## Course Descriptions

**Courses Primarily for First-Year and Sophomore Students**

**MATH 100-0 Quantitative Reasoning** Analyzing topical, real-life problems from a quantitative perspective ("thinking in numbers"). Developing facility with basic algebra, probability, and statistics to research and create complex arguments to answer simple questions such as, 'Should I be worried about arsenic in rice?'

**MATH 104-4 Introduction to Game Theory ** Introduction to the mathematical theory of strategic competition; optimal strategies and equilibria; the Prisoner's Dilemma; bargaining and negotiation; strategic voting; applications to economics and political science. For non-science students seeking a gentle introduction to the subject without the technical details or prerequisites of a more advanced course. Familiarity with high-school mathematics is assumed.

**MATH 105-6 First Year Seminar ** Topics vary.

**MATH 110-0 Introduction to Mathematics I ** Exploration of the beauty and mystery of mathematics through a study of the patterns and properties of the natural numbers 1, 2, 3,... Topics include counting, probability, prime numbers, Euclidean algorithm, and unique factorization. Recommended for students with little mathematical background.

**MATH 111-0 Introduction to Mathematics II** Similar in spirit to 110, with topics chosen from number theory, topology, probability, geometry, cryptology, and algebra. Recommended for students with little mathematical background. 110 is not a prerequisite.

**MATH 202-0 Finite Mathematics** Primarily (but not only) for behavioral sciences. Topics chosen from set theory, combinatorics (the art of counting), finite probability, elementary linear algebra and its applications to linear optimization problems. Among other things, the course will focus on practical applications of these mathematical tools to real life situations, such as, elementary statistics, analyzing survey data, probability tests, supply and demand linear functions and equilibrium prices in economy, minimizing linear cost functions and maximizing linear profit functions in business. This class is popular with psychology, communication, journalism, international studies students to fulfill their math requirements.

**MATH 211-0 Short Course in Calculus** Elements of differential and integral calculus. Examples drawn from the behavioral and social sciences. Students may not receive credit for both 211 and 220. Not suitable for those planning to major in mathematics, the natural sciences, or economics. Does not prepare for 230. 202 is not a prerequisite.

**MATH 212-0 Single Variable Calculus I** Review of trigonometric, exponential, logarithmic, and inverse functions and transformation of graphs. Limits, continuity, derivative of a function, product, quotient and chain rule, mean value theorems, Newton's method, linear approximation and differentials, optimization problems. Students may not receive credit for both 212 and 211 or 220. For students with little or no previous exposure to calculus. Prerequisite: consent of department.

**MATH 213-0 Single Variable Calculus II ** Logarithmic differentiation, implicit differentiation, inverse trigonometric functions, related rates. L'Hôpital's Rule, curve sketching. Fundamental theorem of calculus. Techniques of integration, including integration by substitution and by parts, partial fractions, trigonometric substitutions, numerical integration, areas, and volumes. Students may not receive credit for both 213 and 211 or 224. Prerequisite: 212 or consent of department.

**MATH 214-0 Single Variable Calculus III** Review of trigonometric substitutions and partial fractions. Improper integrals. Applications of integration: computation of arc length and surface area, work, and probability. Sequences and series: the integral and comparison tests, alternating series, power series, ratio test. Taylor's formula and Taylor series. Series solutions of differential equations. Students may not receive credit for both 214 and 224. Prerequisite: 213 or consent of department.

**MATH 220-0 Differential Calculus of One-Variable Functions** Limits, differentiation, linear approximation, optimization, curve sketching, related rates, Newton's method, antiderivatives. Students may not receive credit for both 220 and 211 or 212.

**MATH 224-0 Integral Calculus of One Variable Functions ** Integrals, techniques of integration, volumes, arc length, work, differential equations, sequences and series, Taylor polynomials. Students may not receive credit for both 224 and 213 or 214. Prerequisite: 220.

**MATH 230-0 Differential Calculus of Multivariable Functions** Vector algebra, vector functions, partial derivatives, optimization, Lagrange multipliers. Students may not receive credit for both 230 and 281-1, 285-2, 290-2, or 291-2. Prerequisite: 214 or 224.

**MATH 234-0 Multiple Integration and Vector Calculus** Cylindrical and spherical coordinates, double and triple integrals, line and surface integrals. Change of variables in multiple integrals; gradient, divergence, and curl. Theorems of Green, Gauss, and Stokes. Students may not receive credit for both 234 and 281-1, 285-3, 290-3, or 291-3. Prerequisite: 230.

