Mathematics

Mathematics

COURSES

Course Numbering System: For most courses numbered 20 or above, the last digit in the course number indicates the field of mathematics as follows: probability and statistics, 0; algebra, 1; geometry, 2; analysis, 3; topology, 4; number theory, 5; applications, 6; combinatorics, 8; logic and foundations, 9.

Course Prerequisites: In all cases in which a prerequisite to a course is listed, the honors or advanced placement equivalent of that course may be substituted. For example, wherever Mathematics 13 appears as a prerequisite, Mathematics 14 will serve. Mathematics 11 and 12 also serve in place of Mathematics 13 as a prerequisite.

UNDERGRADUATE COURSES

1. Calculus with Algebra

09F: 11, 12 10F: Arrange

The course description is given under Mathematics 2. This course is open only to students having the permission of the Department. Dist: QDS. The staff.

2. Calculus with Algebra and Trigonometry

10W: 9L, 11 11W: Arrange

Mathematics 1-2 is a two-term sequence. Its purpose is to cover the calculus of Mathematics 3, the standard introduction to calculus, and, at the same time, to develop proficiency in algebra. The sequence is specifically designed for first-year students whose manipulative skill with the techniques of secondary-school algebra is inadequate for Mathematics 3. The objective is to introduce and develop algebraic techniques as they are needed to study the ideas of calculus. The techniques will be taught in class, and the students will be required to practice by solving many drill problems for homework. There will be tutorial-help sessions.

Mathematics 1 will include the concepts of function and graph and the basic ideas and applications of differential and integral calculus, at least as they pertain to polynomial functions. In the second course, Mathematics 2, the study of calculus will be continued so that by the end of the sequence the students will have been introduced to the algebra and calculus of the exponential and logarithm functions and the trigonometric functions and to differential equations.

Prerequisite: Mathematics 1, or permission of the Department. Dist: QDS. The staff.

3. Introduction to Calculus

09F: 9L, 11, 11 10W: 10 10F, 11W: Arrange

This course is the basic introduction to calculus. Students planning to specialize in mathematics, computer science, chemistry, physics, or engineering should elect this course in the fall term. Others may elect it in the winter.

A study of polynomials and rational functions leads to the introduction of the basic ideas of differential and integral calculus. The course also introduces exponential, logarithmic, and trigonometric functions. The emphasis throughout is on fundamental ideas and problem solving.

Mathematics 3 is open to all students who have had intermediate algebra and plane geometry. No knowledge of trigonometry is required. The lectures are supplemented by problem sessions. Dist: QDS. Lahr, Pomerance, Yang (fall), Trout (winter).

4. Applications of Calculus to Medicine and Biology

10W: 10A 11W: Arrange

This course will establish the relevance of calculus to medicine. It will develop mathematical tools extending the techniques of introductory calculus, including some matrix algebra and solution techniques for first order differential equations. These methods will be used to construct simple and elegant models of phenomena such as the mutation of HIV, spread of infectious disease, and biological disposition of drugs and inorganic toxins, enzyme kinetics and population growth.

Prerequisite: Mathematics 3. Note: This is a second-term calculus course, but it does not cover the same material as Mathematics 8, and does not serve as a prerequisite for Mathematics 13. There is a version of this course suitable for major credit: see Mathematics 27. Dist: QDS. Wallace.

5. Exploring Mathematics

09F: 10A 10W: 10 10F, 11W: Arrange

In 09F, Pattern. “Pattern” is a course in math and art taught by a mathematician. In the same way that a social science course might well require writing, this math course requires you to create pieces of art that demonstrate your understanding of the mathematics we are learning. Our main object of study is repeated pattern, such as we commonly see in textiles, tilings, basketry and other contexts. In this course we will see parts of mathematics that are usually reserved for those majoring in the subject. We will also improve our art abilities through the creation, viewing and discussion of our work. Occasionally we will have guest experts come in to critique student work. Dist: QDS. Wallace.

In 10W, The World According to Mathematics. The aim of The World According to Mathematics is to provide the means to see everyday aspects of the world around us through mathematical eyes. The underlying theme of the course is an examination of the notion of representing reality and the fundamental role in this process played by mathematics throughout the history of Western civilization, right up to the present day. Intended for students whose major interests are outside mathematics and the sciences, the course assumes that students have only a background in high school mathematics. By considering specific examples, we will learn to express a substantial number of our experiences in the language of mathematics. We will see that mathematics, especially through mathematical algorithms, is everywhere, whether it be in ISBN book codes or in the swing of a pendulum. Dist: QDS. Lahr.

