The Mathcamp Academic Program

The Mathcamp academic program is designed to provide gifted high school students with the opportunity to study advanced and unusual topics in a challenging and dynamic environment. We will be offering a wide variety of mathematical activities for students of all levels, from beginner to the most advanced. There will be regular courses lasting anywhere from a few days to five weeks, problem-solving sessions under the guidance of experienced problem-solvers, small group tutorials, and lectures by distinguished visitors, with opportunities for follow-up and discussion. Here are a few of the topics that were taught in 2001 or previous summers:

Calculus and analysis:

Derivatives and integrals; Fourier series; real and complex analysis; non-standard analysis; measure theory; wavelets

Geometry and Topology:

Euclidean, hyperbolic, spherical, and projective geometry; geometric transformations; algebraic geometry; point-set topology; classification of surfaces; knot theory; differential topology

Algebra and Number Theory:

Primes and factorization algorithms; congruencies, cryptology, and RSA codes; groups, rings, and fields; matrices and determinants; Galois theory; elliptic curves; algebraic number theory

Combinatorics:

Counting techniques; generating functions; graph theory; Ramsay theorems; partitions

Set Theory and Logic:

Cardinals and ordinals, G�del's incompleteness theorem, complexity

Other:

Fractals; game theory; voting theory; information theory

Problem-solving:

Proof methods and techniques; olympiad problem solving; math relays and team competitions

Many of these topics will be repeated this year, and many others added. An ongoing weekly team problem-solving contest gives students of different levels of experience an opportunity to work together and share newly acquired skills. There will also be lectures on the connections of math to other fields, such as physics and computer science. As you can see, there is a lot more math at Mathcamp than you can possibly expect to learn in five weeks. We expect to guide you in choosing the activities most appropriate to your level and interests. But if you're fascinated by a topic and feel that you don't have the necessary background, we'll work with you to help you learn it.

Academic Materials from Previous Years

• First week's schedule
• Second week's schedule
• Third week's schedule
• Fourth week's schedule
• Fifth week's schedule

• Miscellaneous problems to solve (variety of levels)
• Catalan Numbers
• Probability problems
• Elementary Number Theory problems
• Some information on complex numbers and functions on them
• Complex variables problems
• Projective geometry problems
• Problems on generating functions (hard!)
• Notes and theorems on Measure Theory (tough stuff!)

Courses and academic materials from Mathcamp 2001 will be posted soon.

Meet the Mathcamp Faculty

Titu Andreescu (Director, American Mathematics Competitions)

Problem-solving

Director, American Mathematics Competitions (the organization that administers the AMC->10, AMC->12, AIME, and USAMO contests). Leader of the U.S. IMO team since 1995.

John Conway (Princeton)

Finite Mathematics (and just about everything else!)

One of the most creative thinkers of our time, known for his ground-breaking contributions to such diverse areas as knot theory, geometry of high dimensions, group theory, transfinite arithmetic, and the theory of mathematical games. Outside the mathematical community, he is perhaps best known as the inventor of "The Game of Life".

David DeVidi (University of Waterloo)

Philosophy of mathematics, logic

Dave DeVidi received his PhD from the University of Western Ontario in 1994, and is currently associate professor of philosophy at the University of Waterloo. He is co-author of Logical Options: An Introduction to Classical and Alternative Logics (Broadview Press, 2001), and author or co-author of several other publications in logic, philosophy of mathematics and philosophy of science.

George Hart

Mathematical sculpture

George returns in 2002 to lead more Zometool workshops, in which participants can explore the geometry of three-dimensional space in a big way. Author of Zome Geometry and of Multidimensional Analysis, he has a Ph.D. from MIT but lives a full-time sculptor career. His work can be seen at his website, GeorgeHart.com.

Sanjoy Mahajan (Cambridge University)

Physics

Sanjoy Mahajan majored in physics at Stanford and mathematics at Oxford, and did his PhD in physics at Caltech. During his PhD he became interested in improving physics teaching, which he is trying to do in the Cambridge University physics department. He is writing a textbook on "Lies, Damn Lies, and Approximations".

Jeffrey Shallit (University of Waterloo)

Computer Science

Jeffrey Shallit is Professor of Computer Science at the University of Waterloo. Educated at Princeton University and UC Berkeley, he also taught at the University of Chicago and Dartmouth College. His interests include number theory and theory of computing. With Eric Bach, he is the author of Algorithmic Number Theory (MIT Press, 1996).