**MATH 240-0 Linear Algebra** Basic concepts of linear algebra. Solutions of systems of linear equations; vectors and matrices; subspaces, linear independence, and bases; determinants; eigenvalues and eigenvectors; other topics and applications as time permits. Students may not receive credit for both 240 and 281-3, 285-1, 290-1, or 291-1.

**MATH 250-0 Elementary Differential Equations** Applications of calculus and linear algebra to the solution of ordinary differential equations. Students may not receive credit for both 250 and 281-2, GEN ENG 205-4, or GEN ENG 206-4. Prerequisites: 230; 240 or concurrent registration in 240; or equivalent

**MATH 281-1,2,3 Accelerated Mathematics for ISP: First Year** Fall: Multivariable differential calculus, multiple integration and vector calculus. Winter: Vector integral calculus, differential equations, and infinite series. Spring: Linear algebra, differential equations. Open only to students in ISP.

**MATH 285-1,2,3 Accelerated Mathematics for MMSS: First Year** Fall: Linear algebra. Winter: Continuation of linear algebra; multidimensional calculus. Spring: Multivariable calculus. Prerequisite: first-year standing in MMSS.

**MATH 290-1,2,3 MENU: Linear Algebra and Multivariable Calculus** An introduction to linear algebra covering computations and applications. Topics covered in this course include linear equations, linear transformations, subspaces, dimension, eigenvalues and eigenvectors. This material provides the foundation for the vector calculus material in Math 290-2 and Math 290-3.

**MATH 291-1,2,3 MENU: Intensive Linear Algebra and Multivariable Calculus** The emphasis in this course is on theory and proofs. It is designed to prepare students to take 300-level mathematics courses such as 321 and 331 as sophomores. Linear algebra, similar to 240, but in greater depth. Topics include vectors, inner product, systems of linear equations, and linear transformations.

**Courses Primarily for Sophomore, Junior, and Senior Students**

**MATH 300-0 Foundations of Higher Mathematics ** Introduction to fundamental mathematical ideas - such as sets, functions, equivalence relations, and cardinal numbers - and basic techniques of writing proofs. Students may not receive credit for 300 without prior departmental consent after taking 320-1, 321-1, 330-1, or 331-1. Prerequisite: 240.

**MATH 306-0 Combinatorics & Discrete Mathematics ** Discrete mathematics, inductive reasoning, counting problems, binomial coefficients and Pascal's triangle, Fibonacci numbers, combinatorial probability, divisibility and primes, partitions, and generating functions. Prerequisite: 240 or instructor's consent.

**MATH 308-0 Graph Theory** Introduction to graph theory: graphs, trees, matchings, planar graphs, colorings. Additional topics as time permits. Prerequisites: 291-1, 300, 306, or equivalent.

**MATH 310-1,2,3 Probability and Stochastic Processes ** Probability and Stochastic Processes is a three-quarter sequence in probability and stochastic processes requiring background in calculus but not measure theory. Fall quarter: Probability spaces, combinatorial problems, random variables, moments, special distributions, joint distributions, independence, conditional probability, Weak Law of Large Numbers, Central Limit Theorem. Winter quarter: Theory of Markov Chains including recurrence and transience, state space structure, periodicity, convergence to stationarity, detailed discussion of examples such as random walks and branching processes; Poisson process. Spring quarter: Continuous-time Markov chains, Markovian queues, population growth models; Brownian motion and other diffusion processes; additional topics such as renewal processes as time permits. Corequisite (for 310-1): Math 234-0; or Math 290-3; or Math 291-3; Prerequisite (for Math 310-2): Math 240, Math 310-1

**MATH 311-1,2,3 MENU: Probability and Stochastic** Processes Probability and Stochastic Processes is a three-quarter sequence in probability and stochastic processes requiring background in calculus but not measure theory. The first quarter is a careful introduction to probability spaces, random variables, independence, distributions, and generating functions culminating in the Central Limit Theorem. The second and third quarters largely concern stochastic processes, including discrete and continuous-time Markov chains, Markov Chain Monte Carlo methods, martingales, and diffusion processes. We will use the software package R for simulations (no prior knowledge of R is required.) This sequence covers more topics at a faster pace, and in greater depth than 310-1,2,3. Prerequisite: Math 291-3 or (234 and 300); or Department Consent. Math 320-1 or Math 321-1 recommended

