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Mathematics and Statistics
Member of the Division of Science
Marc Chamberland, Chair, Arnold Adelberg, Christopher French, Eugene Herman, Charles Jepsen, Shonda Kuiper, Emily Moore, Thomas Moore, Holly Mosley, David Romano, Karen Shuman, A. Royce Wolf
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.
The Major
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. 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.
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:
• Completion of two disjoint full-year upper-division sequences in mathematics;
• 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.
Up to 8 credits can be earned for any combination of Mathematics 123, 124, 131, subject to the following constraints:
1. 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.
2. 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 GPA.
115 Introduction to Statistics (Fall or Spring) 4 credits
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. Prerequisite: two years of high school algebra and second semester of first-year standing, or permission of instructor. STAFF.
123 Functions and Differential Calculus (Fall) 4 credits
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. Prerequisite: two years of high school algebra. STAFF.
124 Functions and Integral Calculus (Spring) 4 credits
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. Prerequisite: Mathematics 123 or permission of instructor. WOLF.
131 Calculus I (Fall or Spring) 4 credits
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. Prerequisite: good preparation, including trigonometry, or departmental placement. STAFF.
133 Calculus II (Fall or Spring) 4 credits
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. Prerequisite: Mathematics 124, or 131, or permission of instructor. STAFF.
209 Applied Statistics (Fall or Spring) 4 credits+
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. Prerequisite: Mathematics 133 or permission of instructor. STAFF.
215 Linear Algebra (Fall or Spring) 4 credits+
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. Prerequisite: Mathematics 133 or permission of instructor. STAFF.
218 Combinatorics (Fall or Spring) 4 credits+
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. Prerequisite: Mathematics 215 or permission of instructor. STAFF.
220 Differential Equations (Fall or Spring) 4 credits+
First and second order differential equations; series solutions and Fourier series; linear and nonlinear systems of differential equations; applications. Prerequisite: Mathematics 215 or permission of instructor. STAFF.
271 Problem-Solving Seminar (Fall) 1 credit
Students solve challenging mathematics problems and present solutions. Prepares students to take the Putnam Examination, if they wish. Prerequisite: Mathematics 133, or concurrent registration in Mathematics 133, or permission of instructor. May be repeated for credit. S/D/F only. STAFF.
306 Mathematical Modeling* (Spring) 4 credits+
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. Prerequisite: Mathematics 220 or permission of instructor. CHAMBERLAND.
309 Design and Analysis of Experiments* (Spring) 4 credits+
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. Prerequisite: Mathematics 209, or 336, or permission of instructor. KUIPER.
310 Statistical Modeling* (Spring) 4 credits+
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. Prerequisite: Mathematics 209, or Mathematics 336, or permission of instructor. KUIPER.
314 Topics in Applied Mathematics* (Spring) 4 credits+
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. Prerequisite: Mathematics 220 or permission of instructor. SHUMAN.
316 Foundations of Analysis (Spring) 4 credits+
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. Prerequisite: Mathematics 218, or 220, or permission of instructor. STAFF.
321 Foundations of Abstract Algebra (Fall) 4 credits+
The study of algebraic structures, with emphasis on formal systems such as groups, rings, and fields. Prerequisite: Mathematics 218, or 220, or permission of instructor. STAFF.
324 Number Theory* (Spring) 4 credits+
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. Prerequisite: Mathematics 321 or permission of instructor. WOLF.
326 Field Theory* (Spring) 4 credits+
The study of fields, including such topics as vector spaces and canonical forms, algebraic extensions, finite and cyclotomic fields, geometric constructions and Galois Theory. Prerequisite: Mathematics 321 or permission of instructor. ROMANO.
331 Topology* (Fall) 4 credits+
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. Prerequisite: Mathematics 316 or permission of instructor. FRENCH.
335 Probability and Statistics I (Fall) 4 credits+
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. Prerequisite: Mathematics 215; and 209, or 218, or 220; or permission of instructor. STAFF.
336 Probability and Statistics II (Spring) 4 credits+
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. Prerequisite: Mathematics 335 or permission of instructor. STAFF.
338 Complex Analysis* (Fall) 4 credits+
Theory of analytic functions of a complex variable, based on a preliminary study of the complex number system. Prerequisite: Mathematics 316 or permission of instructor. FRENCH.
444 Senior Seminar (Spring) 4 credits+
Advanced course with varying content, typically with a geometric emphasis. Strongly recommended for students considering further work in mathematics. Requires independent work. Prerequisite: Mathematics 316 and 321, or permission of instructor. May be repeated for credit when content changes. STAFF.
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