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# Mathematics

Professors: McKeon, Susskind; Associate Professor: Johnson; Assistant Professors: Kohli, O'Keefe; Visiting Lecturer: Thompson; Associate Professor Hammond, *chair*

**The Major in Mathematics**

The mathematics major consists of five core courses (210, 212, 226, 301, and 303), as well as the mathematics seminar (495) and additional mathematics courses based on the track or concentration.

Students selecting the general track must take four additional courses: one course from 309, 317, 402, or 404; one elective at the 200 level or higher; and two further electives at the 300 level or higher.

Students selecting the statistics concentration must take five additional courses: 207, 209, 316, 317, and one elective at the 300 level or higher.

All mathematics majors are strongly encouraged to take at least one course in computer science. Students planning to attend graduate school in mathematics or statistics should consult with their adviser to develop an appropriate course of study.

Advisers: C. Hammond, W. Johnson, P. Kohli, K. McKeon, P. Susskind

**The Minor in Mathematics**

The minor in mathematics consists of a minimum of five courses: 113 (or 114), 212, 226, and two additional mathematics courses at the 200 level or higher. Students may, in consultation with their adviser, substitute appropriate courses at the same level or higher. Advanced Placement credit may only be counted toward the minor under exceptional circumstances, with permission of the department.

**The Minor in Applied Statistics**

The interdisciplinary minor in applied statistics is designed to help students develop a broad understanding of methods for collecting, analyzing, and interpreting data. Students learn the fundamental principles of statistics, experiment with techniques for examining and drawing conclusions from data, and study concrete applications of statistics to a variety of fields.

The minor consists of five courses chosen from the following: Mathematics 107, 111, 112, 113, 114, 206, 207, 208, 209, 212, 316, 317; Economics 230, 354; Psychology 201, 202, 407; Biology 203, 208, 298, 307; Botany 315; Computer Science 203, 204, 211. An individual study involving a substantial statistical component may also serve as one of the five courses. Students must take at least one two-course sequence in mathematics (either 207, 209 or 316, 317) and at least one course from a department other than Mathematics. No more than one calculus course (Mathematics 111, 112, 113, 114, or 212) and one introductory statistics course (Mathematics 107, 206, or Psychology 201) may be counted toward the minor; students may not use Advanced Placement credit as a substitute for more than one course. Mathematics majors may not minor in applied statistics, but are encouraged instead to select the statistics concentration within the major.

Students considering graduate study in statistics or in quantitative areas of other fields are encouraged to take courses in mathematics beyond the requirements for the minor, particularly Mathematics 212, 226, and 301.

Advisers: P. Barnes (Biology), P. Kohli (Mathematics), J. Nier (Psychology), Y. Park (Economics)

**Learning Goals in the Mathematics Major**

Mathematics majors are expected to master a significant body of material, including differential and integral calculus of one and several variables, discrete mathematics, and linear algebra. Upper-level courses in abstract algebra, real and complex analysis, and probability provide the theoretical underpinnings for much of modern mathematics, both pure and applied, including techniques and concepts encountered in earlier courses. Students also take a variety of electives, chosen to reflect their own interests, to represent the breadth of the discipline, and to introduce connections to other subjects. These electives may include differential equations, graph theory, mathematical methods for the physical sciences, theory of computation, topology, mathematical statistics, and a variety of other topics. Students may select a specialized course of study that leads to a concentration in statistics. Students are also exposed to a variety of special topics through colloquia and seminar talks sponsored by the department. All students are expected, at some point during their junior or senior year, to give a talk at the departmental seminar on a topic they have independently researched under the guidance of a faculty member. Many majors further develop their mathematical and expository skills by working as student tutors in the Math Help Center.

Mathematics majors acquire a substantial body of mathematical knowledge, become proficient with a wide array of problem-solving techniques, and develop an awareness and appreciation for the vast scope of the discipline. Successful majors are able to employ the techniques they have learned, aided by technology when appropriate, to solve problems in mathematics itself, in statistics, and in a number of other fields, including computer science, the natural and social sciences, engineering, and finance. The techniques and arguments they employ may be geometric, algebraic, analytic, graphical, probabilistic, or statistical, and may include constructing mathematical models. Students also develop the ability to communicate their solutions cogently, both orally and in writing. Most importantly, successful majors learn to construct valid mathematical proofs; that is, to make rigorous arguments to prove or disprove mathematical conjectures. All of these skills help prepare students for a wide variety of potential careers (such as secondary education, financial services, and information technology), as well as graduate study in a number of disciplines (including mathematics, applied mathematics, and statistics).