**MATH 314-0 Probability and Stochastic Processes for Econometrics** This course provides an introduction to probability and statistics. It is a prerequisite for Economics 381-1, Econometrics. The first part of this course covers the structure of probability theory, which is the foundation of statistics, and provides examples of the use of probabilistic reasoning. The course then moves to discussion of the most commonly encountered probability distributions, both discrete and continuous. Next, the course considers random sampling from a population, and the distributions of some sample statistics. From there the course moves to estimation, which is the process of using data (in the best possible way) to learn about the values of unknown parameters of a population or model. Finally, the course treats hypothesis testing, which is a way to decide whether data are consistent or inconsistent with a hypothesis about a feature of a population or model. Prerequisite (or corequisite): Math 234-0; or Math 290-3; or Math 291-3

**MATH 320-1,2,3 Real Analysis ** Axiomatic approach to the real number system. Sequences of real numbers and basic properties of limits of sequences. Real-valued functions of one variable; continuity and uniform continuity. Basic properties of differentiable functions. Construction of the Riemann integral and its basic properties. Prerequisite: Math 291-3; or Math 290-1 and Math 300; or Math 234-0, Math 240-0, and Math 300-0

**MATH 321-1,2,3 MENU: Real Analysis** Rigorous analysis in Euclidean space and on metric spaces. Metric space topology, properties of Euclidean spaces, limits and continuity, differentiation and integration, sequences and series, the inverse and implicit function theorems. Lebesgue integration with applications. 321-1,2 differ from 320-1,2 in two respects: they cover more topics in more depth, and aim at intensive development of students' ability to analyze and create mathematical proofs. Faster than 320, and at a higher level of abstraction. Prerequisite: Department Permission

**MATH 325-0 Complex Analysis **Complex numbers, analytic functions, contour integrals, Cauchy's theorem, Laurent series, residue theorem, conformal mapping, analytic continuation. Students may not receive credit for both 325 and 360-3 or ESAM 311-3. Prerequisites: 234 and 240 or equivalent.

**MATH 327-0 Mechanics for Mathematicians** Fundamental mathematical ideas arising in classical mechanics. Newtonian mechanics. Lagrangian formalism and calculus of variations; motion with constraints; symmetries and conservation laws. Hamiltonian mechanics; Liouville's theorem. No prior knowledge of physics is assumed. Students may not receive credit for 327 after taking Physics 330-1. Prerequisite: a thorough knowledge of linear algebra and vector calculus, as covered, for example, in 234 and 240 or equivalent, plus at least 1 300-level math course. Prerequisite: Math 234-0 and Math 240-0

**MATH 330-1,2,3 Abstract Algebra** 1. Groups and their structure, elementary ring theory; polynomial rings. 2. Continuation of ring theory,. 3. Field theory and Galios theory. Students may not receive credit for corresponding quarters of 330 and 331. Prerequisite: 240 or consent of instructor.

**MATH 331-1,2,3 MENU: Abstract Algebra** 1. Groups and their structure, including the Sylow theorems. 2. Ring theory, polynomial rings. Module theory, including applications to canonical form theorems of linear algebra. 3. Field theory; Galois theory. 331 differs from 330 in two respects: it covers more topics in more depth, and aims at intensive development of students' ability to analyze and create mathematical proofs. Students may not receive credit for corresponding quarters of both 330 and 331. Prerequisites: 240 or equivalent, 300 or equivalent. Prerequisite: Math 291-3; or Math 240-0 and Math 300-0

**MATH 334-0 Linear Algebra: Second Course** Abstract theory of vector spaces and linear transformations. Complex vector spaces, unitary and Hermitian matrices. Jordan canonical form. Selected applications as time permits. Students who took 330-1 (formerly 337-1) prior to 2004-05 may not also take 334 for credit toward the major without departmental consent. Prerequisite: 240 or equivalent. Prerequisite: Math 291-1 or Math 300-0

**MATH 336-1,2 Introduction to the Theory of Numbers** 1. Divisibility and primes, congruences, quadratic reciprocity, Diophantine problems. 2. Additional topics in analytic and algebraic number theory. Prerequisite: 230.

**MATH 340-0 Geometry** Axiomatics for Euclidean geometry. Non-Euclidean geometry. Projective geometry. Introduction of coordinate system from the axioms. Quadrics. Erlangen program. Introduction to plane algebraic curves. Prerequisite: 230 and 300 or equivalent.

