Undergraduate Course Descriptions

Fall 2011 Course Syllabi
Spring 2011 Course Syllabi

MATH-UA 009 Algebra and Calculus
4 points.  Fall and spring terms.
Prerequisite:  High school math or permission of the department.

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



MATH-UA 120 Discrete Mathematics
4 points.  Fall and spring term.
Prerequisite: Calculus I (MATH-UA 121) Refer to Calculus website:http://math.nyu.edu/degree/undergrad/calculus.html

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


MATH-UA 121 Calculus I
4 points.  Fall and spring terms.
Prerequisite: 

   a. SAT score of 750 or higher
   b. ACT/ACTE Math score of 34 or higher.
   c. AB 4 or higher.
   d. BC 3 or higher.
   e. Completion of Algebra and Calculus (MATH-UA 009) with a grade of C or higher.
   d. passing placement exam.

Derivatives, antiderivatives, and integrals of functions of one real variable.  Trigonometric, inverse trigonometric, logarithmic and exponential functions.  Applications, including graphing, maximizing and minimizing functions.  Areas and volumes.


MATH-UA 122 Calculus II
4 points.  Fall and spring terms.
Prerequisite:  Passing MATH-UA 121 Calculus I with a grade of C or better or a BC of 4 or higher or passing placement test.

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


MATH-UA 123 Calculus III
4 points.  Fall and spring terms.
Prerequisite:  Passing MATH-UA 122 Calculus II a grade of C or higher or passing placement test.

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

MATH-UA 125 Mathematical Proof and Proving - identical to MTHED-UE 1049
2 points. Fall and spring Terms
It is suggested that this is taken after Calculus I and during Freshman year.

Will build on students' intuitions and informal logical argumentation and on mathematical concepts with which they are familiar. The course material includes 40-50 carefully constructed case-problems that create the need to prove in an investigative and engaging way, evoking a sense of uncertainty concerning the validity of the explored mathematical statements.

MATH-UA 130 Set Theory - identical to PHIL-UA 73
4 points.

Among the topics to be covered are: 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. If time permits we may also consider some more advanced topics, such as large cardinals or the independence results.

MATH-UA 140 Linear Algebra
4 points.  Fall and spring term.
Prerequisite:  A grade of C or better in MATH-UA 121 Calculus I or the equivalent.

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


MATH-UA 141 Honors Linear Algebra I - identical to G63.2110
4 points.  Fall term.
Prerequisite:  A grade of B or better in MATH-UA 325 Analysis Ior MATH-UA 343 Algebra 1 or the permission of the instructor.

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


MATH-UA 142 Honors Linear Algebra II - identical to G63.2120
4 points.  Spring term.
Prerequisite:  MATH-UA 141 Intensive Linear Algebra I.

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


MATH-UA 211,212 Mathematics for Economics I and II.
4 points. Fall and spring terms,respectively. Includes a recitation section.
Prerequisites: The same as for Calculus I.
Cannot apply both Calculus courses and Math for Economics courses towards your major.

This course is only open to Economics Policy Majors and prospective majors. If an Economics Policy Major decides to double major in Math these courses will replace Calculus I - III and they will need one extra elective.

To be offered in fall 2011 and spring 2012 and every fall and spring thereafter. Elements of calculus and linear algebra are important to the study of economics. This class is designed to provide the appropriate tools for study in the policy concentration. Examples and motivation are drawn from important topics in economics. Topics covered 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 (possibly) dynamic optimization and multivariable integration.

MATH-UA 221 Honors Calculus I:Accelerated Calculus with Linear Algebra I
5 points.Fall term.
Prerequisite: One of the following: (a) a score of 4 or higher on the Advanced Placement Calculus BC exam or 5 on the AB exam; or (b) MATH-UA 121 Calculus I and permission of the instructor.

This is the first semester of a year-long course that covers the essential content of Calculus II, Calculus III and Linear Algebra. The first 1/3 semester discusses sequences and series, Taylor's theorem, and power series. The next 1/3 semester introduces concepts from linear algebra including: linear systems of equations; matrices and LU decomposition; determinants; vector spaces; eigenvalues and eigenvectors. The last 1/3 semester introduces topics from vector calculus including: functions of several variables; vector-valued functions; partial derivatives; various applications including maxima and minima.

MATH-UA 222 Honors Calculus II : Accelerated Calculus with Linear Algebra II
5 points. Spring term
Prerequisite: MATH-UA 221 Honors Calculus I with a B or better.

This is the second semester of a year-long course that covers the essential content of Calculus II, Calculus III and Linear Algebra. Topics covered in the spring are 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).