6. Introduction to Finite Mathematics

Not offered in the period from 09F through 11S

This course is designed for students whose interests lie outside the physical sciences. The course includes an introduction to sets and logic, elementary counting techniques, an introduction to probability, and topics in matrix algebra including the solution of systems of equations and matrix inverses. Illustrative examples and problems will be chosen from the social, managerial, and biological sciences. Computing may be used to illustrate concepts and solve problems. No background in computer programming is assumed. The course is appropriate for the student who plans to take no advanced courses in mathematics. Dist: QDS.

7. First-Year Seminar in Mathematics

Consult special listing

8. Calculus of Functions of One and Several Variables

09F: 10, 11, 12 10W: 12, 2 10S: 11, 2 10F, 11W, 11S: Arrange

This course is a sequel to Mathematics 3 and is appropriate for students who have successfully completed an AB calculus curriculum in secondary school. Roughly half of the course is devoted to topics in one-variable calculus: techniques of integrations, areas, volumes, trigonometric integrals and substitutions, numerical integration, sequences and series including Taylor series.

The second half of the course generally studies scalar valued functions of several variables. It begins with the study of vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength). The rest of the course is devoted to studying differential calculus of functions of several variables. Topics include limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems including the use of Lagrange multipliers.

Prerequisite: Mathematics 3 or equivalent. Dist: QDS. Yang, Gordon, Arkowitz (fall), Vatter, Vatter (winter), the staff (spring).

9. Calculus of Functions of One and Several Variables, Honors Section

Not offered in the period from 09F through 11S

Open to interested students qualified to take Mathematics 8. Dist: QDS.

10. Introductory Statistics

10S: 2 11S: Arrange

An introduction to the basic concepts of statistics. Topics include elementary probability theory, descriptive statistics, the binomial and normal distributions, confidence intervals, basic concepts of tests of hypotheses, chi-square tests, nonparametric tests, normal theory t-tests, correlation, and simple regression. Packaged statistical programs will be used. Because of the large overlap in material covered, no student may receive credit for more than one of the courses Economics 10, Government 10, Mathematics 10, Psychology 10, Social Sciences 10, or Sociology 10 except by special petition. Dist: QDS. Weber.

11. Multivariable Calculus for Two-Term Advanced Placement First-Year Students.

09F: 10, 11 10F: Arrange

This course can be viewed as equivalent to Mathematics 13, but is designed especially for first-year students who have successfully completed a BC calculus curriculum in secondary school. In particular, as part of its syllabus it includes most of the multivariable calculus material present in Mathematics 8.

Topics include vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength), limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems. It continues with multiple integration, vector fields, line integrals, and finishes with a study of Green’s and Stokes’ theorem. Dist: QDS. Sutton, Pauls.

12. Calculus Plus

09F: 10, 2 10F: Arrange

This version of Mathematics 11 is designed for students who are curious about the broader role of calculus within mathematics and the sciences. Non-routine problems and examples will be discussed, and side topics explored. Some of the more routine calculus skills will be left to students to learn on their own or in groups. Open to students who have placed into Mathematics 11. Dist: QDS. Chernov, van Erp.

13. Calculus of Vector-Valued Functions

09F: 11, 11 10W: 2 10S: 9L, 12 10F, 11W, 11S: Arrange

This course is a sequel to Mathematics 8 and provides an introduction to calculus of vector-valued functions. Topics include differentiation and integration of parametrically defined functions with interpretations of velocity, acceleration, arclength and curvature. Other topics include iterated, double, triple and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green’s theorem, curl and divergence, and Stokes’ theorem.

Prerequisite: Mathematics 8 or equivalent. Note: First-year students who have received two terms on the BC exam generally should take Mathematics 11 instead. On the other hand, if the student has had substantial exposure to multivariable techniques, they are encouraged to take a placement exam during orientation week to determine if placement into Mathematics 13 is more appropriate. Dist: QDS. The staff (fall), Yang (winter), van Erp, Orellana (spring).

14. Calculus of Vector-Valued Functions, Honors Section

10W: 2 11W: Arrange

Sections of Mathematics 13 for students who have done satisfactory work in Mathematics 9 or exceptional work in Mathematics 8. Dist: QDS. Webb.

