Reed College Catalog

The mathematics curriculum emphasizes solving problems by rigorous methods that use both calculation and structure. Starting from the first year, students discuss the subject intensely with one another outside the classroom and learn to write mathematical arguments.

The major is grounded in analysis and algebra through the four years of study. A student typically will also take upper-division courses in areas such as computer science, probability and statistics, combinatorics, and the topics of the senior-level courses that change from year to year. In particular, the department offers a range of upper-division computer science offerings, while recent topics courses have covered elliptic curves, polytopes, modular forms, Lie groups, representation theory, and hyperbolic geometry. A year of physics is required for the degree. The yearlong senior thesis involves working closely with a faculty member on a topic of the student’s choice.

The department has a dedicated computer laboratory for majors. Mathematics majors sometimes conduct summer research projects with the faculty, attend conferences, and present papers, but it is more common to participate in a Research Experience in Mathematics (REU) program elsewhere to broaden experience. Many students from the department have enrolled in the Budapest Semester in Mathematics program to study in Hungary.

Graduates from the mathematics department have completed Ph.D.programs in pure and applied mathematics, computer science and engineering, statistics and biostatistics, and related fields such as physics and economics. Graduates have also entered professional careers such as finance, law, medicine, and architecture.

First-year students who plan to take a full year of mathematics can select among Calculus (Mathematics 111), Introduction to Computing (Mathematics 121), Introduction to Number Theory (Mathematics 131), Introduction to Combinatorics (Mathematics 132), or Introduction to Probability and Statistics (Mathematics 141) in the fall, and Introduction to Analysis (Mathematics 112) or Introduction to Probability and Statistics in the spring. The prerequisite for all of these courses except Analysis is three years of high school mathematics. The prerequisite for Analysis is a solid background in calculus, usually the course at Reed or a year of high school calculus with a score of 4 or 5 on the AP exam. Students who intend to go beyond the first-year classes should take Introduction to Analysis. In all cases, it is recommended to consult the academic adviser and a member of the mathematics department to help determine a program.

The mathematics department’s web page can be found at academic.reed.edu/math.

Requirements for the Major

  1. Mathematics 111 or the equivalent, 112, 211, and 212.
  2. Mathematics 321, 331, and 332.
  3. Four additional units in mathematics courses numbered higher than 300 (excluding Mathematics 470).
  4. Physics 100 or the equivalent.
  5. Mathematics 470.

Mathematics 111 - Calculus

Full course for one semester. This includes a treatment of limits, continuity, derivatives, mean value theorem, integration—including the fundamental theorem of calculus, and definitions of the trigonometric, logarithmic, and exponential functions. Prerequisite: three years of high school mathematics. Lecture-conference.

Mathematics 112 - Introduction to Analysis

Full course for one semester. Field axioms, the real and complex fields, sequences and series. Complex functions, continuity and differentiation; power series and the complex exponential. Prerequisite: Mathematics 111 or equivalent. Lecture-conference.

Mathematics 121 - Introduction to Computing

Full course for one semester. An introduction to computer science, covering topics such as elementary data structures, algorithms, computability, floating point computations, and programming in a high-level language. Prerequisite: three years of high school mathematics. Lecture-conference and lab.

Mathematics 131 - Introduction to Number Theory

Full course for one semester.  A rigorous introduction to the theorems of elementary number theory. Topics may include: axioms for the integers, Euclidean algorithm, Fermat's Little Theorem, unique factorization, primitive roots, primality testing, public-key encryption systems, Gaussian integers. Prerequisite: three years of high school mathematics or consent of instructor. Lecture-conference. Not offered 2009-10.

Mathematics 132 - Introduction to Combinatorics

Full course for one semester. Permutations and combinations, finite mathematical structures, inclusion-exclusion principle, elements of the theory of graphs, permutation groups, and the rudiments of Pólya theory will be discussed. Prerequisite: three years of high school mathematics. Lecture-conference.

Mathematics 141 - Introduction to Probability and Statistics

Full course for one semester. The basic ideas of probability including properties of expectation, the Law of Large Numbers, and the Central Limit Theorem are discussed. These ideas are applied to the problems of statistical inference, including estimation and hypothesis testing. The linear regression model is introduced, and the problems of statistical inference and model validation are studied in this context. A portion of the course is devoted to statistical computing and graphics. Prerequisite: three years of high school mathematics. Lecture-conference and lab.

Mathematics 211 - Multivariable Calculus I

Full course for one semester. A development of the basic theorems of multivariable differential calculus, optimization, and Taylor series. Inverse and implicit function theorems may be included. Prerequisite: Mathematics 112 or consent of the instructor. Lecture-conference.

Mathematics 212 - Multivariable Calculus II

Full course for one semester. A study of line, multiple, and surface integrals, including Green’s and Stokes’s theorems, linear differential equations. Differential geometry of curves and surfaces or Fourier series may be included. Prerequisite: Mathematics 211 or consent of the instructor. Lecture-conference.

Mathematics 311 - Complex Analysis

Full course for one semester. A study of complex valued functions: Cauchy’s Theorem and residue theorem, Laurent series, and analytic continuation. Prerequisite: Mathematics 212. Lecture-conference.

Mathematics 321 - Real Analysis

Full course for one semester. A careful study of continuity and convergence in metric spaces. Sequences and series of functions, uniform convergence, normed linear spaces. Prerequisite: Mathematics 212. Mathematics 331 must be taken before or at the same time as this course. Lecture-conference.