MATH-UA 224 Vector Analysis
4 points.  Spring term.
Prerequisite:  Prerequisite:  Passing MATH-UA 325 Analysis I with a grade of C or better.

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' and Stokes' theorems on manifolds.

MATH-UA 228 Earth's Atmosphere and Ocean: Fluid Dynamics & Climate

4 points.
Prerequisites: Calculus I (or equivalent), with a grade of B- or better, though completion of Calculus III (multivariate calculus) is preferred and recommended. Students should also have some familiarity with introductory physics (even at the advanced high school level).

An introduction to the dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction. This is the core of climate science. Lectures will be guided by consideration of observations and experiments, but the goal 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), climate and climate variability (including El Nino and anthropogenic warming).

MATH-UA 233 Theory of Probability
4 points.  Fall term.
Prerequisite:  MATH-UA 122 Calculus II and MATH-UA 123 with a grade of C or better and/or the equivalent.

An introduction to the mathematical treatment 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.


MATH-UA 234 Mathematical Statistics
4 points.  Spring term.
Prerequisite:  MATH-UA 233 Theory of Probability with a grade of C or better and/or the equivalent.  Not open to students who have taken MATH-UA 235 Probability and Statistics.

An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the 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; analysis of variance.  Applications to the sciences.


MATH-UA 235 Probability and Statistics
4 points.  Spring term.
Prerequisite:  MATH-UA 122 Calculus II with a grade of C or better and/or the equivalent.

A combination of MATH-UA 233 Theory of Probability and MATH-UA 234 Mathematical Statistics at a more elementary level, so as to afford the student some acquaintance 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 approximation; application to coin-tossing, radioactive decay, etc.  In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.


MATH-UA 240 Combinatorics
4 points.  Spring term of even years.
Prerequisite:  MATH-UA 122 Calculus II with a grade of C or better and/or the equivalent.

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.

MATH-UA 243 Intro to Cryptography Introduction to Cryptography MATH-UA 243 Identical to CSCI-UA 480
 4 points. Spring term.
Prerequisite: CSCI-UA 310 with a grade of C or better or permission of the instructor.

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, EIGamal); digital signatures (RSA, Fiat-Shamir); authentication applications (identification, zero-knowledge) and others time permitting.


MATH-UA 246 Abstract Algebra
4 points. Spring term 2010
Prerequisite: MATH-UA 122 Calculus II and MATH-UA 140 Linear Algebra with a grade of C or better. Course not open to math majors and/or students who have taken Algebra I MATH-UA 343.

An 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.

NOTES: This course does not count toward the math major because of its considerable overlap with the more intensive Algebra I, MATH-UA 343, required as part of the majors program in Mathematics. It is, however, accepted toward the math minor, and is a strongly recommended course in the Steinhardt Math Ed major.


MATH-UA 248 Theory of Numbers
4 points.  Fall term.
Prerequisite:  MATH-UA 122 Calculus II with a grade of C or better and/or the equivalent.

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


MATH-UA 250 Mathematics of Finance
4 points.  Fall term.
Prerequisite:  MATH-UA 123 Calculus III, (and an introductory course in probability or statistics, MATH-UA 233 Theory of Probability, MATH-UA 235 Probability and Statistics, V31.0018 Statistics, V31.0120 Analytical Statistics or equivalent) with a grade of C+ or better.

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


MATH-UA 251 Introduction to Mathematical Modeling
4 points.  Spring term.
Prerequisite:  MATH-UA 121 Calculus I, MATH-UA 122 Calculus II and MATH-UA 123 Calculus III with a grade of C or better or permission of the instructor.

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


MATH-UA 252 Numerical Analysis
4 points. Spring term.
Prerequisite:  MATH-UA 123 Calculus III, MATH-UA 140 Linear Algebra with a grade of C or better.

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 gives an introduction to this subject for mathematics majors.  Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.


MATH-UA 255 Mathematics in Medicine and Biology - identical to G23.1501
4 points.  Fall term.
Prerequisite:  MATH-UA 121 Calculus I and V23.0011 Principles of Biology I or permission of the instructor.

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


MATH-UA 256 Computers in Medicine and Biology - identical to G23.1502
4 points.  Spring term.
Prerequisite:  MATH-UA 255 Mathematics in Medicine and Biology, or permission of the instructor. Familiarity with a programming language is recommended. The language used in the course will be MATLAB, but prior experience with MATLAB is not required.

Introduces students to the use of computer simulation as a tool for investigating biological phenomena. The course requirement is to construct three computer models during the semester, to report on results to the class, and to hand in a writeup describing each project. These projects can be done individually, or as part of a team. Topics discussed in class are the circulation of the blood, gas exchange in the lung, electrophysiology of neurons and neural networks, the renal countercurrent mechanism, cross-bridge dynamics in muscle, and the dynamics of epidemic and endemic diseases. Projects are normally chosen from this list, but may be chosen otherwise by students with other interests.