In summary, students will be able to:

- Acquire a comprehensive knowledge of the fundamental concepts underlying the discipline of mathematics, as well as material from specific courses of their own selection.
- Use mathematical methods and skills to solve a wide variety of problems, both within mathematics and in other disciplines.
- Analyze and prove mathematical statements, effectively communicating their ideas both orally and in writing.
- Become fluent with increasing levels of mathematical abstraction.
- Master sophisticated techniques from advanced courses.
- Attend and participate in talks from both local and visiting mathematics faculty on advanced topics.
- Research new topics independently, analyze them, and present them in a cogent way to their peers and professors.

**Courses **

**MATHEMATICS 105 INTRODUCTION TO MATHEMATICAL THOUGHT **Mathematics as a creative and evolving discipline. Traditional and modern mathematical concepts presented by surveying different areas in mathematics or focusing on a particular theme such as number theory or mathematics and politics. Focus on mathematical concepts rather than on drill.

Not open to students who have received credit for a college-level mathematics course. Enrollment limited to 30 students. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 107 INTRODUCTION TO STATISTICS **An introduction to basic statistical methods and concepts. Topics include exploratory data analysis, experimental design, sampling, inference for means and proportions, regression, and categorical data. Statistical software used to analyze real data. Students may not receive credit for both Courses 107 and 206.

Students with previous credit for a 200- or 300-level course in mathematics must receive permission of the instructor. Enrollment limited to 30 students. This course satisfies General Education Area 2. *Staff*

**MATHEMATICS 108 MATHEMATICS OF MONEY **An introduction to elementary mathematical concepts relating to finance, including simple and compound interest, annuities, mortgages, and bonds. Emphasis on the use of mathematics both to understand financial topics and to draw conclusions about them.

Not open to students who have received credit for a college-level mathematics course. Enrollment limited to 30 students. This course satisfies General Education Area 2. *Staff*

**MATHEMATICS 110 NETWORKS AND THEIR APPLICATIONS** An introduction to the use of networks (discrete graphs) as modeling tools in a wide variety of applications. Examples include representing chemical compounds, routing snowplows, scheduling courses, sequencing traffic lights, representing data in a computer, describing interpersonal relationships, and solving puzzles and games.

Enrollment limited to 25 students. This course satisfies General Education Area 2. *K. McKeon*

**MATHEMATICS 111 CALCULUS A: CALCULUS WITH PRECALCULUS **An introduction to differential and integral calculus, in which the relevant precalculus background is also developed. Topics include functions, limits, derivatives and integrals, along with applications to rates of change, velocity, acceleration, optimization, and area.

Students are encouraged to have a departmental interview to determine the appropriate level at which to enter the calculus sequence. Course 111 is a suitable starting point for students who have had no previous exposure to calculus or who do not have a strong background in mathematics. Enrollment limited to 30 students. Offered both semesters. This course satisfies General Education Area 2. *Staff*

**MATHEMATICS 112 CALCULUS B: DERIVATIVES AND INTEGRALS **A continuation of the study of differential and integral calculus. Topics include the mean value theorem and l′Hospital′s rule; extremal problems and curve sketching; definite and indefinite integrals; techniques of integration; and applications of the definite integral to problems relating to area and volume.

* Prerequisite:* Course 111 or permission of the instructor. Students are encouraged to have a departmental interview to determine the appropriate level at which to enter the calculus sequence. Course 112 is a suitable starting point for students who have had previous exposure to calculus but have not received Advanced Placement credit. Enrollment limited to 30 students. Offered both semesters. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 113 CALCULUS C: INTEGRALS AND SERIES **A continuation of the study of integral calculus and an introduction to sequences, series, parametric equations, and polar coordinates. Specific topics include trigonometric substitution, partial fractions decomposition, and improper integrals, as well as convergence tests, power series, and Taylor polynomials. Additional topics may include arc length, surface area, probability, and elementary differential equations.