16. Linear Programming (Identical to Computer Science 16)

Offered only as Computer Science 16 effective 07F

17. An Introduction to Mathematics Beyond Calculus

10W: 11 11W: Arrange

Gives prospective Mathematics majors an early opportunity to delve into topics outside the standard calculus sequence. Specific topics will vary from term to term, according to the interests and expertise of the instructor. Designed to be accessible to bright and curious students who have mastered BC Calculus, or its equivalent. This course counts toward the Mathematics major, and is open to all students, but enrollment may be limited, with preference given to first-year students.

Prerequisite: Mathematics 8, or placement into Mathematics 11. Dist: QDS. Shemanske.

19. Discrete Mathematics in Computer Science

09F: 12 10W: 10 10F, 11W: Arrange

This course integrates discrete mathematics with algorithms and data structures, using computer science applications to motivate the mathematics. It covers logic and proof techniques, induction, set theory, counting, asymptotics, discrete probability, graphs, and trees.

Mathematics 19 is identical to Computer Science 19 and may substitute for it in any requirement.

Prerequisite: Computer Science 5, Engineering Sciences 20, or advanced placement. Dist: QDS. Winkler (fall), Zomorodian (winter).

20. Discrete Probability

09F: 2 10X, 10F, 11S: Arrange

Basic concepts of probability are introduced in terms of finite probability spaces and stochastic processes having a finite number of outcomes on each experiment. The basic theory is first illustrated in terms of simple models such as coin tossing, random walks, and casino games. Also included are Markov chain models and their applications in the social and physical sciences. The computer will be used to suggest and motivate theoretical results and to study applications in some depth. There is an honors version of this course: see Mathematics 60.

Prerequisite: Mathematics 8. Dist: QDS. The staff..

22. Linear Algebra with Applications

09F: 2 10S: 10 10X, 10F, 11S: Arrange

This course presents the fundamental concepts and applications of linear algebra with emphasis on Euclidean space. Significant goals of the course are that the student develop the ability to perform meaningful computations and to write accurate proofs. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. Applications may be drawn from areas such as optimization, statistics, biology, physics, and signal processing.

Students who plan to take either Mathematics 63 or Mathematics 71 are strongly encouraged to take Mathematics 24.

Prerequisite: Mathematics 8. Dist: QDS. Rockmore (fall), Luca (spring).

23. Differential Equations

09F: 10 10W: 11 10S: 10, 12 10F, 11W, 11S: Arrange

This course is a survey of important types of differential equations, both linear and nonlinear. Topics include the study of systems of ordinary differential equations using eigenvectors and eigenvalues, numerical solutions of first and second order equations and of systems, and the solution of elementary partial differential equations using Fourier series.

Prerequisite: Mathematics 13. Dist: QDS. Van Erp (fall), Wallace (winter), Chu, Sutton (spring).

24. Linear Algebra

10W: 10 10S: 10 11W, 11S: Arrange

This course is an introduction to the fundamental concepts of linear algebra in abstract vector spaces. The topics and goals of this course are similar to those of Mathematics 22, but with an additional emphasis on mathematical abstraction and theory.

(Mathematics 24 can be substituted for Mathematics 22 as a prerequisite for any course or program.)

Dist: QDS. Groszek (winter), Sutton (spring).

25. Number Theory

09F: 11 10F: Arrange

The great mathematician C. F. Gauss once wrote “Mathematics is the queen of sciences and number theory is the queen of mathematics.” Number theory is that part of mathematics dealing with the integers and certain natural generalizations. Topics include modular arithmetic, unique factorization into primes, linear Diophantine equations, and Fermat’s Little Theorem. Discretionary topics may include cryptography, primality testing, partition functions, multiplicative functions, the law of quadratic reciprocity, historically interesting problems.

Prerequisite: Mathematics 8. Dist: QDS. Shemanske.

26. Numerical Methods in Computation (Identical to Computer Science 26 and Engineering Sciences 91)

09F, 10F: 12

A study and analysis of important numerical and computational methods for solving engineering and scientific problems. The course will include methods for solving linear and nonlinear equations, doing polynomial interpolation, evaluating integrals, solving ordinary differential equations, and determining eigenvalues and eigenvectors of matrices. The student will be required to write programs and run them on the computer.

Prerequisite: Mathematics 23, and Computer Science 5 or 14. Dist: QDS.