MATH-UA 262 Ordinary Differential Equations
4 points.  Fall and spring terms.
Prerequisite:  MATH-UA 122 Calculus II, MATH-UA 123 Calculus III and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

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


MATH-UA 263 Partial Differential Equations
4 points.  Spring term.
Prerequisite:  MATH-UA 262 Ordinary Differential Equations with a grade of C or better or the equivalent.

Many laws of physics are formulated as partial differential equations.  This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity.  Non-linear conservation laws and the theory of shock waves are discussed.  Further applications to physics, chemistry, biology, and population dynamics.


MATH-UA 264 Chaos and Dynamical Systems
4 points.  Fall term.  Spring term.
Prerequisite:  MATH-UA 122 Calculus II and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

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

MATH-UA 270 Transformations and Geometries
4 points.  Fall term of odd years.
Prerequisite:  MATH-UA 123 Calculus III with a grade of C or better or the equivalent. Also, MATH-UA 140 Linear Algebra with the grade of C or better is strongly suggested.

An axiomatic and algebraic study of Euclidean, non-Euclidean, affine, and projective geometries.  Special attention to group theoretic methods. 


MATH-UA 282 Functions of a Complex Variable
4 points.  Spring term.
Prerequisite:  MATH-UA 123 Calculus III plus one higher level course such as MATH-UA 140 Linear Algebra with the grade of C or better.

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


MATH-UA 325 Analysis I
4 points.  Fall and spring term.
Prerequisite:  MATH-UA 123 Calculus III and MATH-UA 140 Linear Algebra or the equivalent.

The real number system.  Convergence of sequences and series.  Rigorous study of functions of one real variable: continuity, connectedness, compactness, metric spaces, power series, uniform convergence and continuity.


MATH-UA 326 Analysis II
4 points.  Spring term.
Prerequisite:  MATH-UA 325 Analysis I or permission of the department.

Functions of several variables.  Limits and continuity.  Partial derivatives.  The implicit function theorem.  Transformation of multiple integrals.  The Riemann integral and its extensions.


MATH-UA 343 Algebra 1
4 points.  Fall term and Spring terms
Prerequisite:  MATH-UA 123 and MATH-UA 140 Linear Algebra with a grade of C or better and/or the equivalent. Additionally, it is suggested for students to have taken MATH-UA 325 Analysis I as a prerequisite.

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


MATH-UA 344 Algebra 2
4 points.  Spring term.
Prerequisite:  MATH-UA 343 Algebra I with a grade of C or better.

Extension fields, roots of polynomials.  Construction with straight-edge and compass.  Elements of Galois theory.


MATH-UA 375 Topology
4 points.  Offered on request.
Prerequisite:  MATH-UA 325 Analysis I or permission of the department.

Set-theoretic preliminaries.  Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.


MATH-UA 377 Differential Geometry
4 points.  Spring term of odd years.
Prerequisite:  MATH-UA 326 Analysis II or permission of the department.

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


MATH-UA 393-0394 Honors I, II
4 points each term.  Fall and spring term.
Prerequisite:  Approval of the director of the honors program.

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.


MATH-UA 394 Honors II
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.


MATH-UA 395-396 Special Topics I, II
4 points each term.  Offered on request.
Prerequisite:  Permission of the department.
Covers topics not offered regularly; experimental courses and courses offered on student demand.  Detailed course descriptions are available during preregistration.

Spring 2012 Matrices and Hamiltonian Systems
 The goal of the course is to describe and analyze the behavior of certain distinguished Hamiltonian flows on matrices. This includes, for example, the celebrated Toda flow, which is a completely integrable Hamiltonian system. The Toda flow is intimately related to the QR algorithm for the computation of eigenvalues of symmetric matrices.

Outline of course:

1. Basics of symmetric matrix theory: eigenvalues, eigenvectors, Householder transformation and Jacobi matrices.

2. Basics of Hamiltonian mechanics: mechanics in even dimensional Euclidean space. Integrable systems. Examples of flows on matrices, including the Toda flow.

3. The QR and Toda algorithms. Computing eigenvalues and the long-time behavior of the QR and Toda algorithms.



MATH-UA 997-998 Independent Study
2 or 4 points each term.  Fall and spring terms.
Prerequisite:  Permission of the department.

To register for this course a student must complete an application form for Independent Study and have the approval of a faculty sponsor and the Director of Undergraduate Studies.