# Mathematics Course Listings

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110. Concepts of Mathematics.

An introduction to significant ideas of mathematics, intended for students who will not specialize in mathematics or science. Topics are chosen to display historical perspective, mathematics as a universal language and as an art, and the logical structure of mathematics. This course is intended or non-majors; it does not count toward either the major or minor in mathematics; students who have passed a calculus course (MATH 135, 136 or 205) may not receive course credit for MATH 110.

123. Mathematics of Art.

This course explores the connections between mathematics and art: how mathematics can provide a vocabulary for describing and explaining art; how artists have used mathematics to achieve artistic goals; and how art has been used to explain mathematical ideas. This course is intended for non-majors; it does not count toward either the major or minor in mathematics. Offered as scheduling allows.

134. Precalculus.

A development of skills and concepts necessary for the study of calculus. Topics include the algebraic, logarithmic, exponential, and trigonometric functions; Cartesian coordinates; and the interplay between algebraic and geometric problems. This course is intended for students whose background in high school was not strong enough to prepare them for calculus; it does not count toward the major or minor in mathematics. Students who have passed a calculus course (MATH 135, 136 or 205) may not receive course credit for MATH 134.

135. Calculus I.

The study of differential calculus. The focus is on understanding derivatives as a rate of change. Students also develop a deeper understanding of functions and how they are used in modeling natural phenomena. Topics include limits; continuity and differentiability; derivatives; graphing and optimization problems; and a wide variety of applications.

136. Calculus II.

The study of integral calculus. Topics include understanding Riemann sums and the definition of the definite integral; techniques of integration; approximation techniques; improper integrals; a wide variety of applications; and related topics. Prerequisite: MATH 135 or the equivalent.

205. Multivariable Calculus.

This course extends the fundamental concepts and applications of calculus, such as differentiation, integration, graphical analysis, and optimization, to functions of several variables. Additional topics include the gradient vector, parametric equations, and series. Prerequisite: MATH 136 or the equivalent.

206. Vector Calculus.

A direct continuation of Mathematics 205, the main focus of this course is the study of smooth vector fields on Euclidean spaces and their associated line and flux integrals over parameterized paths and surfaces. The main objective is to develop and prove the three fundamental integral theorems of vector calculus: the Fundamental Theorem of Calculus for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. Prerequisite: MATH 205. Offered as scheduling allows.

217. Linear Algebra.

A study of finite dimensional linear spaces, systems of linear equations, matrices, determinants, bases, linear transformations, change of bases and eigenvalues.

230. Differential Equations.

An introduction to the various methods of solving differential equations. Types of equations considered include first order ordinary equations and second order linear ordinary equations. Topics may include the Laplace transform, numerical methods, power series methods, systems of equations and an introduction to partial differential equations. Applications are presented. Prerequisite: MATH 136.

250. Mathematical Problem-Solving.

Students meet once a week to tackle a wide variety of appealing math problems, learn effective techniques for making progress on any problem, and spend time writing and presenting their solutions. Participation in the Putnam mathematics competition in early December is encouraged but not required. This course is worth 0.25 credit, meets once per week, and is graded pass/fail. Since topics vary from semester to semester, students may repeat this course for credit.

280. A Bridge to Higher Mathematics.

This course is designed to introduce students to the concepts and methods of higher mathematics. Techniques of mathematical proof are emphasized. Topics include logic, set theory, relations, functions, induction, cardinality, and others selected by the instructor.

302. Symbolic Logic

A study of elementary symbolic logic. Topics include sentential and predicate logic. Prerequisite MATH 280 or CS 220 or PHIL 202. *Also offered as PHIL 302 and CS 302*.

305. Real Analysis.

A rigorous introduction to fundamental concepts of real analysis. Topics may include sequences and series, power series, Taylor series and the calculus of power series; metric spaces, continuous functions on metric spaces, completeness, compactness, connectedness; sequences of functions, pointwise and uniform convergence of functions. Prerequisites: MATH 205 and 280. Offered in fall semester.

306. Complex Analysis.

Topics include algebra, geometry and topology of the complex number field, differential, and integral calculus of functions of a complex variable. Taylor and Laurent series, integral theorems, and applications. Prerequisites: MATH 205 and 280. Offered in spring semester.

315. Group Theory.

An introduction to the abstract theory of groups. Topics include the structure of groups, permutation groups, subgroups, and quotient groups. Prerequisite: MATH 280. Offered in spring semester.