27. Advanced Calculus and Dynamics in Biology and Medicine

10W: 10A 11W: Arrange

This course will prepare students to read the technical literature in mathematical biology, epidemiology, pharmacokinetics, ecological modeling and related areas. Topics include systems of nonlinear ordinary differential equations, equilibria and steady state solutions, phase portraits, bifurcation diagrams, and some aspects of stability analysis. Emphasis is placed on the student’s ability to analyze phenomena and create mathematical models. This interdisciplinary course is open to mathematics majors, biology majors, and students preparing for a career in medicine.

Prerequisite: Mathematics 22. Note: Students without the mathematical prerequisites can take this course as Mathematics 4: no student may take both Mathematics 4 and 27 for credit, and only Mathematics 27 is eligible to count towards the major in mathematics. Dist: QDS. Wallace.

28. Introduction to Combinatorics

10W: 12 11W: Arrange

Beginning with techniques for counting—permutations and combinations, inclusion-exclusion, recursions, and generating functions—the course then takes up graphs and directed graphs and ordered sets, and concludes with some examples of maximum-minimum problems of finite sets. Topics in the course have application in the areas of probability, statistics, and computing.

Prerequisite: Mathematics 8, or Mathematics 3 and 6. Dist: QDS. Elizalde.

29. Introduction to Computability

11S: Arrange

What does it mean for a function to be computable? This course examines several different mathematical formalizations of the notion of computability, inspired by widely varying viewpoints, and establishes the surprising result that all these formalizations are equivalent. It goes on to demonstrate the existence of noncomputable sets and functions, and to make connections to undecidable problems in other areas of mathematics. The course concludes with an introduction to relative computability. This is a good companion course to Computer Science 39; the two share only the introduction of Turing machines. Offered in alternate years.

Prerequisite: None, but the student must be willing to learn to work abstractly and to read and write proofs. Dist: QDS.

30. Introduction to Linear Models

Not offered in the period from 09F through 11S

This course provides an introduction to the most common model used in statistical data analysis. Simple linear regression, multiple regression, and analysis of variance are covered, as well as statistical model-building strategies. Regression diagnostics, analysis of complex data sets and scientific writing skills are emphasized. Methods are illustrated with data sets drawn from the health, biological, and social sciences. Computations require the use of a statistical software package such as STATA. Offered in alternate years.

Prerequisite: Mathematics 10, another elementary statistics course, or permission of the instructor. Dist: TAS.

31. Topics in Algebra

09F: 12 10X, 10F: Arrange

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography. As a result of the variable syllabus, this course may not serve as an adequate prerequisite for Mathematics 81. Students who contemplate taking Mathematics 81 should consider taking Mathematics 71 instead of this course.

Prerequisite: Mathematics 22. Dist: QDS. The staff.

32. The Shape of Space

10W: 12

Topics in intuitive geometry and topology, for example: how to turn a sphere inside out; knots, links, and their invariants; polyhedra in 2, 3, and 4 dimensions; the classification of surfaces; curvature and the Gauss-Bonnet theorem; spherical and hyperbolic geometry; Escher patterns and their quotients; the shape of the universe. Offered in alternate years.

Prerequisite: Mathematics 22 or 24. Dist: QDS. Doyle.

33. Mathematics in the Sciences and Engineering

Not offered in the period from 09F through 11S

This course provides an overview of the mathematical tools used for analyzing common problems in science and engineering. Particular attention will be given to problems involving linear operators. Topics include partial differential equations, Fourier analysis, linear spaces and operators (in particular, matrix operators), and the calculus of variations. Both analytical and numerical methods will be covered.

Prerequisite: Mathematics 23. Dist: TAS.

35. Real Analysis

10W: 11 11W: Arrange

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Some applications of the theory may be presented. Mathematics 63 presents similar material, but from a more sophisticated point of view. This course may not serve as an adequate prerequisite for either Mathematics 73 or 83. Students who contemplate taking one of these two advanced courses should consider taking Mathematics 63 instead of this course.

Prerequisite: Mathematics 13 and permission of the instructor, or Mathematics 22. Dist: QDS. Lahr.

36. Mathematical Models in the Social Sciences (Identical to Mathematics and Social Science 36)

09F: 2A 10F: Arrange

Disciplines such as anthropology, economics, sociology, psychology, and linguistics all now make extensive use of mathematical models, using the tools of calculus, probability, game theory, network theory, often mixed with a healthy dose of computing. This course introduces students to a range of techniques using current and relevant examples. Students interested in further study of these and related topics are referred to the courses listed in the Mathematics and Social Sciences program.