**MATH 342-0 Introduction to Differential Geometry** Curves and surfaces in three-dimensional space. Prerequisites: 234 and 240 or equivalent. Prerequisite: Math 230-0 and Math 240; or Math 290-3; or Math 291-3

**MATH 344-1,2 Introduction to Topology ** 1. Basic concepts: topologies, connectedness, compactness, separation axioms. Geometric concepts, including simplicial complexes and manifolds. 2. Fundamental groups. Language of categories. Covering spaces. Prerequisite (for Math 344-1): Math 320-1 or Math 321-1. Prerequisite (for Math 344-2): Math 344-1 and (Math 330-1 or Math 331-1)

**MATH 351-0 Fourier Analysis & Boundary Value Problems** Expansion in orthogonal functions with emphasis on Fourier series. Applications to solution of partial differential equations arising in physics and engineering. May not receive credit for both 351 and 381 ir both 351 and ESAM 311-2. Prerequisites: 240 and 250 or equivalent.

**MATH 353-0 Qualitative Differential Equations** Qualitative theory of ordinary differential equations. Linear systems, phase portraits, periodic solutions, stability theory, Lyapunov functions, chaotic differential equations. Prerequisite: Math 240-0 and 250-0; or Math 360-1; or equivalent

**MATH 354-1,2 Chaotic Dynamical Systems** 1. Chaotic phenomena in deterministic discrete dynamical systems, primarily through iteration of functions of one variable. 2. Iteration of functions of two and more variables, including the study of the horseshoe map, attractors, and the Henon map. Complex analytic dynamics, including the study of the Julia set and Mandelbrot set. Prerequisite: 240

**MATH 360-1,2 MENU: Applied Analysis** 1. Linear ordinary differential equations and their applications. 2. Systems of linear ordinary differential equations, qualitative analysis of ordinary differential equations, linear partial differential equations, Laplace transform, Fourier series, orthogonal functions, and applications. Prerequisites: 290-1,2,3 or 291-1,2,3.

**MATH 366-1 Mathematical Models in Finance** Cash flow computations. Basic financial concepts (stocks, bonds, options, arbitrage, hedging) and put-call parity. Binomial tree models. Risk-neutral valuation. Random walk and Brownian motion as a tool of modeling fluctuations. Options pricing. Application of the central limit theorem. The Black-Scholes formula and partial differential equation. Numerical approximations. Prerequisites: 240 plus any 1 of 310-1, 383, 385, IEMS 202, or ECON 381-1. Some acquaintance with basic differential equations is desirable but not required.

**MATH 368-0 Introduction to Optimization** Methods and concepts of linear and nonlinear optimization theory, going beyond the treatment of optimization in calculus. Topics not usually covered in real analysis, including Kuhn-Tucker Theory, convexity conditions, and linear programming. Fulfills a prerequisite for the Kellogg managerial analytics certificate. Prerequisites: 285-3, 290-3, or 291-3; or both 240 and 300.

**MATH 370-0 Mathematical Logic** Mathematical formulation and rigorous discussion of logical systems, particularly the propositional calculus and the functional calculi of first and second order. Well-formed formulae, formal languages, proofs, tautologies, effective procedures, deduction theorems, axiom schemata. Prerequisite: consent of instructor.

**MATH 381-0 Fourier Analysis & Boundary Value Problems for ISP** Fourier series. Hilbert spaces and orthogonal functions. Parseval theorem. Poisson summation formula and lattice points. Fourier integrals: Gaussian functions. Fourier inversion formula. Convolution. Sturm-Liouville theory. Application to partial differential equations. Heat and wave equations. Prerequisite: Math 281-1,2,3; Physics 125-1,2,3

**MATH 382-0 Complex Analysis and Group Theory for ISP** Complex analysis, elements of group theory. May not receive credit of both 325 and 382. Ordinarily taken only by students in ISP; permission required otherwise. Prerequisites: 281-1,2,3; Physics 125-1,2,3

**MATH 385-0 Probability and Statistics for MMSS ** Probability theory and its social science applications. May not receive credit for both 385 and aby if 310-1, 311-1, 314, or Stat 320-0, 383. Prerequisite: second-year standing in MMSS

**MATH 395-0 Undergraduate Seminar** Topics of modern mathematics and relationships among different branches of mathematics. Open only to superior students by consent of department. May be taken for one unit of credit at a time, but may be repeated for credit with change of topic. Prerequisite: Department Permission.

**MATH 399-0 Independent Study** Open on approval of department to undergraduates who are qualified to do independent work under the direction of a faculty advisor. Students must file a plan of study with the department before enrollment in 399. Prerequisite: Department Permission.