## Course Offerings

### Mathematics (2012 - 2014)

Algebra and Calculus

MATH-UA 9 *Prerequisites: high school mathematics or permission of the department. Offered every term. 4 points.*

An intensive course in intermediate algebra and trigonometry. Topics include algebraic, exponential, logarithmic, and trigonometric functions and their graphs.

Discrete Mathematics

MATH-UA 120 *Prerequisite: Calculus I (MATH-UA 121) with a grade of C or better, or permission of the department. Offered every term. 4 points.*

A first course in discrete mathematics. Sets, algorithms, and induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

Calculus Tracks

Calculus Tracks

Two calculus tracks are available: the standard track of* Calculus I*, *II*, and *III *(MATH-UA 121-123) and the *Honors I*, *II* track (MATH-UA 221, 222). Pursuing the honors track requires that the student know the material from* Calculus I* (MATH-UA 121), because the honors track covers material from *Calculus II *and *III *(MATH-UA 122, 123) as well as from* Linear Algebra* (MATH-UA 140). The honors courses MATH-UA 221, 222 count as the equivalent of two mathematics courses. Switching tracks is not encouraged. A student who intends to take the full calculus sequence should be prepared to continue on the same track for the whole sequence.

Calculus I

MATH-UA 121 *Prerequisite: a score of 650 or higher on the mathematics portion of the SAT, a score of 650 or higher on the SAT Subject Test in Mathematics 1, a score of 650 or higher on the SAT Subject Test in Mathematics 2, an ACT mathematics score of 30 or higher, a score of 3 or higher on the AP Calculus AB exam, an AB subscore of 3 or higher on the AP Calculus BC exam, a score of 3 or higher on the AP Calculus BC exam, a grade of C or higher in Algebra and Calculus (MATH-UA 9), or a passing score on a departmental placement exam. Offered every term. 4 points.*

Derivatives, antiderivatives, and integrals of functions of one variable. Applications include graphing, maximizing, and minimizing functions. Definite integrals and the fundamental theorem of calculus. Areas and volumes.

Calculus II

MATH-UA 122 *Prerequisite: Calculus I (MATH-UA 121) or equivalent with a grade of C or better, a score of 4 or higher on the AP Calculus AB or BC exam, or a passing score on a departmental placement exam. Offered every term. 4 points.*

Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

Calculus III

MATH-UA 123 *Prerequisite: Calculus II (MATH-UA 122) or equivalent with a grade of C or better, a score of 5 on the AP Calculus BC exam, or a passing score on a departmental placement exam. Offered every term. 4 points.*

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

**Introduction to Mathematical Proofs**

MATH-UA 125 *Prerequisite: Calculus I (MATH-UA 121). Offered every spring. 2 points.*

The course will introduce elements of mathematical proof. The focus is on three main themes and key ideas related to proof and proving: 1. The meaning of mathematical statements – universal and existential; 2. The roles of examples in determining the validity of mathematical statements; 3. The various forms and methods of mathematical proofs, including Direct (deductive) proof; proof by exhaustion of all cases; indirect proof (proof by contradiction or by contrapositive); proof by induction; disproof by counterexample. Formal construction of proofs will be introduced gradually. This is a problem-based course. Lessons are structured around activities that engage students in doing proofs that are meaningful to them and based on mathematical topics with which they are familiar.

Set Theory

MATH-UA 130 *Identical to PHIL-UA 73. 4 points.*

The axioms of set theory; Boolean operations on sets; set-theoretic representation of relations, functions, and orderings; the natural numbers; theory of transfinite cardinal and ordinal numbers; the axiom of choice and its equivalents; and the foundations of analysis. May also cover such advanced topics as large cardinals or independence results.

Linear Algebra

MATH-UA 140 *Prerequisite: Calculus I (MATH-UA 121) with a grade of C or better or equivalent. Offered every term. 4 points.*

Systems of linear equations. Gaussian elimination, matrices, determinants, and Cramer's rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, quadratic forms.

Honors Linear Algebra I

MATH-UA 141 *Identical to MATH-GA 2110. Prerequisite: a grade of B or better in Analysis I (MATH-UA 325) and/or Algebra I (MATH-UA 343) or the equivalent. MATH-GA 2110 is offered every semester but called Linear Algebra I in the fall and summer sessions. 4 points.*

Linear spaces, subspaces, and quotient spaces; linear dependence and independence; basis and dimension. Linear transformation and matrices, dual spaces and transposition. Solving linear equations. Determinants. Quadratic forms and their relation to local extrema of multivariable functions.