Prerequisite: Mathematics 13, 20. Dist: TAS. Pauls.

38. Graph Theory

10S: 12 11S: Arrange

The theory of graphs has roots in both practical and recreational mathematics. Today there are major applications of graph theory in management science (operations research) and computer science. This course is a survey of the theory and applications of graphs. Topics will be chosen from among connectivity, trees, and Hamiltonian and Eulerian paths and cycles; isomorphism and reconstructability; planarity, duality, and genus; independence and coloring problems, including interval graphs, interval orderings and perfect graphs, color-critical graphs and the four-color theorem; matchings; network flows, including applications to matchings, higher connectivity, and transportation problems; matroids and their relationship with optimization.

Prerequisite: Mathematics 22 (or Computer Science 25 and permission of the instructor). Dist: QDS. Elizalde.

39. Logic

Not offered in the period from 09F through 11S

This course begins with a brief treatment of sentential logic and then concentrates on first-order logic. Both proof theory and model theory are taken up. The course ends with a proof of the Gödel incompleteness theorem. Connections with the philosophy of mathematics are discussed. There is an honors version of this course: see Mathematics 69. Offered in alternate years.

Prerequisite: one of Mathematics 22, 28, and 29, or Philosophy 10 by permission of the instructor. Dist: QDS.

40. Topics in Applied Probability

Not offered in the period from 09F through 11S

This is a continuation of Mathematics 20 (60). The course studies probability models chosen from queueing theory, genetics, statistical physics, and gambling. Additional probability concepts such as continuous probability and stochastic processes will be discussed in the context of these models. Offered in alternate years.

Prerequisite: Mathematics 13 and 20, or permission of the instructor. Dist: QDS.

42. Differential Geometry I

11W: Arrange

This course will cover curves and surfaces in Euclidean 3-dimensional space. Topics include curvature and torsion of curves, the Frenet-Serret equations, Gaussian and mean curvature of surfaces, geodesics and parallel transport, isometries and Gauss’s Theorem Egregium, the Riemann Curvature tensor. One or more of the following topics will be studied if time permits: vector fields, tangent bundles, hypersurfaces, connections, and curvature. Offered in alternate years.

Prerequisite: Mathematics 22 or permission of the instructor, and Mathematics 23. Dist: QDS.

43. Functions of a Complex Variable

10S: 10 11S: Arrange

This course covers the differential and integral calculus of complex variables including such topics as Cauchy’s theorem, Cauchy’s integral formula and their consequences; singularities, Laurent’s theorem, and the residue calculus; harmonic functions and conformal mapping. Applications will include two-dimensional potential theory, fluid flow, and aspects of Fourier analysis.

Prerequisite: Mathematics 13. Dist: QDS. Gordon.

46. Introduction to Applied Mathematics

10S: 2 11S: Arrange

Develops tools to analyze phenomena in the physical and life sciences, from cell aggregation to vibrating drums to traffic jams. Focus is on applied linear and nonlinear partial differential equations: methods for Laplace, heat and wave equations (Fourier transform, Green’s functions, eigenfunction expansions), Burger’s and reaction-diffusion equation. Further topics may include linear and integral operators, nonlinear optimization, linear programming, asymptotics, boundary layers, or inverse problems. Students will develop numerical skills with a package like MATLAB/Octave.

Prerequisite: Mathematics 22 and 23, or permission of the instructor. Dist: TAS. Chernov.

50. Probability and Statistical Inference

10W: 12 11W: Arrange

Introduction to continuous probability and statistical inference for data analysis. Includes the theory of estimation and the theory of hypothesis testing using normal theory t-tests and nonparametric tests for means and medians, tests for variances, chi-square tests, and an introduction to the theory of the analysis of variance and regression analysis. Analysis of explicit data sets and computation are an important part of this hands-on statistics course.

Prerequisite: Mathematics 13 and 20, or permission of the instructor. Dist: QDS. Demidenko.

53. Chaos!

09F: 10A

Chaotic dynamical systems are everywhere: weather patterns, swinging pendula, population dynamics, even human heart rhythms. With a balance of theory and applications, this course will introduce: flows, fixed points, bifurcations, Lorenz equations, Lyapunov exponent, one-dimensional maps, period-doubling, Julia sets, fractal dimension. Optional topics may include: Hamiltonian systems, symbolic dynamics. Numerical explorations will involve a package like MATLAB/Octave, and students will present a final project investigating a related topic. Offered in alternate years.