Yan Soibelman (Kansas State University)

Algebra, Geometry, and Physics

Yan Soibelman's mathematical interests range from classical analysis to homological algebra, symplectic geometry and string theory. He is the author of the book "Algebras of functions on quantum group". One of his earliest results (obtained at the age of 18) was a topological generalization of the well-known fact that there exist plane figures which cannot be decomposed into two pieces of smaller diameter. Currently he is working with Maxim Kontsevich on foundations of deformation theory, with applications to algebra, geometry and physics. Yan has been a visiting professor of many research centers all over the globe, and is a Fellow of the Clay Mathematics Institute (see his web page at here).

William Stein (Harvard University)

Number Theory, Computation

William Stein is a Postdoctoral Fellow at Harvard University who loves doing systematic computations with the types of objects that appear in the proof of Fermat's Last Theorem.

Sergei Tabachnikov (Penn State University)

Differential Topology, Dynamical Systems

Sergei Tabachnikov is actively involved in a number of projects on mathematical education; among them, he is the Director of the Mathematics Advanced Study Semesters program (MASS) at Penn State University, where he is a professor. Sergei has authored a book on mathematical billiards, and regularly visits various research institutes around the world.

Josh Tenenbaum (Stanford)

Cognitive science and artificial intelligence

Josh Tenenbaum teaches psychology and computer science at Stanford University. In his research, he builds mathematical models of human and machine learning, reasoning and perception. He also works on neural networks, information theory, and statistical inference. Read more at http://www-psych.stanford.edu/~jbt.

Faculty members for Mathcamp 2002, listed along with their affiliation and mathematical interests, will include:

Mira Bernstein (Wellesley)

Algebraic geometry
Executive Director, Mathcamp

Mira Bernstein, the "top banana" of Mathcamp 2000, returns this year. With her specialization in algebraic geometry, she can answer burning questions like "How do you look at infinity without blinking?" and "How many lines cross four randomly chosen lines in space?" She is also interested in games, voting theory, information theory, and linguistics.

David Savitt (McGill University)

Number theory, arithmetic geometry
Academic coordinator, Mathcamp

Originally from Vancouver, Canada, David Savitt was the first-ever counselor at Mathcamp, and this year he will be working at his seventh Mathcamp. Dr. Savitt received his PhD at Harvard University in 2001 (where his work focused on an extension of the results which led to the proof of Fermat's Last Theorem) and is now a postdoctoral researcher at McGill University in Montreal, Canada and at Universite Paris-Sud in Orsay, France.

Dorin Andrica (Babes-Bolyai University)

Computer science, problem solving

A professor of mathematics and computer science at Babes-Bolyai University in Romania, has maintained a strong interest in mathematics education and elementary mathematics throughout his career. In addition to numerous research articles and books, Professor Andrica has published hundreds of original problems in various journals and competitions. He is a member of the council for the Romanian National Olympiad, and will teach problem-solving at Mathcamp.

Mark Krusemeyer (Carleton College)

Problem solving

Mark Krusemeyer, who has been at Mathcamp every year since 1997, is Professor of Mathematics at Carleton College. He is one of the authors of the Wohascum County Problem Book, published by the MAA (Mathematical Association of America), and he has been a problem poser for the William Lowell Putnam Mathematical Competition, a annual contest for undergraduates across the U.S.A. and Canada. He has received an award for Distinguished College or University Teaching of Mathematics from the North Central Section of the MAA. His non-mathematical interests include early music (at Carleton, he also teaches recorder), hiking, bridge, and table tennis.

Mark Sapir (Vanderbilt)

Algorithmic problems in algebra

Mark Sapir is a leading expert in group theory and algorithmic problems in different areas of mathematics. His contributions include split systems, S-machines and diagram groups. He has written more than seventy research papers and monographs, as well as several books for Russian high schools (with more than 3,000,000 copies printed). His other interests include educational software and Web design.

G. Rubin Thomas

Graph theory
Founder, Mathcamp

George Rubin Thomas, founder of Mathcamp and chairman and founder of the Mathematics Foundation of America, is a former mathematics professor. He now lives and works in London, Ontario.

Mary Pat Campbell (NYU), mentor
Kenny Easwaran (Stanford), mentor
Brenda Fine (British Columbia), mentor
Maren Friesen (British Columbia), junior counselor
Julian Gilbey (Queen Mary, Univ. of London), mentor
Tivon Jacobson (NYU), mentor
Yvonne Lai (MIT), junior counselor
Moses Liskov (MIT), mentor
Ari Nieh (Berkeley), mentor
Sam Payne (Michigan), mentor
Igor Pavlovsky (MIT), mentor
Mike Shulman (Caltech), junior counselor
Lori Thomas (Harvey Mudd), junior counselor
Simona Topor Pop (Harvard), junior counselor
Chris Tuffley (Berkeley), mentor,
Dan Zaharopol (MIT), junior counselor