PROMYS
Encyclopedia
The Program in Mathematics for Young Scientists (PROMYS) is a six-week number theory
program currently under the direction of Professor Glenn Stevens at Boston University
beginning in early July and ending in mid-August.
The program is open to high school
students, although college students and above can return as counselors. Each year, there are roughly 45 first year students and 15 returning students, in addition to undergraduate counselors and junior counselors. The program centers around a daily number theory lecture from Professor Stevens, which is mandatory for everyone. First-year students are typically also involved in an Exploration Lab, an open-ended problem in number theory or a closely related area of math that is accessible to first-year students. Exploration Lab topics have included relationships between Pick's theorem
and the Euler characteristic
, linear fractional transformations, Linear Diophantine equations, integer partitions and compositions
, finite difference
s, and Chebyshev polynomials
.
Returning students, some of whom are attending the program for the third or even fourth time, typically take advanced courses sponsored by the Clay Mathematics Institute
participate in a Research Lab, although all courses and labs are open to all students. Advanced courses are taught by professors at Boston University and other schools, including Brandeis University
and Syracuse University
. In addition, there are several minicourses and guest lectures on a wide variety of mathematical topics.
Research topics have included Ducci sequences (also known as the n-numbers game), dragon curve
s, arithmetrees, and tropical algebraic geometry
. In 2008 the advanced courses featured Galois theory
, representation theory of finite groups
, and geometry
. Through dealing a usually intuitive subject with mathematical rigor, the geometry course exposes its students to many areas of mathematics, including set theory
, algebra
, and non-Euclidean geometry
. 2009 featured advanced courses in linear algebra
, combinatorics
, and geometry. The 2010 advanced courses were in abstract algebra
, modular forms, and geometry & symmetry.
Students are required to do problem sets every weekday and one extra long problem set on the weekend. Every Friday night, the counselors organize "an event so fun that it is necessary and sufficient," hence the name "mandatory fun." These events include a picnic, casino night, a talent show, animation night, a "dance," and an ice cream social. Also during the weekends, students usually participate in games of Egyptian Rat Screw, Illuminati
, or Mafia
lasting entire nights.
The counselors also hold informal seminars of various topics, such as the p-adic number system
. During some nights a guest lecture is organized where mathematicians and people working in math-related subjects such as Wolfram Group lecture on what they find interesting and worth sharing.
The predecessor to PROMYS was the Ross program
, of which Professor Stevens and many of the advanced course instructors are graduates.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
program currently under the direction of Professor Glenn Stevens at Boston University
Boston University
Boston University is a private research university located in Boston, Massachusetts. With more than 4,000 faculty members and more than 31,000 students, Boston University is one of the largest private universities in the United States and one of Boston's largest employers...
beginning in early July and ending in mid-August.
The program is open to high school
High school
High school is a term used in parts of the English speaking world to describe institutions which provide all or part of secondary education. The term is often incorporated into the name of such institutions....
students, although college students and above can return as counselors. Each year, there are roughly 45 first year students and 15 returning students, in addition to undergraduate counselors and junior counselors. The program centers around a daily number theory lecture from Professor Stevens, which is mandatory for everyone. First-year students are typically also involved in an Exploration Lab, an open-ended problem in number theory or a closely related area of math that is accessible to first-year students. Exploration Lab topics have included relationships between Pick's theorem
Pick's theorem
Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
and the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
, linear fractional transformations, Linear Diophantine equations, integer partitions and compositions
Composition (number theory)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Any integer...
, finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
s, and Chebyshev polynomials
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
.
Returning students, some of whom are attending the program for the third or even fourth time, typically take advanced courses sponsored by the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
participate in a Research Lab, although all courses and labs are open to all students. Advanced courses are taught by professors at Boston University and other schools, including Brandeis University
Brandeis University
Brandeis University is an American private research university with a liberal arts focus. It is located in the southwestern corner of Waltham, Massachusetts, nine miles west of Boston. The University has an enrollment of approximately 3,200 undergraduate and 2,100 graduate students. In 2011, it...
and Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...
. In addition, there are several minicourses and guest lectures on a wide variety of mathematical topics.
Research topics have included Ducci sequences (also known as the n-numbers game), dragon curve
Dragon curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.-Heighway dragon:...
s, arithmetrees, and tropical algebraic geometry
Tropical geometry
Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry. Its leading ideas had appeared in different guises in previous works of George M...
. In 2008 the advanced courses featured Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, representation theory of finite groups
Representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
, and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. Through dealing a usually intuitive subject with mathematical rigor, the geometry course exposes its students to many areas of mathematics, including set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, and non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. 2009 featured advanced courses in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, and geometry. The 2010 advanced courses were in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, modular forms, and geometry & symmetry.
Students are required to do problem sets every weekday and one extra long problem set on the weekend. Every Friday night, the counselors organize "an event so fun that it is necessary and sufficient," hence the name "mandatory fun." These events include a picnic, casino night, a talent show, animation night, a "dance," and an ice cream social. Also during the weekends, students usually participate in games of Egyptian Rat Screw, Illuminati
Illuminati (game)
Illuminati is a standalone card game made by Steve Jackson Games , inspired by The Illuminatus! Trilogy by Robert Anton Wilson and Robert Shea. The game has ominous secret societies competing with each other to control the world through sinister means, including legal, illegal, and even mystical...
, or Mafia
Mafia (game)
Mafia is a party game created in the USSR by Dimitry Davidoff in 1986, modelling a battle between an informed minority and an uninformed majority. Players are secretly assigned roles: either "mafia", who know each other; or "townspeople", who know only the number of mafia amongst them. In the...
lasting entire nights.
The counselors also hold informal seminars of various topics, such as the p-adic number system
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
. During some nights a guest lecture is organized where mathematicians and people working in math-related subjects such as Wolfram Group lecture on what they find interesting and worth sharing.
The predecessor to PROMYS was the Ross program
Ross Mathematics Program
The Ross Mathematics Program is an intensive eight-week residential summer camp for high school students interested in mathematics. The program is currently run by Professor Daniel Shapiro and is based out of the Ohio State University.-History:...
, of which Professor Stevens and many of the advanced course instructors are graduates.