Prerequisite: Mathematics 22 and 23, or permission of the instructor. Dist: QDS. Barnett.

54. Topology I

10X: Arrange

This course begins with the definitions of topological space, open sets, closed sets, neighborhoods, bases and subbases, closure operator, continuous functions, and homeomorphisms. The course will study constructions of spaces including subspaces, product spaces, and quotient spaces. Special categories of spaces and their interrelations will be covered, including the categories defined by the various separation axioms, first and second countable spaces, compact spaces, and connected spaces. Subspaces of Euclidean and general metric spaces will be among the examples studied in some detail.

Prerequisite: Mathematics 13 and 22. Dist: QDS.

56. Numerical Analysis (Identical to Computer Science 56)

Not offered in the period from 09F through 11S

This course introduces the student to the concepts of modern numerical analysis. The main emphasis will be on developing effective numerical methods to solve problems in ordinary and partial differential equations. Other topics will be chosen from optimization, approximation, Fourier Transform, and Monte Carlo methods. The specific content will depend in part on the instructor. Offered in alternate years.

Prerequisite: Mathematics 33 and Computer Science 5 or 14, or permission of the instructor. Dist: QDS.

60. Probability (Honors Section of Mathematics 20)

10S: 9L

This course is a more theoretical introduction to probability theory than Mathematics 20. In addition to the basic content of Mathematics 20, the course will include other topics such as continuous probability distributions and their applications. Offered in alternate years.

Prerequisite: Mathematics 13, or permission of the instructor. Dist: QDS. Winkler.

63. Real Analysis

10W: 11 11W: Arrange

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Students may not take both Mathematics 35 and 63 for credit.

Prerequisite: Mathematics 22 or 24, or Mathematics 13 and permission of the instructor. Dist: QDS. Van Erp.

66. Mathematical Topics in Modern Physics

11S: Arrange

This introductory course presents mathematical topics that are relevant to issues in modern physics. It is mainly designed for two audiences: mathematics majors who would like to see modern physics and the historical motivations for theory in their coursework, and physics majors who want to learn mathematics beyond linear algebra and calculus. Possible topics include (but are not limited to) introductory Hilbert space theory, quantum logics, quantum computing, symplectic geometry, Einstein’s theory of special relativity, Lie groups in quantum field theory, etc. No background in physics is assumed. Offered in alternate years.

Prerequisites: Mathematics 24, or Mathematics 22 and permission of the instructor. Dist: QDS.

68. Algebraic Combinatorics

09F: 2

This course covers the use of abstract algebra in studying the existence, construction, enumeration, and classification of combinatorial structures. The theory of enumeration, including both Polya Theory and the Incidence Algebra, and culminating in a study of algebras of generating functions, will be a central theme in the course. Other topics that may be included if time permits are the construction of block designs, error-correcting codes, lattice theory, the combinatorial theory of the symmetric group, and incidence matrices of combinatorial structures. Offered in alternate years.

Prerequisite: Mathematics 28 and 31, or Mathematics 71, or permission of the instructor. Dist: QDS. Vatter.

69. Logic (Honors Section of Mathematics 39)

11W: Arrange

This course begins with a study of relational systems as they occur in mathematics. First-order languages suitable for formalizing such systems are treated in detail, and several important theorems about such languages, including the compactness and Lowenheim-Skolem theorems, are studied. The implications of these theorems for the mathematical theories being formulated are assessed. Emphasis is placed on those problems relating to first-order languages that are of fundamental interest in logic. Offered in alternate years.

Prerequisite: experience with mathematical structures and proofs, as offered by such courses as Mathematics 71, 54, or 24; or permission of the instructor. Dist: QDS.

70. Mathematical Statistics

Not offered in the period from 09F through 11S

This course will be a continuation of the study of the theory of statistical inference that was begun in Mathematics 50. Topics will include the mathematical development of normal theory t-tests and nonparametric tests for means and medians, tests for variances, chi-square tests, and an introduction to the theory of the analysis of variance and regression analysis. Offered in alternate years.

Prerequisite: Mathematics 50. Dist: QDS.

71. Algebra

09F: 10 10F: Arrange

The sequence Mathematics 71 and 81 is intended as an introduction to abstract algebra. Mathematics 71 develops basic theorems on groups, rings, fields, and vector spaces.

