Mathematics

Mathematics

Math Short Terms

Course descriptions for some past mathematical short terms – Autumn or Winter – are:

Quantum Mechanics for Everyone
Heisenberg's Uncertainty Principle, Schrodinger's Cat, Wave-Particle Duality. You may have heard these terms bandied about, but what exactly are they and why do physicists take these ideas seriously? In this course we will investigate the peculiar world of the very small, known as Quantum Mechanics, where objects behave probabilistically, counter to all of our usual intuitions. To do so, we will look at the actual experiments that physicists have constructed, and what we will see is that very small objects can act like waves and also like particles, amongst other oddities. We'll keep the math to a minimum, focusing on the experimental outcomes, though a small amount of probability will creep in (requiring no more than high school algebra skills).

Introduction to Cryptology
Since antiquity, humans have sent coded messages to each other, and this course offers a peek into the world of cryptography and codes, which has become ubiquitous in today's society, from banking systems to smart phones. Students will start by learning number theory, which is a branch of mathematics that deals with counting numbers. Then they will be introduced to important mathematical structures called Finite Fields upon which a few simple classical cryptosystems are built. Lastly, students will study a few public key systems which are used widely in our modern daily lives. Throughout the winter term, students will have opportunities to learn how to use Mathematica to write simple computation programs to solve elementary number theory problems and to encode and decode messages.

Mathematical Approaches to Contemporary Problems
This course will offer students new insights and a fundamental understanding of how mathematics contributes to solving important national and global problems in the environmental, behavioral, and natural sciences.  Starting with a real problem concerning an important process of general interest, we will formulate simplifying assumptions about how the process evolves over time.  We will build equations to model the sequence representing the values of the characteristic under study and will develop and interpret solutions in the context of the original problem.  We can then either accept the model as adequate or modify it in order to obtain a better model.  Important applications include fitting simple models to medicine dosages, repeated loan payments, oil consumption and reserves, predicting population levels.  We also look at chaotic processes.  These models use basic mathematical functions studied in high school. 

Elementary Mathematical Models
Applied mathematics creates equations or inequalities (models) expressing interrelationships among quantities we want to predict, explain or optimize and other quantities which influence them.  Examples include relating the price of commodity to demand and supply, the search for maximum profit, deciding how much to order and when in order to minimize cost, predicting population levels from predator-prey interactions and explaining propogation of  inherited genetic characteristics through succesive generations.  Students model change using difference equations, create models by applying concepts of proportionality, geometric similarity, Monte Carlo simulation, and practice fitting simple models to collected data and writing up their conclusions.  Prerequiste:  background in high school algebra.

Natural & Invented Languages
Over the years numerous attempts have been made to improve human language by making it simpler, less prone to error and misunderstanding, more regular, and fairer for all speakers.  All of the most ambitious attempts, even by the leading minds of the day, have failed spectacularly.  What are the components of language and why are they so resistant to human tinkering and seeming improvement?  In this course we will investigate the basic features of human language including phonetics, phonology, syntax, and semantics, using the scientific tools of modern linguistics, and in doing so try to answer this vexing question.  In addition to learning about the components of language each student will construct their own invented language, producing grammars for each component of language we study, describing how they interact to yield a viable human language.

The Fascinating World of Chaos and Fractals
Chaos, fractals, and dynamical systems are three of the hottest areas in contemporary mathematics.  Through hands-on computer experimentation the student will be taken on a tour of the exciting and beautiful world of chaos and fractals.  The student will learn how to generate fascinating computer images of fractals, Julia sets, and the famous Mandelbrot set.

How to play games, mathematically
Game theory studies optimal strategies when more than one player interact, or when one player has more than one choice to deal with a problem.  It has found many applications in a variety of subjects such as economics, computer science, psychology, philosophy, political science, and military conflicts. In this course we will play a dozen or so different games and analyze the mathematics behind them.  We will consider the real world applications of game theory too.

Mathematical Recreations of Lewis Carroll
This course is based on two works by the mathematician-author of Alice in Wonderland, Lewis Carroll.  In Symbolic Logic, a diagrammatic method of drawing conclusions from propositions is developed.  Modern puzzle fans will delight in the wit and humor of Carroll’s problems.  In The Game of Logic, Carroll demonstrates how logic can become a fascinating board game.  Tricky and amusing problems are presented and solved.

Mathematics & Art
An investigation into mathematical visualization in art.  Topics covered include Escher hyperbolic patterns, topological morphing, self-similarity, tessellation, cardinality and versions of infinity.  The art of M.C. Escher, Helaman Ferguson, Alexander Pankin, George Hart, and Charles O. Perry, among others, are studied.  Students will actively produce mathematical art using computers in addition to paint and collage mediums.  Moreover, some time will be spent building mathematical structures with Lego Building Blocks.

Mathematics & Strength
An in-depth study on the nature of strength and the mathematical models that describe the biomechanics of strength.  Some time will be spent studying sport specific strength training programs and, in particular, evaluating the mathematics behind the concepts of specialization and periodization and their effect on sports mastery.  In addition to lectures and readings, each student must participate in a strength training program which will be conducted in the Gamble Weight Room.  Costs of this project are extensive as a set of exercise resistance bands (~$50) and wrist wraps (~$25) must be purchased from the college bookstore.  These costs are in addition to the textbook.  Prerequisites: MA 131M

Number Theory and Cryptography
One of the most important applications of modern mathematics in our current times is the use of cryptography in securing our network systems of communications.  Although the idea dates back to ancient times only after the appearance of the RSA system can one start to build a really safe way to transmit data over long distances via the internet.  The backbone of the RSA system is Fermat's Little Theorem in number theory.  In this course we are going to study topics related to this and some other cryptosystems.  We will use Maple and Mathematica to do some real world research and problem solving so you will get some hands-on experience.  The prerequisite is Linear Algebra MA 236N or something equivalent.

Our Graduates

Many of our graduates go on to graduate school at such institutions as MIT, Stanford, Dartmouth, Brown, Duke, UCLA, Vanderbilt and more. Click here for a list of just a few of these institutions.