* Prerequisite:* Course 112 or permission of the instructor. Students are encouraged to have a departmental interview to determine the appropriate level at which to enter the calculus sequence. Course 113 is a suitable starting point for students who have received Advanced Placement credit for the Calculus AB examination. Enrollment limited to 30 students. Offered both semesters. This course satisfies General Education Area 2. *Staff*

**MATHEMATICS 115 MATHEMATICS FROM A CULTURAL PERSPECTIVE **Seminar focusing on the practice of mathematics within different cultural groups and societies, in either historical or contemporary settings. Groups are defined according to ethnic, geographic, or social criteria. Specific themes chosen from concepts such as infinity, number, symbols, and the geometric.

Enrollment limited to 16 students. This course satisfies General Education Area 2 and is a designated Writing course. *Staff *

**MATHEMATICS 120, 220, 320 SERVICE-LEARNING PRACTICUM IN MATHEMATICS **Service in an area school to enhance understanding of a concurrent mathematics course by working with students at an area school for a minimum of two hours per week. Specific projects to teach the students about the subject of the related mathematics course are developed in consultation with the professor and schoolteacher. Students electing Course 120, 220, 320 must concurrently enroll in a four credit mathematics course. Two credit hours. This course may be taken for credit two times.

Permission of the instructor. *Staff*

**MATHEMATICS 205 ENVIRONMENTAL MODELING **An introduction to the use of mathematics to understand and describe issues relating to the environment. Applications to geophysics (atmospheric carbon content, surface water runoff, pollutant dispersion, resource depletion) and biology (population growth, harvesting, extinction) will be considered. Students will both develop and implement mathematical models. This is the same course as Environmental Studies 205.

*Prerequisite:* Any calculus course (111, 112, 113, 114, or 212) or permission of the instructor. Enrollment limited to 16 students. *Staff*

**MATHEMATICS 206 INTRODUCTION TO STATISTICAL METHODS **An introduction to statistical methodology for students who have taken a semester or more of calculus. Topics include descriptive statistics; probability distribution of random variables; point and interval estimation of mean and proportion; statistical hypothesis testing; linear regression; and introductory aspects of the design and analysis of experiments. Weekly computer labs allow students to apply statistical techniques to analyze real data. Students may not receive credit for both Courses 107 and 206.

* Prerequisite: *Any calculus course (111, 112, 113, or 212) or permission of the instructor. Enrollment limited to 30 students. Offered first semester. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 207 ADVANCED REGRESSION TECHNIQUES **An introduction to simple linear regression, multiple linear regression, ordinary least squares estimation, model diagnostics, transformation of variables, weighted least squares estimation, variable selection, collinearity, and logistic regression. The course employs a case-study approach, with extensive use of statistical software to analyze real data and interpret the results of the analysis

* Prerequisite:* Any introductory statistics course (Course 107, 206, or Psychology 201); or Course 112, 113, or 212; or permission of the instructor. Enrollment limited to 30 students. Offered first semester. This course satisfies General Education Area 2. *P. Kohli *

**MATHEMATICS 208 DESIGN AND ANALYSIS OF EXPERIMENTS **An introduction to simple comparative designs, factorial designs, block designs, and post-hoc comparisons. Additional topics chosen from nested designs, repeated measures, and random effects models. The course employs a case-study approach, with extensive use of statistical software to design experiments that are optimally suited to specific conditions.

*Prerequisite:* Any statistics course (Course 107, 206, 207, 209, 317, or Psychology 201) or permission of the instructor. Enrollment limited to 30 students. This course satisfies General Education Area 2. *P. Kohli *

**MATHEMATICS 209 INTRODUCTION TO TIME SERIES ANALYSIS **An introduction to the theory and methods of modern time series analysis. Topics include univariate time series; stationary and non-stationary processes; linear time series models, including AR, MA, ARMA and ARIMA models; estimation of the mean and autocovariance function; statistical tests for white noise; forecasting methods; and applications to economics, biology, astronomy, physics, social sciences, and related areas.

*Prerequisite:* Course 207 or Economics 230. Enrollment limited to 30 students. This course satisfies General Education Area 2. *P. Kohli*

**MATHEMATICS 210 DISCRETE MATHEMATICS **An introduction to topics in discrete mathematics, including set theory, logic, equivalence relations, mathematical induction, combinatorics, graphs, trees, algorithm analysis, and elementary number theory. Applications to computer science will be considered.

