James D. Fix
Computer science, cryptography, and private data analysis.
Albert Y. Kim
Algebraic topology and category theory.
Algebraic geometry. On leave fall 2014.
Algebraic geometry, number theory, and quantum computation.
Number theory and complex analysis.
Differential geometry and ergodic theory.
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 PhDprograms 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), 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
- Mathematics 111 or the equivalent, 112, 211, and 212.
- Mathematics 321, 331, and 332.
- Four additional units in mathematics courses numbered higher than 300 (excluding Mathematics 470).
- Physics 101 and 102 or the equivalent.
- 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.
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 and linear differential equations. Differential geometry of curves and surfaces or Fourier series may be included. Prerequisite: Mathematics 112 and 211 or consent of the instructor. Lecture-conference.
Mathematics 221 - Computer Science Fundamentals II
Full course for one semester. A second course in computer science, an introduction to advanced structures and techniques. The course will develop the foundations of computing, providing an introduction to theoretical models of computation and also to practical computer system construction. Selected topics include: digital design, from gates to processors; the construction of interpreters, including language parsing and run-time systems; parallelism and concurrency; and universality. There will be significant programming projects exploring a number of these topics, and students will be introduced to the advanced programming techniques and data structures that support their construction. Prerequisite: Mathematics 121. Lecture-conference.
Mathematics 241 - Case Studies in Statistical Analysis
Full course for one semester. Applied statistics class with an emphasis on data analysis. The course will be problem driven with a focus on collecting and manipulating data, using exploratory data analysis and visualization tools, identifying statistical methods appropriate for the question at hand, and communicating the results in both written and presentation form. Prerequisite: Mathematics 141. Some knowledge of R or programming is strongly recommended. 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 331. Lecture-conference. Offered in alternate years.
Not offered 2014—15.
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 - Topics in 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 2014—15.
Mathematics 342 - Topology
Full course for one semester. An introduction to basic topology, followed by selected topics such as topological manifolds, embedding theorems, and the fundamental group and covering spaces. Prerequisite: Mathematics 332, which may be taken concurrently. Lecture-conference.
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.
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.
Not offered 2014—15.
Mathematics 374 - Divisor Theory of Graphs
Full course for one semester. This course will study graphs as discrete versions of Riemann surfaces and, dually, as dynamical systems exhibiting self-organized criticality (the Abelian sandpile model). Topics will include: the discrete Laplacian, Baker and Norine’s recent Riemann-Roch theory of graphs, Smith normal form, the matrix-tree theorem, Dhar’s burning algorithm, the Tutte polynomial, acyclic orientations, and the sandpile group. Depending on interest, additional topics may include spanning tree bijections, harmonic morphisms, pattern formation, matroids and duality, domino tilings, simplicial homology and higher-dimensional critical groups, Abelian networks, and tropical geometry. Prerequisite: Mathematics 331 or consent of the instructor. Lecture-conference.
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 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.
Not offered 2014—15.
Mathematics 385 - Computer Graphics
Full course for one semester. Introduction to computer image synthesis and mathematical modeling for computer graphics applications. Topics include image processing, 2-D and 3-D modeling techniques such as curve and surface representation, geometric algorithms for intersection and hidden surface removal, 3-D rendering, and animation. Prerequisite: Mathematics 121 and 211. Mathematics 331 recommended. Lecture-conference. Offered in alternate years.
Mathematics 387 - Computability and Complexity
Full course for one semester. Introduction to models of computation including finite automata, formal languages, and Turing machines, culminating in universality and undecidability. An introduction to resource-bounded models of computation and algorithmic complexity classes, including NP and PSPACE, and the notions of relative hardness and completeness. Prerequisite: Mathematics 121 and three semesters of college mathematics. Lecture-conference.
Mathematics 389 - Computer Systems
Full course for one semester. A study of the design and implementation of computing systems, focusing on aspects whose underpinnings are firmly based in algorithms and applied logic or whose implementation offers interesting problems in those areas. A survey of computer architecture and the hardware-software interface, compilation and run time, and concurrent and networked programming. An introduction to theoretical approaches to problems related to the synchronization and coordination of independently executing processes. Prerequisite: Mathematics 121 and additional programming experience equivalent to Mathematics 382, 384, or 385. Lecture-conference.
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.
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.
Not offered 2014—15.
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.