MATH W 1003x or y College Algebra and Analytic Geometry

For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits.
Prerequisites: Score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate gade on the General Studies Mathematics Placement Examination.
3 points

MATH V 1101x or y Calculus I

The Help Room on the 3rd floor of Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the instructors and teaching assistants. (SC)
Prerequisites: see Courses for First-Year Students. Functions, limits, derivatives, introduction to integrals. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 1102x or y Calculus II

Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC)
Prerequisites: MATH V1101 or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 1201x or y Calculus III

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)
Prerequisites: MATH V1101 with a grade of B or better or Math V1102, or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 1202x or y Calculus IV

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)
Prerequisites: MATHV1102, V1201, or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 1207x-V1208y Honors Mathematics A-B

The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)
Prerequisites: (see Courses for First-Year Students). Recitation Section Required. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
4 points

MATH V 2000x An Introduction to higher Mathematics

Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training.
General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH BC 2001x Perspectives in Mathematics

Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.
Prerequisites: Some calculus or permission of the instructor.
1 point

MATH BC 2001x Perspectives in Mathematics
Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.
Prerequisites: Some calculus or permission of the instructor.
1 point

MATH BC 2006y Combinatorics

Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.
Corequisites: MATH V2010 is helpful as corequisite, not required. Not offered in 2009-2010.
3 points

MATH V 2010x or y Linear Algebra

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)
Prerequisites: V1201, or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 2020x Honors Linear Algebra

A more extensive treatment of the material in Math V2010, with increased emphasis on proof. Not to be taken in addition to Math V2010 or Math V1207-V1208.
Prerequisites: Math V1201
3 points

MATH V 2500x or y Analysis and Optimization

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)
Prerequisites: Math V1102-Math V1201 or the equivalent and MATH V2010. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3007y Complex Variables

Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)
Prerequisites: MATH V1202. An elementary course in functions of a complex variable. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3020x Number Theory and Cryptography

Congruences. Primitive roots. Quadratic residues. Contemporary applications.
Prerequisites: one year of calculus. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3021y Combinatorial Number Theory

Advanced topics in number theory. Continued fractions. Approximations by rational numbers. Transcendental numbers. Arithmetic functions. Partitions of numbers and their generating functions. Stress on the combinatorial and algorithmic aspects of number theory. Contemporary applications.
Prerequisites: MATH V3020 or MATH W4041. Not offered in 2009-2010.
3 points

MATH V 3025y Making, breaking codes

A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory.
Prerequisites: Calculus I, II, III and Linear Algebra. Not offered in 2009-2010.
3 points

MATH V 3027x Ordinary Differential Equations

Equations of order one; systems of linear equations. Second-order equations. Series solutions at regular and singular points. Boundary value problems. Selected applications.
Prerequisites: MATH V1201 or the equivalent. Corequisites: MATH V2010. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3028y Partial Differential Equations

. Introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems.
Prerequisites: MATH V3027 and MATH V2010 or the equivalent General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3050y Discrete Time Models In Finance

Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.
Prerequisites: MATH V1102, V1201(or V1101, V1102, V1201), V2010. Recommended: MATH V3027(or MATH E1210) and SIEO W3600.
3 points

MATH V 3386x Differential Geometry

Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.
Prerequisites: MATH V1202 or the equivalent.
3 points

MATH V 3901x-V3902y Supervised Readings In Mathematics

Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor.
Prerequisites: the written permission of the staff member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the director of undergraduate studies. The written permission must be deposited with the director of undergraduate studies before registration is completed. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
2-3 points.

MATH V 3951x-V3952y Undergraduate Seminars In Mathematics

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow.
Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the permission of the director of undergraduate studies. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH V 3997x-V3998y Supervised individual research

For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member. .
Prerequisites: The written permission of the faculty member who agrees to act as a supervisor, and the permission of the director of the undergraduate studies.
3 points

MATH W 4007y Analytic Number Theory

A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms.
Prerequisites: Math V3007
3 points

MATH W 4032y Fourier Analysis

Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines.
Prerequisites: three terms of calculus and linear algebra or four terms of calculus. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH W 4041xy-W4042xy Introduction To Modern Algebra

The second term of this course may not be taken without the first. Prerequisite: Math V1102-Math V1202 and MATH V2010, or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.
General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH W 4043y Advanced Topics In Algebra: Algebraic Number Theory

Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function.
Prerequisites: MATH W4041-W4042 or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH W 4044x Representations of Finite Groups

Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups.
Prerequisites: Math V2010 and Math W4041 or the equivalent.
3 points

MATH W 4045y Algebraic Curves

Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem.
Prerequisites: Mathematics W4041,W4042 and Mathematics V3007.
3 points