Honors Linear Algebra II

MATH-UA 142 *Identical to MATH-GA 2120. Prerequisite: Honors Linear Algebra I (MATH-UA 141). Offered in the spring. 4 points.*

Special theory, eigenvalues, and eigenvectors; Jordan canonical forms. Inner product and orthogonality. Self-adjoint mappings, matrix inequalities. Normed linear spaces and linear transformation between them. Positive matrices. Applications.

Mathematics for Economics I, II

MATH-UA 211, 212 *Prerequisites for MATH-UA 211: same as for Calculus I (MATH-UA 121). Prerequisite for MATH-UA 212: completion of MATH-UA 211 with a grade of C or higher. Intended for declared and prospective majors in economics on the policy track. Cannot apply both standard or honors calculus courses and Mathematics for Economics courses towards the mathematics major. Economics policy majors pursuing a double major in mathematics may substitute MATH-UA 211, 212 for the regular calculus sequence and must complete one extra elective in mathematics. Offered every term. 4 points per term.*

Elements of calculus and linear algebra with examples and motivation drawn from important topics in economics. Topics include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization; series, including geometric and Taylor series; ordinary differential equations; matrix algebra; eigenvalues; and (if time permits) dynamic optimization and multivariable integration.

**Mathematics for Economics III**

MATH-UA 213. *Prerequisite: Mathematics for Economics II (MATH-UA 212). Offered every semester. 4 points. *

Further topics in vector calculus. Vector spaces, matrix analysis. Linear and nonlinear programming with applications to game theory. This course will provide economics students who have taken Mathematics for Economics I (MATH-UA 211) and Mathematics for Economics II (MATH-UA 212) with the tools to take higher-level mathematics courses.

Honors Calculus I: Accelerated Calculus with Linear Algebra

MATH-UA 221 *Prerequisite: one of the following: (a) a score of 4 or higher on the Advanced Placement Calculus BC exam or of 5 on the AB exam; or (b) Calculus I (MATH-UA 121). Requires permission of the instructor. Offered in the fall. 5 points.*

First semester of a yearlong sequence that covers the content of Calculus II and III (MATH-UA 122, 123) as well as Linear Algebra (MATH-UA 140). Sequences and series; Taylor's theorem; power series; linear systems of equations; matrices and LU decomposition; determinants; vector spaces; eigenvalues and eigenvectors; functions of several variables; vector-valued functions; partial derivatives; various applications including maxima and minima.

Honors Calculus II: Accelerated Calculus with Linear Algebra

MATH-UA 222 *Prerequisite: Honors Calculus I (MATH-UA 221) with a B or better. Offered in the spring. 5 points.*

Second semester of a yearlong sequence that covers the content of Calculus II and III (MATH-UA 122, 123) as well as Linear Algebra (MATH-UA 140). Multidimensional differentiation (e.g. differentials, gradients, Taylor expansions, applications); multidimensional integration (e.g. double and triple integrals, Green's theorem, divergence theorem, applications); differential equations (e.g. first-order linear equations, second-order linear equations, applications); and additional topics in linear algebra (e.g. inner products, orthogonality, applications).

Vector Analysis

MATH-UA 224 *Prerequisite: a grade of C or better in Analysis I (MATH-UA 325). Offered in the spring. 4 points.*

Brief review of multivariate calculus: partial derivatives, chain rule, Riemann integral, change of variables, line integrals. Lagrange multipliers. Inverse and implicit function theorems and their applications. Introduction to calculus on manifolds: definition and examples of manifolds, tangent vectors and vector fields, differential forms, exterior derivative, line integrals and integration of forms. Gauss's and Stokes's theorems on manifolds.

Earth's Atmosphere and Ocean: Fluid Dynamics and Climate

MATH-UA 228 *Identical to ENVST-UA 360. Prerequisite: Calculus I (MATH-UA 121) or equivalent with a grade of B- or better and familiarity with introductory physics (at least at the advanced high school level). Recommended: Calculus III (MATH-UA 121). 4 points.*

Introduction to dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction. Goal of the lectures is to develop an understanding of the unifying principles of planetary fluid dynamics. Topics include the global energy balance, convection and radiation (the greenhouse effect), effects of planetary rotation (the Coriolis force), structure of the atmospheric circulation (the Hadley cell and wind patterns), structure of the oceanic circulation (wind-driven currents and the thermohaline circulation), and climate and climate variability (including El Nino and anthropogenic warming).