316. Ring Theory.

An introduction to the abstract theory of algebraic structures including rings and fields. Topics may include ideals, quotients, the structure of fields, Galois theory. Prerequisite: MATH 280. Offered in fall semester.

318. Graph Theory.

Graph theory deals with the study of a finite set of points connected by lines. Problems in such diverse areas as transportation networks, social networks and chemical bonds can be formulated and solved by the use of graph theory. The course includes theory, algorithms, applications, and history. Prerequisite: MATH 217 or 280. Offered every other year. *Also offered as CS 318*.

319. Geometry.

This course presents a selection of nice results from Euclidean geometry, such as the Euler line, the nine-point circle and inversion. Students explore these topics dynamically using geometric construction software. A portion of the course is also devoted to non-Euclidean geometry, such as spherical, projective or hyperbolic geometry. This course is especially recommended for prospective secondary school teachers. Prerequisite: MATH 217 or MATH 280. Offered as scheduling allows.

321. Financial Mathematics.

An introduction to the modern mathematics of finance. This course develops the mathematics necessary in an arbitrage-free model to understand the connections and pricing of stocks, bonds, futures, forwards and a variety of derivatives and options. The course derives the Capital Asset Pricing Model for portfolio optimization and covers the theory behind the Black-Scholes model for pricing options. Offered as scheduling allows.

323. History of Mathematics.

This seminar is primarily for juniors and seniors and covers topics in the history of mathematics. Offered as scheduling allows.

325. Probability.

This course covers the theory of probability and random variables, counting methods, discrete and continuous distributions, mathematical expectation, multivariate random variables, functions of random variables and limit theorems. Prerequisite: MATH 205. Offered in the fall semester. *Also offered as STATS 325.*

333. Mathematical Methods of Physics.

Important problems in the physical sciences and engineering often require powerful mathematical methods for their solution. This course provides an introduction to the formalism of these methods and emphasizes their application to problems drawn from diverse areas of classical and modern physics. Representative topics include the integral theorems of Gauss and Stokes, Fourier series, matrix methods, selected techniques from the theory of partial differential equations and the calculus of variations with applications to Lagrangian mechanics. The course also introduces students to the computer algebra system Mathematica as an aid in visualization and problem-solving. Prerequisites: MATH 205 and PHYS 152. Offered in fall semester. *Also offered as PHYS 333*.

341. Number Theory.

The theory of numbers addresses questions concerning the integers, such as “Is there a formula for prime numbers?” This course covers the Euclidean algorithm, congruences, Diophantine equations and continued fractions. Further topics may include magic squares, quadratic fields or quadratic reciprocity. Prerequisite: MATH 217 or MATH 280 or permission of the instructor. Offered as scheduling allows.

360. Combinatorics.

This class covers several problem-solving methods different from those seen in most other math classes. In particular, we'll focus on the art of the combinatorial proof. This method lets us take something that may seem abstract and translate it into something concrete that we can count. Other methods and topics covered may include bijections, set theory, Fibonacci identities, Catalan numbers, and combinatorial games. Prerequisite: MATH 280. Offered as scheduling allows.

370. Topology.

An introduction to topology. Topics may include the general notion of a topological space, subspaces, metrics, continuous maps, connectedness, compactness, deformation of curves (homotopy) and the fundamental group of a space. Prerequisite: MATH 280. Offered as scheduling allows.

371. Dynamical Systems and an Introduction to Chaos

An introduction to discrete dynamical systems. The focus of this course is the dynamics of real functions. Topics include fixed point theory, orbit analysis, bifurcations, Sharkovsky’s Theorem, symbolic dynamics, and mathematical chaos. Additional topics, time permitting, include fractals, the dynamics of complex functions, the Mandelbrot set, and Julia sets. Prerequisites: MATH 280. Offered as scheduling allows.

380. Theory of Computation.

The basic theoretical underpinnings of computer organization and programming. Topics include the Chomsky hierarchy of languages and how to design various classes of automata to recognize computer languages. Application of mathematical proof techniques to the study of automata and grammars enhances understanding of both proof and language. Prerequisites: MATH 280. Offered as scheduling allows. *Also offered as CS 380. *

289, 389. Independent Study.

Permission required.

450. SYE: Senior Seminar.

Permission required.

489. SYE: Senior Project for Majors.

Permission required.

498. SYE: Senior Honors Project for Majors.

Permission required.