MATH W 4050y Topics In Geometry and Topology

Advanced topics in geometry and topology chosen by the instructor from the following list. Non-Euclidean geometry (e.g., hyperbolic, elliptic, projective), combinatorial topology, algebraic topology, knot theory, braid theory, Morse theory, dynamical systems, foliations, graph theory.
Prerequisites: Math W4041 Not offered in 2009-2010.
3 points

MATH W 4051x Topology

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.
Prerequisites: MATH V1202, MATH V2010, and rudiments of group theory (e.g., MATH W4041). MATH V1208 or W4061 is recommended, but not required. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH W 4052x Introduction to Knot Theory

The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants.
Prerequisites: Math V2010 or equivalent and Math W4041. Recommended: Math W4051 or equivalent.
3 points

MATH W 4053y Introduction to Algebraic Topology

The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces.
Prerequisites: MATH V21010, MATH W4041, MATH W4051 Not offered in 2009-2010.
3 points

MATH W 4061xy-W4062xy Introduction To Modern Analysis

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.
Prerequisites: The second term of this course may not be taken without the first. Prerequisites: MATH V1202 or the equivalent and V2010. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH W 4065x Honors Complex Variables

A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.
Prerequisites: MATH V1207 and Math V1208 or MATH W4061.
3 points

MATH W 4071x Introduction To the Mathematics of Finance

The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.
Prerequisites: MATH V1202, V3027, STAT W4150, SEIO W4150, or their equivalents. General Education Requirement: Quantitative and Deductive Reasoning (QUA).
3 points

MATH G 4073x Quantitative Methods In Investment Management

Surveys the field of quantitative investment strategies from a "buy side" perspective, through the eyes of portfolio managers, analysts and investors. Financial modeling there often involves avoiding complexity in favor of simplicity and practical compromise. All necessary material scattered in finance, computer science and statistics is combined into a project-based curriculum, which give students hands-on experience to solve real world problems in portfolio management. Students will work with market and historical data to develop and test trading and risk management strategies. Programming projects are required to complete this course.

MATH W 4081y Introduction To Differentiable Manifolds

The implicit function theorem. Concept of a differentiable manifold. Tangent space and tangent bundle, vector fields, differentiable forms. Stoke's theorem, tensors. Introduction to Lie groups.

MATH G 4151x Analysis and Probability

Measure theory; elements of probability; elements of Fourier analysis; Brownian motion.
4.5 points

MATH W 4386x-W4387y Geometrical Concepts In Physics

Material from topology and differential geometry with illustrations of their use in electrodynamics, general relativity, and Yang-Mills theory. In particular topological and differential manifolds, tensors, vector bundles, connections, and Lie groups are covered.
Prerequisites: MATH V1202 or the equivalent and V2010. Not offered in 2009-2010.
3 points

MATH W 4391x-W4392y Quantum Mechanics: An Introduction for Mathematicans and Physicists

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.
Prerequisites: Math V1202 or the equivalent and Math V2010.
3 points

MATH E 1210x or y Ordinary Differential Equations

Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.
Prerequisites: MATH V1201 or the equivalent.
3 points

APMA BC 2001x Perspectives in Mathematics

Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.
Prerequisites: Some calculus or permission of the instructor.
1 point

MATH BC 2001x Perspectives in Mathematics

Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.
Prerequisites: Some calculus or permission of the instructor.
1 point

APMA E 4101y Introduction to nonlinear dynamical systems

Introduction to nonlinear differential equations and dynamical systems and their chaotic behavior. Topics include Lagrangian and Hamiltonian formulation of mechanics, phase space dynamics. Critical points and stability. Systems of linear differential equations. Solving simple nonlinear differential equations. Onset of chaos. Systems with dissipation, limit points and cycles, strange attractors.

APMA E 4101x Introduction to Dynamical Systems

An introduction to the analytic and geometric theory of dynamical systems; basic existence, uniqueness and parameter dependence of solutions to ordinary differential equations; constant coefficient and parametrically forced systems; Fundamental solutions; resonance; limit points, limit cycles and classification of flows in the plane (Poincare-Bendixson Therem); conservative and dissipative systems; linear and nonlinear stability analysis of equilibria and periodic solutions; stable and unstable manifolds; bifurcations, e.g. Andronov-Hopf; sensitive dependence and chaotic dynamics; selected applications. - <.>
Prerequisites: APMA E2101 (or MATH E1210) and APMA E3101
3 points

APMA E 4400y Introduction to biophysical modeling.

Introduction to physical and mathematical models of cellular and molecular biologoy. Physics at the cellular schale (viscosity, heat, diffusion, statistical mechanics). RNA transcription and regulation of genetic expression. Genetic and biochemical networks. Bioinformatics as applied to reverse-engineering of naturally-occurring networks and to forward-engineering of synthetic biological networks. Mathematical and physical aspects of functional genomics.
Prerequisites: Advanced calculus or the instructor's approval.
3 points