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A Summer Program for Mathematically Talented High-School Students from around the World

The Mathcamp Academic Program

The Mathcamp academic program gives students the opportunity to study advanced and unusual topics in mathematics in a challenging and fun environment. Throughout the day, there are activities at all levels, beginning to most advanced, on all sorts topics in pure and applied math, and in lots of different formats:

  • courses taught by regular faculty, lasting anywhere from a few days to five weeks.
  • lectures and seminars by distinguished visitors.
  • math contests and problem-solving sessions
  • hands-on workshops and individual projects.

There is so much going on at every hour of the day and night that no one could possibly do it all! With the help of a faculty advisor, students design their own schedule and select the activities most appropriate to their level and interests. However, if you're really fascinated by a topic, don't worry about not having all the necessary background: we'll work with you to help you learn it!


Classes

The Mathcamp schedule is so packed that it may seem daunting at first. However, it is important to remember that none of the classes are mandatory. We assume that you are at Mathcamp because you want to do math, but exactly how much time you spend in class is up to you. In fact, we discourage students from going to too many classes. Some of your most important learning experiences will take place outside of class: thinking about what you've learned, doing homework, talking to the visiting lecturers, solving problems with friends, or working on your independent project. Our goal is to put an enormous amount of fascinating mathematics within your reach, so that you can do as much as you want, in the way that is most enjoyable and productive for you.

To get an idea of what classes at Mathcamp are like, check out the course offerings from this year's Mathcamp at Current Summer Academics. (Previous years' schedules are available here.)

You'll notice there are usually at least 4 classes (often more!) offered at any one time, so you can choose the ones that interest you most. (Read the detailed weekly class descriptions to find out more.) Each class has a "star rating" to indicate its difficulty: from * (accessible to everyone) to **** (very advanced). The "one-star" classes aren't any less interesting than the "four-star" classes -- they just move at a more relaxed pace and assume less mathematical experience. (We like to say that in a "one-star" class, every student has the right to follow what's going on all the time; in a "four-star" class, the strongest students have the right to be challenged at all times.) You can choose the level you are most comfortable with or experiment by taking classes on many different levels; your academic advisor will help you choose courses that best match your background and interests.

Course offerings vary from year to year, depending on the interests of the students and faculty. Some of the topics taught in previous years have included:

  • Discrete Mathematics
    • Combinatorics (enumerative, algebraic, geometric)
    • Generating functions and partitions
    • Graph theory
    • Ramsey theory
    • Probability
    • Finite geometries
    • Polytopes and polyhedra
    • Combinatorial Game Theory
  • Algebra and Number Theory
    • Linear algebra
    • Groups, rings, and fields
    • Primes and factorization algorithms
    • Congruences and quadratic reciprocity
    • Galois theory
    • Algebraic number theory
    • Analytic number theory
    • Fermat's Last Theorem for polynomials
    • p-adic numbers
    • Geometry of numbers
  • Geometry and Topology
    • Euclidean and non-Euclidean (hyperbolic, spherical, projective, inversive) geometries
    • Geometric transformations
    • Algebraic geometry
    • Point-set topology
    • Combinatorial topology
    • Knot theory
    • The Brouwer Fixed-Point Theorem
  • Calculus and Analysis
    • Topics in calculus
    • Fourier analysis
    • Complex analysis
    • Real analysis
    • Dynamical systems
  • Set Theory, Logic, and Foundations
    • Cardinals and ordinals
    • Gödel's Incompleteness Theorem
    • The Banach-Tarski Paradox
    • Model Theory
    • Category Theory
  • Computer Science
    • Theoretical CS
    • Complexity theory
    • Information Theory
    • Cryptography
    • Algorithms
  • Connections to Other Fields
    • Relativity and quantum mechanics
    • Neural networks
    • Mathematical biology
    • Game theory
    • Voting theory
    • Bayesian statistics
  • Discussions
    • Philosophy of Mathematics
    • Math Education
    • How to Give a Math Talk
    • College And Beyond
  • Problem Solving
    • Proof methods
    • Elementary and advanced techniques
    • Contest problems of various levels of difficulty
    • Relays and team competitions

Projects

Students at Mathcamp are encouraged to work on a project, supervised by one of the mentors or faculty. Projects range in scope from creative applications of simple techniques to advanced problems connected to faculty research. Usually the mentors and faculty propose project ideas, and students can choose one they want to work on. However, if a student wants to propose a project of his or her own, there is usually someone on the staff willing to serve as an advisor.

Selected projects from previous years:

  • Billiard-ball geometry
  • Information theory and psychology
  • Periodicity of Fibonacci numbers mod n
  • Knight tours on an m-by-n chessboard
  • Non-convex polyhedra
  • Cellular automata
  • Cops and robbers on a graph
  • Constructing the regular 17-gon
  • Admissible covers of algebraic curves
  • Mathematical Finance
  • Algorithmic composition of music
  • Intelligent ways of searching the web
  • Probability in sports
  • The elasticity equation of string
  • Digital signal processing
  • Light paths in universes with alternate physics

One of the projects from Mathcamp 2004 resulted in a research paper coauthored by a student and two staff members (to appear in the Journal of Discrete and Computational Geometry).

You can take a look at specific projects from the last few years:

  • Mathcamp 2007: staff project proposals [PDF].
  • Mathcamp 2006: staff project proposals [PDF].
  • Mathcamp 2005: staff project proposals [PDF].
  • Mathcamp 2004: staff project proposals [PDF].

Past Course Offerings

Here are some weekly academic schedules and class descriptions from previous Mathcamps:

  • MATHCAMP 2008:
    • Week 1: Schedule [PDF] and class descriptions [PDF].
    • Week 2: Schedule [PDF] and class descriptions [PDF].
    • Week 3: Schedule [PDF] and class descriptions [PDF].
    • Week 4: Schedule [PDF] and class descriptions [PDF].
    • Week 5: Class proposals [PDF] (students voted to decide which would be offered), and the final schedule [PDF].
  • Mathcamp 2007:
    • Week 1: Schedule [PDF] and class descriptions [PDF].
    • Week 2: Schedule [PDF] and class descriptions [PDF].
    • Week 3: Schedule [PDF] and class descriptions [PDF].
    • Week 4: Schedule [PDF] and class descriptions [PDF].
    • Week 5: Class proposals [PDF] (students voted to decide which would be offered), and the final schedule [PDF].
  • Mathcamp 2006:
    • Week 1: Schedule [PDF] and class descriptions [PDF].
    • Week 2: Schedule [PDF] and class descriptions [PDF].
    • Week 3: Schedule [PDF] and class descriptions [PDF].
    • Week 4: Schedule [PDF] and class descriptions [PDF].
    • Week 5: Class proposals [PDF] (students voted to decide which would be offered), and the final schedule [PDF].
  • Mathcamp 2005:
    • Week 1: Schedule [PDF] and class descriptions [PDF].
    • Week 2: Schedule [PDF] and class descriptions [PDF].
    • Week 3: Schedule [PDF] and class descriptions [PDF].
    • Week 4: Schedule [PDF] and class descriptions [PDF].
    • Week 5: Class proposals [PDF] and the final schedule [PDF].