Mathematics 322 - Ordinary Differential Equations

Full course for one semester. An introduction to the theory of ordinary differential equations. Existence and uniqueness theorems, global behavior of solutions, qualitative theory, numerical methods. Prerequisite: Mathematics 212 and Mathematics 331. Lecture-conference. Offered in alternate years.

Mathematics 331 - Linear Algebra

Full course for one semester. A brief introduction to field structures, followed by presentation of the algebraic theory of finite dimensional vector spaces. Geometry of inner product spaces is examined in the setting of real and complex fields. Prerequisite: Mathematics 112 and 211 or consent of the instructor. Lecture-conference.

Mathematics 332 - Abstract Algebra

Full course for one semester. An elementary treatment of the algebraic structure of groups, rings, fields, and/or algebras. Prerequisite: Mathematics 331. Lecture-conference.

Mathematics 341 - Geometry

Full course for one semester. Topics in geometry selected by the instructor. Possible topics include the theory of plane ornaments, coordinatization of affine and projective planes, curves and surfaces, differential geometry, algebraic geometry, and non-Euclidean geometry. Prerequisite: Mathematics 331 or consent of the instructor. Lecture-conference. Offered in alternate years. Not offered 2009-10.

Mathematics 351 - Mathematical Logic

Full course for one semester. This course will be concerned with one or more of the following areas of mathematics: recursive function theory, model theory, computability theory, and general theory of formal systems. Prerequisite: two years of college mathematics. Lecture-conference. Offered in alternate years.

Mathematics 361 - Number Theory

Full course for one semester. A study of integers, including topics such as divisibility, theory of prime numbers, congruences, and solutions of equations in the integers. Prerequisite: Mathematics 331 or consent of the instructor. Mathematics 332 is recommended. Lecture-conference. Offered in alternate years. Not offered 2009-10.

Mathematics 372 - Combinatorics

Full course for one semester. Emphasis is on enumerative combinatorics including such topics as the principle of inclusion-exclusion, formal power series and generating functions, and permutation groups and Pólya theory. Selected other topics such as Ramsey theory, inversion formulae, the theory of graphs, and the theory of designs will be treated as time permits. Prerequisite: Mathematics 211. Lecture-conference. Offered in alternate years.

Mathematics 382 - Algorithms and Data Structures

Full course for one semester. An introduction to computer science covering the design and analysis of algorithms. The course will focus on various abstract data types and associated algorithms. The course will include implementation of some of these ideas on a computer. Prerequisites: Mathematics 121 and Mathematics 211 or consent of the instructor. Lecture-conference.

Mathematics 384 - Programming Language Design and Implementation

Full course for one semester. A study of the organization and structure of modern programming languages. This course will survey key programming language paradigms including functional, object-oriented, and logic- and constraint-based languages. A formal approach will be taken to understanding the fundamental concepts underlying these paradigms, including their syntax, semantics, and type systems. The course will cover selected topics in the implementation of language systems such as parsers, interpreters, and compilers, and of run-time support for high-level languages. Prerequisite: Mathematics 121. Lecture-conference. Offered in alternate years.

Mathematics 391 - Probability

Full course for one semester. A development of probability theory in terms of random variables defined on discrete sample spaces. Special topics may include Markov chains, Stochastic processes, and measure-theoretic development of probability theory. Prerequisite: Mathematics 212. Lecture-conference.

Mathematics 392 - Mathematical Statistics

Full course for one semester. Theories of statistical inference, including maximum likelihood estimation and Bayesian inference. Topics may be drawn from the following: large sample properties of estimates, linear models, multivariate analysis, empirical Bayes estimation, and statistical computing. Prerequisite: Mathematics 391 or consent of the instructor. Lecture-conference. Offered in alternate years.

Mathematics 411 - Topics in Advanced Analysis

Full course for one semester. Topics selected by the instructor. Prerequisite: Mathematics 321 or consent of the instructor. Lecture-conference.

Mathematics 412 - Topics in Algebra

Full course for one semester. Topics selected by the instructor, for example, commutative algebra, Galois theory, algebraic geometry, and group representation theory. Prerequisite: Mathematics 332 or consent of the instructor. Lecture-conference.

Mathematics 441 - Topics in Computer Science Theory

Full course for one semester. Exploration of topics from advanced algorithm design and theoretical computer science including complexity theory, cryptography, computational geometry, and randomized algorithms, as selected by the instructor. Prerequisite: Mathematics 382 or consent of the instructor. Lecture-conference. Offered in alternate years. Not offered 2009-10.

Mathematics 442 - Topics in Computer Science Systems

Full course for one semester. A study of the design and implementation techniques used in a particular area of computer science as selected by the instructor. Students will implement a working system in that area. Recent offerings have covered distributed and networked systems, compilers, and computer game design. Prerequisite: Mathematics 382 or consent of the instructor. Lecture-conference. Offered in alternate years.

Mathematics 443 - Topics in Advanced Algorithms

Full course for one semester. Exploration of topics from advanced algorithm design such as combinatorial optimization, computational geometry, parallel and distributed algorithms, and randomized algorithms, as selected by the instructor. Prerequisite: Mathematics 382 or consent of the instructor. Lecture-conference. Not offered 2009-10.

Mathematics 470 - Thesis

Full course for one year.

Mathematics 481 - Independent Study

One-half course for one semester. Independent reading primarily for juniors and seniors. Prerequisite: approval of the instructor and the division.