Mathematics and Statistics
Faculty:
Study of the mathematical sciences develops logical thinking and quantitative ability; mathematical skills in rigorous deductive analysis and in the use of data are germane to many disciplines. The curriculum of the department is divided into two basic parts: mathematics and statistics. Each provides a combination of fundamental theory and widely applicable material of interest to all students of liberal arts. The curriculum further prepares majors who plan careers in pure or applied mathematics, probability or statistics, or in the natural or social sciences, in teaching, or in other professions. Depending on their background and interests, students may enter the study of mathematics at different points. Those with good preparation normally start in 131, while those with less preparation may start in 123, and those with advanced standing in 133 or 215. Thereafter, the student’s intellectual curiosity, interests, and abilities and the needs of various disciplines determine the particular mathematics courses selected. Several courses make use of the department’s network of workstations for graphics, computation, data analysis, and numeric experimentation. Mathematics majors pursue many interests. All are encouraged to study in depth at least one field, such as physics or economics, in which mathematics is applied extensively. Some enjoy working on challenging problems, such as those presented in the Putnam Examination or the Mathematical Contest in Modeling, both of which are national mathematics competitions; many present talks to the Mathematics and Statistics Student Seminar. Visiting lecturers extend the curriculum beyond the classroom, as do opportunities for students to do summer research in mathematics.
A minimum of 32 credits in the department. Required are at least four courses in mathematics at the 300–400 level, including Math 316 and 321 (the “Foundations” courses) and at least one of the year sequences: Math 321-324, 321-326, 316-331, 316-338, 335-336. Courses numbered below 123 do not satisfy major requirements. Courses numbered 297, 299, 397, 399, 499 (MAPs) and plus-2’s do not count toward the major. With departmental approval, 4 credits of computer science may count toward the mathematics major. Strongly recommended: A working knowledge of a modern computer programming language; coursework in another department in which mathematics or statistics is used in a substantial way; and (for students considering graduate work in mathematics) a reading knowledge of French, German, or Russian. Up to 8 credits can be earned for any combination of Mathematics 123, 124, 131, subject to the following constraints:
- Upon successful completion (grade C or better) of either Mathematics 124 or Mathematics 131, no further credits may be earned in any of these three courses.
- If a student completes all three of Mathematics 123, 124, and 131, the student’s credit is canceled in the first of these courses in which the student earned a grade of F or D. Also, the grade for that course will no longer be counted in computing the student’s G.P.A.
To be considered for honors in mathematics, graduating seniors, in addition to meeting the College’s general requirements for honors, must demonstrate excellence in the major. The department applies the following criteria:
- Excellent performance in two areas of upper division mathematics and statistics courses: Algebra (321-324 or 321-326), Analysis (316-331 or 316-338), Probability and Statistics (335-336), Applied Statistics (309-310), and Applied Mathematics (306-314).
- Participation in local activities related to mathematics, judged to be excellent by members of the department. Such activities might include completing the senior seminar, giving Mathematics and Statistics student seminar talks, actively participating in the Problem-Solving Seminar, doing independent projects in mathematics, or carrying out summer research under the direction of members of the department;
- Performance in the study or use of mathematics, judged to be excellent by mathematicians outside the department. Evidence of such performance might include an outstanding score in the Putnam Competition or the Iowa Mathematics Competition, a score at or above the 75th percentile on the Graduate Record Examination in Mathematics, an award in the Mathematical Competition in Modeling, passing actuarial exams, or refereed talk at a mathematical conference or colloquium, a paper accepted by a refereed mathematical journal, or summer research conducted elsewhere.
Individual or small group instruction in math skills at the precalculus level. Specific content for the course determined for the individual student based on his or her math background and needs. This course is open to any student who has not completed a math course at the 124 level or higher. Students must have permission of instructor to register. May be repeated once for credit, with permission of the director. Note: instruction in basic math skills is available without credit for all students. S/D/F only.
Also listed as Social Studies 115. Introduces the notions of variability and uncertainty and such common statistical concepts as point and interval estimation and hypothesis testing. Data-oriented, with real-world examples chosen from the social and biological sciences. The computer is used for data analysis and to illustrate probabilistic and statistical concepts.
An introductory course in mathematics and the first in a two-course sequence. This first semester is an introduction to the differential calculus of functions of one variable with an extensive review of precalculus topics such as algebra and functions. This review, together with an emphasis on developing problem-solving skills, is designed to help students learn to do mathematics at the college level. Mathematics 123-124 has the same calculus content as Mathematics 131.
A continuation of Mathematics 123. An introduction to the integral calculus of functions of one variable. Topics include the definite integral, techniques of integration, and applications of the integral. Successful completion of this course prepares students for Mathematics 133.