Theory of Probability

MATH-UA 233 *Prerequisite: a grade of C or better in Calculus III (MATH-UA 123) or equivalent. Not open to students who have taken Probability and Statistics (MATH-UA 235). Offered in the fall. 4 points.*

Introduction to the mathematical techniques of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains, applications.

Mathematical Statistics

MATH-UA 234 *Prerequisite: a grade of C or better in Theory of Probability (MATH-UA 233) or equivalent. Not open to students who have taken Probability and Statistics (MATH-UA 235). Offered in the spring. 4 points.*

Introduction to the mathematical foundations and techniques of modern statistical analysis used in the interpretation of data in quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression, and analysis of variance. Applications to the sciences.

Probability and Statistics

MATH-UA 235 *Prerequisite: a grade of C or better in Calculus II (MATH-UA 122) or equivalent. Not open to students who have taken Theory of Probability (MATH-UA 233). Offered in the spring. 4 points.*

Combination of MATH-UA 233 and MATH-UA 234 at a more elementary level to acquaint students with both probability and statistics in a single term. In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal distribution; application to coin-tossing; radioactive decay. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.

Combinatorics

MATH-UA 240 *Prerequisite: Calculus II (MATH-UA 122) with a grade of C or better, or equivalent. Offered every other spring. 4 points.*

Techniques for counting and enumeration, including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph theoretic problems.

Introduction to Cryptography

MATH-UA 243 *Identical to CSCI-UA 480. Prerequisite: Basic Algorithms (CSCI-UA 310) with a grade of C or better, or permission of the instructor. Offered in the spring. 4 points.*

An introduction to both the principles and practice of cryptography and its application to network security. Topics include symmetric-key encryption (block ciphers, modes of operations, AES), message authentication (pseudorandom functions, CBC-MAC), public-key encryption (RSA, ElGamal), digital signatures (RSA, Fiat-Shamir), and authentication applications (identification, zero-knowledge).

Abstract Algebra

MATH-UA 246 *Prerequisites: Calculus II (MATH-UA 122) and Linear Algebra (MATH-UA 140) with grades of C or better. Not open to mathematics majors and/or students who have taken Algebra I (MATH-UA 343). Offered in the spring. 4 points.*

Introduction to the main concepts, constructs, and applications of modern algebra. Groups, transformation groups, Sylow theorems, and structure theory; rings, polynomial rings, and unique factorization; introduction to fields and Galois theory. Although not acceptable for the mathematics majors, it is accepted toward the mathematics minor and is a strongly recommended course for the Steinhardt mathematics education major.

Theory of Numbers

MATH-UA 248 *Prerequisite: Calculus II (MATH-UA 122) with a grade of C or better or equivalent. Offered in the fall. 4 points.*

Divisibility and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

Mathematics of Finance

MATH-UA 250 *Prerequisites: Calculus III (MATH-UA 123) and one of the following: Theory of Probability (MATH-UA 233), Probability and Statistics (MATH-UA 234), Statistics (ECON-UA 18), or Analytical Statistics (ECON-UA 20) with a grade of C+ or better and/or permission of the instructor. Offered in the fall. 4 points.*

Introduction to the mathematics of finance. Topics: linear programming with application to pricing. Interest rates and present value. Basic probability, random walks, central limit theorem, Brownian motion, log-normal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.

Introduction to Mathematical Modeling

MATH-UA 251 *Prerequisite: a grade of C or better in Calculus III (MATH-UA 123) or permission of the instructor. Offered in the spring. 4 points.*

Formulation and analysis of mathematical models. Mathematical tools include dimensional analysis, optimization, simulation, probability, and elementary differential equations. Applications to biology, economics, and other areas of science. The necessary mathematical and scientific background is developed as needed. Students participate in formulating models as well as in analyzing them.