Prerequisite: Mathematics 22 or 24. Dist: QDS. Arkowitz.

72. Topics in Geometry

11S: Arrange

This course develops one or more topics in geometry. Possible topics include hyperbolic geometry; Riemannian geometry; the geometry of special and general relativity; Lie groups and algebras; algebraic geometry; projective geometry. Offered in alternate years.

Prerequisite: Mathematics 71, or permission of the instructor. Depending on the specific topics covered, Mathematics 31 may not be an acceptable prerequisite; however, in consultation with the instructor, Mathematics 31 together with some outside reading should be adequate preparation for the course.. Dist: QDS.

73. Multi-Variable Analysis

10S: 10 11S: Arrange

This course develops aspects of the general theory of differentiation and integration in Euclidean space. Primary topics include the Implicit and Inverse Function Theorems, differential forms, and Stokes’ Theorem.

Prerequisite: Mathematics 63. In general, Mathematics 35 is not an acceptable prerequisite; however in consultation with the instructor, Mathematics 35 together with some outside reading should be adequate preparation for the course. Dist: QDS. Webb.

74. Topics in Topology

10S: 2

This course develops one or more topics in topology. Possible topics include classification of surfaces, fundamental group and covering spaces, knot theory, combinatorial topology, and fixed point theory. Offered in alternate years.

Prerequisite: Mathematics 31/71 and 54, or permission of the instructor. Dist: QDS. Trout.

75. Applied Topics in Number Theory and Algebra

10S: 11

Provides some applications of number theory and algebra. Specific topics will vary; two possibilities are cryptology and coding theory. The former allows for secure communication and authentication on the Internet, while the latter allows for efficient and error-free electronic communication over noisy channels. Students may take Math 75 for credit more than once. Offered in alternate years.

Prerequisite: Mathematics 25 or 22/24 or Math 31/71, or permission of the instructor. Dist: QDS. Yang.

76. Topics in Applied Mathematics

10W: 10A 11W: Arrange

The numerical nature of twenty-first century society means that applied mathematics is everywhere: animation studios, search engines, hedge funds and derivatives markets, and drug design. Students will gain an in-depth introduction to an advanced topic in applied mathematics. Possible subjects include digital signal and image processing, quantum chaos, computational biology, cryptography, coding theory, waves in nature, inverse problems, information theory, stochastic processes, machine learning, and mathematical finance.

Prerequisite: Mathematics 22, 23, or permission of the instructor. Dist: QDS. Rockmore.

81. Abstract Algebra

10W: 10 11W: Arrange

This course is the second term of the basic algebra sequence begun in Mathematics 71. While the content of this course varies somewhat from year to year, the topics treated will usually be chosen from among permutation groups, Sylow theory, factorization theory in commutative rings, Galois theory, modules, Wedderburn-Artin theory of semi-simple rings, Noetherian rings, integral extensions, and Dedekind domains.

Prerequisite: Mathematics 71. In general, Mathematics 31 is not an acceptable prerequisite; however, in consultation with the instructor, Mathematics 31 together with some outside reading should be adequate preparation for the course. Dist: QDS. Elizalde.

83. Fourier Analysis and Integration

Not offered in the period from 09F through 11S

From its beginnings in the eighteenth century, Fourier analysis has branched in many directions that are central to applied mathematics. The core of the course consists of the main ideas of one-dimensional Fourier analysis of both periodic and non-periodic phenomena, coupled with an introduction to Lebesgue integration sufficient for understanding the contemporary foundations of the subject. Additional topics are drawn from such areas as signal processing, probability limit laws, and number theory. Offered in alternate years.

Prerequisite: Mathematics 63. In general, Mathematics 35 is not an acceptable prerequisite, however, in consultation with the instructor, Mathematics 35 together with some out side reading should be adequate preparation for the course. Dist: QDS.

86. Mathematical Finance I

09F: 11 10F: Arrange

Financial derivatives can be thought of as insurance against uncertain future financial events. This course will take a mathematically rigorous approach to understanding the Black-Scholes-Merton model and its applications to pricing financial derivatives and risk management. Topics may include: arbitrage-free pricing, binomial tree models, Ito calculus, the Black-Scholes analysis, Monte Carlo simulation, pricing of equities options, and hedging.

