and analysis of DNA chips to study expression of genes. The level of mathematics will be at the undergraduate level and often is accessible to biology students.
Nearly everyone who has taken an interest in the history of mathematics becomes fascinated with some facet of ancient mathematics. But only a few have the mathematical preparation, historical sensitivities, and linguistic skills to do original work. The speakers at this short course will give an expository survey
of their special area of ancient mathematics. They will discuss some areas of current research, point out open questions, and provide guidelines to help you delve into the expository and research literature. Those of you who have taught history of mathematics will undoubtedly learn that some of what you read in older literature has been superseded by modern scholarship. Thus you will have much to carry back to your classroom. Speakers and their talks include:
De Witt Sumner
Vera Pless and W. Cary Huffman
Kiran Bhutani, Catholic University
Paul Blanchard, Boston College
Robert Almgren, University of Chicago
Arthur Benjamin, Harvey Mudd College and A. Brent Morris, National Security Agency
Akram Aldroubi, Vanderbilt University
Dennis Healy, DARPA/Dartmouth College
Herb Hethcote, University of Iowa