Numerical Analysis

MATH-UA 252 *Prerequisite: a grade of C or better in both Calculus III (MATH-UA 123) and Linear Algebra (MATH-UA 140) or equivalent. Offered in the spring. 4 points.*

In numerical analysis, one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course introduces the subject to mathematics majors. Theory and practical examples using Matlab are combined to explore topics ranging from simple root-finding procedures to differential equations and the finite element method.

Mathematics in Medicine and Biology

MATH-UA 255 *Identical to BIOL-GA 1501. Prerequisites: Calculus I (MATH-UA 121) and Principles of Biology I (BIOL-UA 11) or permission of the instructor. Offered in the fall. 4 points.*

Intended primarily for premedical students with interest and ability in mathematics. Topics of medical importance using mathematics as a tool, including control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, muscle, cardiac catheterization, and computer diagnosis. Material from the physical sciences is introduced as needed and developed within the course.

Computers in Medicine and Biology

MATH-UA 256 *Identical to BIOL-GA 1502. Prerequisite: Mathematics in Medicine and Biology (MATH-UA 255) with a grade of C or better, or permission of the instructor. Familiarity with a programming language such as Pascal, Fortran, or BASIC is recommended. Offered in the spring. 4 points.*

Introduces the student of biology or mathematics to the use of computers as tools for modeling physiological phenomena. The student constructs two computer models selected from the following list: circulation, gas exchange in the lung, control of cell volume, and the renal countercurrent mechanism. The student then uses the model to conduct simulated physiological experiments.

**Introduction to Computer Simulation**

MATH-UA 257 *Prerequisites: Calculus I (MATH-UA 121) and General Physics I (PHYS-UA 11). Co-requisite: Calculus II (MATH-UA 122). Offered every spring. 4 points.*

In this course, students will learn how to do computer simulations of such phenomena as orbits (Kepler problem and N-body problem), epidemic and endemic disease (including evolution in response to the selective pressure of a malaria), musical stringed instruments (piano, guitar, and violin), and traffic flow in a city (with lights, breakdowns, and gridlock at corners). The simulations are based on mathematical models, numerical methods, and Matlab programming techniques that will be taught in class. The use of animations (and sound where appropriate) to present the results of simulations will be emphasized.

Ordinary Differential Equations

MATH-UA 262 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) and Linear Algebra (MATH-UA 140) or equivalent. Offered every term. 4 points.*

First- and second-order equations. Series solutions. Laplace transforms. Introduction to partial differential equations and Fourier series.

Partial Differential Equations

MATH-UA 263 *Prerequisite: a grade of C or better in Ordinary Differential Equations (MATH-UA 262) or equivalent. Offered in the spring. 4 points.*

Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples of such laws as embodied in the wave equation, the diffusion equation, and Laplace's equation. Nonlinear conservation laws and the theory of shock waves. Applications to physics, chemistry, biology, and population dynamics.

Chaos and Dynamical Systems

MATH-UA 264 *Prerequisite: a grade of C or better in both Calculus II (MATH-UA 122) and Linear Algebra (MATH-UA 140) or equivalent. Offered in the fall. 4 points.*

Topics include dynamics of maps and of first-order and second-order differential equations: stability, bifurcations, limit cycles, and dissection of systems with fast and slow timescales. Geometric viewpoint, including phase planes, is stressed. Chaotic behavior is introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications are drawn from physics and biology. Homework and projects are assigned, as well as a few computer lab sessions. (Programming experience is not a prerequisite.)

Transformations and Geometries

MATH-UA 270 *Prerequisite: a grade of C or better in Calculus III (MATH-UA 123) or equivalent. Strongly recommended: Linear Algebra (MATH-UA 140). Offered alternate years in the fall. 4 points.*

This is a thorough course in planar Euclidean geometry. Emphasis is placed on development of students' proof-writing and problem-solving skills. It begins with a study of the basic structures (e.g.,\ angles, lines, arcs) and concepts (e.g.,\ construction, congruence, similarity) known to Euclid and builds toward modern results. The second half of the course will focus on isometries of the plane, their classification, and applications of complex numbers and conformal maps to geometry. Time permitting, contrasts will be made with some non-Euclidean geometries.

Functions of a Complex Variable

MATH-UA 282 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) and Linear Algebra (MATH-UA 140) or equivalent. Offered in the spring. 4 points.*

Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy's theorem and the Cauchy integral formula. Singularities, residues, Taylor and Laurent series. Fractional linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow, and more.