* Prerequisite:* Any calculus course (111, 112, 113, 114, or 212) or Computer Science 110. Enrollment limited to 30 students. Offered first semester. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 212 MULTIVARIABLE CALCULUS **An introduction to vectors in Euclidean spaces, functions of several variables, partial derivatives, multiple integrals, vector fields, and line integrals, culminating with a treatment of Green′s theorem. Applications include curvature, tangent planes, volumes, and extremal problems with and without constraints.

* Prerequisite: * Course 113 or 114, or permission of the instructor. Students are encouraged to have a departmental interview to determine the appropriate level at which to enter the calculus sequence. Course 212 is a suitable starting point for students who have received Advanced Placement credit for the Calculus BC examination. Enrollment limited to 30 students. Offered both semesters. This course satisfies General Education Area 2. *Staff*** **

**MATHEMATICS 225 ORDINARY DIFFERENTIAL EQUATIONS **Techniques for solving first order differential equations and higher order linear equations, including methods involving power series and Laplace transforms. Applications may include exponential growth and decay, physical vibrations, electrical circuits, planetary motion, falling bodies, and population growth.

* Prerequisite:* Course 113, 114, or 212; or permission of the instructor. Enrollment limited to 30 students. Offered second semester. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 226 LINEAR ALGEBRA **An introduction to standard topics in linear algebra, including systems of linear equations, matrices, determinants, vectors, vector spaces, linear transformations, eigenvalues, and eigenvectors. Applications to calculus, geometry, economics, and the physical sciences may be considered.

* Prerequisite:* Course 113, 114, or 212; or permission of the instructor. Enrollment limited to 30 students. Offered both semesters. This course satisfies General Education Area 2. *Staff *

**MATHEMATICS 232 MATHEMATICS OF FINANCE **An introduction to mathematical techniques used to define and analyze securities and investments, including concepts such as cash flow, investments, markets, arbitrage, dynamics, risk aversion, pricing, and hedging.

* Prerequisite: *Course 113, 114, or 212; or permission of the instructor. Enrollment limited to 30 students. This course satisfies General Education Area 2. *P. Susskind *

**MATHEMATICS 301 REAL ANALYSIS I **An introduction to the rigorous study of real analysis. Topics include elementary set theory, the real number system, sequences, series, basic topological properties, continuous functions, and derivatives. Additional topics may include metric spaces, uniform convergence, special functions, Riemann integrals, and Stieltjes integrals. Emphasis on understanding and writing mathematical proofs.

* Prerequisite:* Course 212 and either 225 or 226, or permission of the instructor. Enrollment limited to 30 students. Offered first semester. This is a designated Writing course. *Staff *

**MATHEMATICS 303 ABSTRACT ALGEBRA I **An introduction to abstract algebraic structures. Topics include groups, subgroups, permutation groups, cosets, homomorphisms, factor groups, rings, rings of polynomials, and fields. Emphasis on understanding and writing mathematical proofs.

* Prerequisite:* Courses 210 and 226, or permission of the instructor. Enrollment limited to 30 students. Offered second semester. This is a designated Writing course. *Staff *

**MATHEMATICS 305, 306 SELECTED TOPICS **A study of topics selected from any area of pure mathematics, applied mathematics, or statistics. Topics vary from year to year and may include number theory, chaos and dynamical systems, numerical analysis, or statistical computing. Computer software may be used for research and experimentation. May be repeated for credit.

Prerequisites vary depending on the choice of topics, and will be communicated to students by the department. Enrollment limited to 30 students. *Staff*

**MATHEMATICS 309 COMPLEX ANALYSIS **An introduction to functions of a complex variable, with particular emphasis on the theory of analytic functions. Topics include the algebraic and geometric structure of the complex number system; the extension of exponential, trigonometric, and logarithmic functions to complex arguments; differentiation and integration in the complex plane; series representations for analytic and meromorphic functions; and the calculus of residues.

* Prerequisite:* Course 301; or 212 and either 225 or 226; or permission of the instructor. Enrollment limited to 30 students. *Staff*

**MATHEMATICS 310 GRAPH THEORY **Structure and properties of graphs and their applications. Topics include traversability, trees, connectivity, network flow, graph coloring, chromatic number, and planarity. Discussion of the application of graph theory to computer science, transportation, scheduling, communication, chemistry, and a variety of other fields.