The normal first course in mathematics and the first in a two-course sequence. An introduction to the differential and integral calculus of functions of one variable. Also introduces a few concepts and methods of differential equations.
A continuation of Mathematics 131. Topics include functions of more than one variable: partial and total derivatives, multiple integrals, vector-valued functions, parametrized curves, and applications to differential equations.
The course covers the application of basic statistical methods such as univariate graphics and summary statistics, basic statistical inference for one and two samples, linear regression (simple and multiple), one- and two-way ANOVA, and categorical data analysis. Students use statistical software to analyze data and conduct simulations. A student who takes Mathematics 209 cannot receive credit for Mathematics 115.
A unified study of the concepts underlying linear systems and linear transformations and of the techniques for using them. Topics: matrix algebra, rank, orthogonality, vector spaces and dimension, eigenvectors and eigenvalues. Typical applications: fitting lines and curves to data, Markov processes, linear differential equations.
An introduction to the basic objects, numbers, and techniques of combinatorics. Includes combinations, permutations, partitions, and graphs; binomial and other coefficients; inclusion-exclusion, recurrence relations, and generating functions and series. The course will emphasize proof writing and basic logic.
First and second order differential equations; series solutions and Fourier series; linear and nonlinear systems of differential equations; applications.
A study of Euclidean geometry and various non-Euclidean geometries. All topics will be motivated from an axiomatic standpoint, then specific models satisfying certain sets of axioms will be studied. The course will emphasize proof writing and basic logic.
The course covers divisibility questions among integers, elementary properties of prime numbers, the unique factorization theorem of positive integers, congruences, the Chinese remainder theorem, Fermat’s little theorem, Diophantine equations, and quadratic reciprocity. The course will emphasize proof writing and basic logic.
Students solve challenging mathematics problems and present solutions. Prepares students to take the Putnam Examination, if they wish. May be repeated for credit. S/D/F only.
An introduction to the process and techniques of modeling “real-world” situations, using topics from linear algebra and differential equations. Appropriate mathematics, including numerical methods, developed when needed. Models drawn from both the social and natural sciences. Offered in Alternate years.
In addition to a short review of hypothesis testing, confidence intervals, and 1-way ANOVA, this course incorporates experiments from several disciplines to explore design and analysis techniques. Topics include factorial designs, block designs (including latin square and split plot designs), random, fixed, and mixed effects models, crossed and nested factors, contrasts, checking assumptions, and proper analysis when assumptions are not met. Offered in Alternate years.
This course will focus on investigative statistics labs emphasizing the process of data collection and data analysis relevant for science, social science, and mathematics students. These labs will incorporate current events and interdisciplinary research, taking a problem-based approach to learn how to determine which statistical techniques are appropriate. Topics will typically include nonparametric tests, designing an experiment, and generalized linear models. Offered in alternate years.
Topics include, but are not limited to, one of the following: Chaos and Fractals (one- and two-dimensional discrete dynamics, iterated function systems, fractal dimension), Fourier analysis (fast Fourier transform, Fourier series, wavelets), or partial differential equations (heat and wave equation, eigenfunction expansions). May be repeated for credit. Offered in alternate years.
A thorough study of the topology of the real line and of limits of functions of one real variable. This theory is then used to develop the theory of the derivative and integral of functions of one real variable and also sequences and series of real numbers and functions.
The study of algebraic structures, with emphasis on formal systems such as groups, rings, and fields.
The primary subject matter of this course is elementary number theory from an algebraic viewpoint. Topics include congruencies, quadratic reciprocity, sums of powers and Diophantine analysis. An introduction to algebraic number theory, emphasizing algebraic integers and unique factorization, is included.
The study of fields, including such topics as vector spaces and canonical forms, algebraic extensions, finite and cyclotomic fields, geometric constructions and Galois theory. Offered in alternate years.
General and/or metric topology. Fundamental theorems on continuous mappings and on compact and connected sets. Particular emphasis on the real line and Euclidean n-space. Offered in alternate years.
An introduction to the mathematical theory of probability and statistical inference. Discrete and continuous distributions will be considered. The limit theorems of probability, including the Law of Large Numbers and the Central Limit Theorem, will be introduced.
A systematic treatment of mathematical statistics based on probability theory. Topics will include: principles of estimation and hypothesis testing, regression, sampling distributions, decision theory, and nonparametric inference. Applications will be given.
Theory of analytic functions of a complex variable, based on a preliminary study of the complex number system. Offered in alternate years.
Advanced course with varying content, typically with a geometric emphasis. Strongly recommended for students considering further work in mathematics. Requires independent work. May be repeated for credit when content changes.
* Indicates courses not offered every year.