Analysis I

MATH-UA 325 *Prerequisites: a grade of C or better in Calculus III (MATH-UA 123) and Linear Algebra (MATH-UA 140) or equivalent. Offered every term. 4 points.*

The real number system. Convergence of sequences and series. Rigorous study of functions of one real variable. Continuity, connectedness, compactness, metric spaces.

Analysis II

MATH-UA 326 *Prerequisite: a grade of C or better in Analysis I (MATH-UA 325) or permission of the department. Offered in the spring. 4 points.*

Rigorous study of functions of several variables. Limits and continuity. Differentiable functions. The implicit function theorem. Transformation of multiple integrals. Riemann integral.

Algebra I

MATH-UA 343 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) and Linear Algebra (MATH-UA 140) or equivalent. Strongly recommended: Analysis I (MATH-UA 325). Offered in the fall. 4 points.*

Groups, homomorphisms, automorphisms, and permutation groups. Rings, ideals, and quotient rings, Euclidean rings, and polynomial rings.

Algebra II

MATH-UA 344 *Prerequisite: a grade of C or better in Algebra I (MATH-UA 343). Offered in the spring. 4 points.*

Extension fields and roots of polynomials. Construction with straight edge and compass. Unique

factorization in rings. Elements of Galois theory.

Topology

MATH-UA 375 *Formerly MATH-UA 275. Prerequisite: a grade of C or better in Analysis I (MATH-UA 325) or permission of the department. Offered in the fall. 4 points.*

Metric spaces, topological spaces, compactness, connectedness. Covering spaces and homotopy groups.

Differential Geometry

MATH-UA 377 *Prerequisite: a grade of C or better in Analysis II (MATH-UA 326) or permission of the department. Offered in the fall. 4 points.*

The differential properties of curves and surfaces. Introduction to manifolds and Riemannian geometry.

Honors I

MATH-UA 393 *Prerequisite: approval of the director of the honors program. Offered in the fall. 4 points.*

A lecture/seminar course on advanced topics selected by the instructor and the audience, alternating between pure and applied, fall and spring. Topics vary yearly. Detailed course descriptions are available during preregistration.

Honors II

MATH-UA 394 *Prerequisite: approval of the director of the honors program. Offered in the spring. 4 points.*

The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem. Nehari, Conformal Mapping; Ahlfors, Complex Analysis.

Special Topics I, II

MATH-UA 395, 396 *4 points per term.*

Topics vary yearly. Detailed course descriptions are available during preregistration. Covers topics not offered regularly, such as experimental courses and courses offered on student demand.

Independent Study

MATH-UA 997, 998 *Prerequisite: permission of the department. 2 or 4 points per term.*

To register for this course, a student must seek out a faculty sponsor and draft a brief research proposal to be approved by the director of undergraduate studies.

Graduate Courses Open to Undergraduates

Graduate Courses Open to Undergraduates

Qualified students may take certain courses in the Graduate School of Arts and Science provided they first obtain permission from the director of undergraduate studies. A few such courses are listed below. If these courses are offered toward fulfillment of the requirement for the baccalaureate degree, no advanced credit is allowed for them in the graduate school. These are all 3-point courses, unless cross-listed as an undergraduate 4-point course.

Numerical Methods

MATH-GA 2010, MATH-GA 2020

Scientific Computing

MATH-GA 2043

Linear Algebra or Linear Algebra I, II

MATH-GA 2111 *(for students who have not taken MATH-UA 142) or MATH-GA 2110, 2120*

Algebra

MATH-GA 2130, MATH-GA 2140

Number Theory

MATH-GA 2210, MATH-GA 2220

Topology

MATH-GA 2310, MATH-GA 2320

Differential Geometry I, II

MATH-GA 2350, MATH-GA 2360

Real Variables

MATH-GA 2430, MATH-GA 2440

Complex Variables

MATH-GA 2450, 2460

Ordinary Differential Equations

MATH-GA 2470

Introduction to Applied Mathematics

MATH-GA 2701, 2702

Game Theory, Linear Programming

MATH-GA 2731, MATH-GA 2742

Mathematical Topics in Biology

MATH-GA 2850, 2851

Stochastic Calculus

MATH-GA 2902

Probability

MATH-GA 2911, 2912

Mathematical Statistics

MATH-GA 2962