* Prerequisite:* Course 210. Enrollment limited to 30 students. *K. McKeon*

**MATHEMATICS 311 ADVANCED LINEAR ALGEBRA **A continuation of the material introduced in Course 226, with emphasis on the underlying mathematical theory. Topics include invariant subspaces, inner product spaces, orthonormal bases, orthogonal projections, linear functionals, adjoints, self-adjoint and normal operators, and the spectral theorem.

* Prerequisite*: Course 226. Enrollment limited to 30 students. *C. Hammond*

**MATHEMATICS 312 MATHEMATICAL METHODS FOR THE PHYSICAL SCIENCES **Topics important in both advanced mathematics and the sciences, principally physics. These may include complex functions and power series; multiple integration; change of variables; the Jacobian; elementary Fourier analysis; series solutions of differential equations; orthogonal bases, e.g., Legendre polynomials, and special functions; partial differential equations, e.g. Laplace's, Poisson's, diffusion or heat flow equations; integral transforms; and physical examples.

* Prerequisite:* Course 225 and one of Course 226 or Course 212, or permission of the instructor*. *Enrollment limited to 30 students. *P. Susskind*

**MATHEMATICS 314 EUCLIDEAN AND NONEUCLIDEAN GEOMETRY **A study of Euclidean and one or more non-Euclidean geometries. The geometric theory, its historical setting, its physical and philosophical implications will all be treated. The purpose of the course will be to clarify the role of Euclidean geometry in mathematics, to introduce the ideas of axiom systems and their central role in mathematics, and to shed further light on the nature of mathematics.

* Prerequisite*: Course 113 or Course 226, and permission of the instructor. Enrollment limited to 30 students. *P. Susskind*

**MATHEMATICS 315 TOPOLOGY **An introduction to point-set topology, with emphasis on connections to analysis and geometry. Topics include topological spaces, product spaces, continuous functions, metric spaces, connectedness, compactness, countability conditions, and separation axioms.

* Prerequisite:* Courses 210 and 301, or permission of the instructor. Enrollment limited to 30 students. *C. Hammond*

**MATHEMATICS 316 PROBABILITY **A study of the theory relating to problems of randomness and uncertainty. Topics include conditional probabilities, random variables, discrete and continuous distributions, expected value and variance, joint distributions, and the law of large numbers. Applications to a variety of disciplines will be considered. Emphasis on preparation for Course 317.

* Prerequisite:* Courses 210 and 212, or permission of the instructor. Enrollment limited to 30 students. Offered every third semester. *P. Kohli, K. McKeon*

**MATHEMATICS 317 MATHEMATICAL STATISTICS **An introduction to methods of statistical inference, with emphasis on the underlying mathematical theory. Topics include estimation, hypothesis testing, and modes of convergence.

* Prerequisite:* Course 316. Enrollment limited to 30 students. *P. Kohli *

**MATHEMATICS 323 THEORY OF COMPUTATION **An introduction to the classical and contemporary theory of computation, including abstract automata theory, formal languages, computability by Turing machines and recursive functions, computability and decidability, and computational complexity. This is the same course as Computer Science 323.

* Prerequisite:* Course 210. Enrollment limited to 30 students. *P. Susskind, C. Chung *

**MATHEMATICS 402 REAL ANALYSIS II **A continuation of topics from Course 301.

* Prerequisite:* Course 301. Enrollment limited to 16 students. *Staff *

**MATHEMATICS 404 ABSTRACT ALGEBRA II** A continuation of topics from Course 303.

* Prerequisite:* Course 303. Enrollment limited to 16 students. *Staff *

**MATHEMATICS 495 SEMINAR IN MATHEMATICS **Lectures and discussions on topics of current interest to the mathematical community. These discussions will be led by Connecticut College faculty, advanced students, and visiting mathematicians.

* Prerequisite:* Course 301 or 303, and permission of the instructor. One meeting per week throughout the semester. Two credit hours. This course may be taken for credit two times. Enrollment limited to 16 students.

**MATHEMATICS 291, 292 INDIVIDUAL STUDY** Independent work with a selected faculty member. Course may be taken for either two or four credits.

**MATHEMATICS 391, 392 INDIVIDUAL STUDY **Independent work with a selected faculty member. Course may be taken for either two or four credits.** **

**MATHEMATICS 491, 492 INDIVIDUAL STUDY **Independent work with a selected faculty member. Course may be taken for either two or four credits.

**MATHEMATICS 497-498 HONORS STUDY**