Prerequisites: Mathematics 20/60 and Mathematics 23, as well as some programming experience (e.g., Computer Science 5). Dist: QDS. Chu.

87. Reading Course

All terms: Arrange

Advanced undergraduates occasionally arrange with a faculty member a reading course in a subject not occurring in the regularly scheduled curriculum.

88. Special Topics

Not offered in the period from 09F through 11S

From time to time a section of Mathematics 88 may be offered in order to provide an advanced course in a topic which would not otherwise appear in the curriculum. Consult the advisor to majors for details about topics to be covered. Dist: QDS.

89. Seminar in Logic

10W: 2

A study of selected topics in logic, such as model theory, set theory, recursive function theory, or undecidability and incompleteness. Offered in alternate years.

Prerequisite: Mathematics 39 or 69. Dist: QDS. Groszek.

96. Mathematical Finance II

11S: Arrange

This course is a continuation of Mathematics 86 with an emphasis on the mathematics underlying fixed income derivatives. Topics may include: stochastic calculus, Radon-Nikodym derivative and change of measure, Girsanov’s theorem, the Martingale representation theorem, interest rate models (e.g, H-J-M, Ho-Lee, Vasicek, C-I-R), interest rate derivatives, interest rate trees and model calibration, and credit derivatives. Offered in alternate years

Prerequisites: Mathematics 86. Dist: QDS.

97. Undergraduate Research

All terms: Arrange

Open only to students who are officially registered in the Honors Program. Permission of the adviser to majors and thesis adviser required. This course does not serve for major credit nor for distributive credit, and may be taken at most twice.

98. Senior Seminar

Not offered in the period from 09F through 11S

This course satisfies the college’s requirement for a culminating experience. The topic of the seminar will be chosen by the instructor. After an introduction by the instructor, students will prepare and present short talks on various aspects of the topic in order to develop and refine their ability to present mathematics orally. Each student will then make a formal oral presentation and prepare a written report on a topic chosen by the student and instructor. Students will prepare drafts of their report for feedback from the seminar participants and revise their work in light of this feedback. Students also doing an Honors Project may submit their project in lieu of the final written report.

GRADUATE COURSES

A qualified honors major may apply to the course instructor for permission to elect a graduate course. This listing covers 100-level offerings for 2009 fall through 2010 spring only. Courses marked with an asterisk (*) are not offered in this period.

*100. Topics in probability theory.

101. Topics in algebra.

09F: Arrange. Shemanske.

102. Topics in geometry

10S: Arrange. Doyle.

103. Topics in analysis

09F: Arrange. Chernov.

*104. Topics in topology

*105. Topics in number theory

*106. Topics in applied mathematics

107. Supervised Tutoring

Tutoring or assisting with teaching under the supervision of a faculty member.

108. Topics in combinatorics

10S: Arrange. Orellana.

*109. Topics in mathematical logic

*110. Probability theory

111. Algebra

10S: Arrange. Webb.

112. Geometry

10W: Arrange. Pauls.

113. Analysis

10W: Arrange. Williams.

*114. Algebraic topology

*115. Number Theory

*116. Applied mathematics

*118. Combinatorics

*119. Mathematical Logic

*120. Current problems in probability theory

*121. Current problems in algebra

*122. Topics in analysis

*123. Current problems in analysis

124. Current problems in topology

09F: Arrange. Gordon.

125. Current problems in number theory

10S: Arrange. Pomerance.

126. Current problems in applied mathematics

09F: Arrange. Graham.

127. Reading Course

Advanced graduate students may elect a program of supervised reading continuing the topics of their course work.

*128. Current problems in combinatorics

*129. Current problems in mathematical logic

137. Independent Reading

Advanced graduate students may, with the approval of the advisor to graduate students, engage in an independent reading program.

*147. Teaching Seminar

A seminar to help prepare graduate students for teaching. (This course does not count toward the general College requirements for the master’s degree.)

148. Independent Project

A graduate student may, with the approval of the advisor to graduate students, engage in an independent study project. Groups of graduate students may, for example, prepare joint work including reading and informal seminars aimed at mastering a certain topic.

149. Supervised Teaching

Teaching under the supervision of a faculty member.

156. Graduate Research

Research under the guidance of a staff member.

157. Thesis Research

Research under the guidance of the student’s thesis advisor. Open to candidates for the Ph.D. degree.

158. Independent Research

Advanced graduate students may, with the approval of the advisor to graduate students, engage in an independent research project.