Mathematics Courses of Instruction

MATH UN1101 Calculus I. 3 points.

Prerequisites: (see Courses for First-Year Students). Functions, limits, derivatives, introduction to integrals, or an understanding of pre-calculus will be assumed.

The Help Room in 333 Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the teaching assistants. (SC)

MATH UN1102 Calculus II. 3 points.

Prerequisites: MATH UN1101 or the equivalent.

Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC) 

MATH UN1201 Calculus III. 3 points.

Prerequisites: MATH UN1101 or the equivalent

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) 

MATH UN1202 Calculus IV. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

MATH UN1207 Honors Mathematics A. 4 points.

Prerequisites:  (see Courses for First-Year Students).  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)

MATH UN1208 Honors Mathematics B. 4 points.

Prerequisites: (see Courses for First-Year Students).

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)

MATH UN2000 An Introduction to Higher Mathematics. 3 points.

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. CC/GS: Partial Fulfillment of Science Requirement.  BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

MATH BC2001 Perspectives in Mathematics. 1 point.

Prerequisites: some calculus or the instructor's permission.

Intended as an enrichment to the mathemathics curriculum of the first 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.

MATH BC2006 Combinatorics. 3 points.

Corequisites: MATH V2010 is helpful as a corequisite, but not required.

Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.

MATH UN2010 Linear Algebra. 3 points.

Prerequisites: MATH UN1201 or the equivalent.

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

MATH V2020 Honors Linear Algebra. 3 points.

CC/GS: Partial Fulfillment of Science Requirement
Not offered during 2019-20 academic year.

Prerequisites: MATH V1201.

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-Math V1208.

MATH UN2030 Ordinary Differential Equations. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent.

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.

MATH UN2500 Analysis and Optimization. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010.

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)

MATH UN3007 Complex Variables. 3 points.

Prerequisites: MATH UN1202 An elementary course in functions of a complex variable.

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)

MATH UN3020 Number Theory and Cryptography. 3 points.

Prerequisites: one year of calculus.

Prerequisite: One year of Calculus. Congruences. Primitive roots. Quadratic residues. Contemporary applications.

MATH UN3025 Making, Breaking Codes. 3 points.

Prerequisites: (MATH UN1101 and MATH UN1102 and MATH UN1201) and and MATH UN2010.

A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory.

MATH UN3027 Ordinary Differential Equations. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent.
Corequisites: MATH UN2010

Equations of order one; systems of linear equations. Second-order equations. Series solutions at regular and singular points. Boundary value problems. Selected applications. 

MATH UN3028 Partial Differential Equations. 3 points.

Prerequisites: MATH UN3027 and MATH UN2010 or the equivalent

Introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems. 

MATH UN3050 Discrete Time Models in Finance. 3 points.

Prerequisites: (MATH UN1102 and MATH UN1201) or (MATH UN1101 and MATH UN1102 and MATH UN1201) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600).

Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.

MATH UN3386 Differential Geometry. 3 points.

Prerequisites: MATH UN1202 or the equivalent.

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.

MATH UN3901 Supervised Readings in Mathematics I. 2-3 points.

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.  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.

MATH UN3902 Supervised Readings in Mathematics II. 2-3 points.

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.  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.

MATH UN3951 Undergraduate Seminars in Mathematics I. 3 points.

Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

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.

MATH UN3952 Undergraduate Seminars in Mathematics II. 3 points.

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

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. Prerequisite: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

MATH V3997 Supervised Individual Research. 3 points.

Prerequisites: the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.

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.

MATH V3998 Supervised Individual Research. 3 points.

Prerequisites: the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.

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.

MATH UN1003 College Algebra and Analytic Geometry. 3 points.

Prerequisites: score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate grade on the General Studies Mathematics Placement Examination.

Columbia College students do not receive any credit for this course and must see their CSA advising dean. 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.

MATH GU4007 Analytic Number Theory. 3 points.

Prerequisites: MATH UN3007

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.

MATH GU4032 Fourier Analysis. 3 points.

Prerequisites: three terms of calculus and linear algebra or four terms of calculus.

Prerequisite: three terms of calculus and linear algebra or four terms of calculus. 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. 

MATH GU4041 Introduction to Modern Algebra I. 3 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent

The second term of this course may not be taken without the first. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

MATH GU4042 Introduction to Modern Algebra II. 3 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent.

The second term of this course may not be taken without the first. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

MATH GU4043 Algebraic Number Theory. 3 points.

Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent

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.

MATH GU4044 Representations of Finite Groups. 3 points.

Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent.

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.

MATH GU4045 Algebraic Curves. 3 points.

Prerequisites: (MATH GU4041 and MATH GU4042) and MATH UN3007

Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem.

MATH W4046 Introduction to Category Theory. 3 points.

CC/GS: Partial Fulfillment of Science Requirement
Not offered during 2019-20 academic year.

Prerequisites: MATH W4041.

Categories, functors, natural transformations, adjoint functors, limits and colimits, introduction to higher categories and diagrammatic methods in algebra.

MATH GU4051 Topology. 3 points.

Prerequisites: (MATH UN1202 and MATH UN2010) and rudiments of group theory (e.g., MATH GU4041). MATH UN1208 or MATH GU4061 is recommended, but not required.

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.

MATH GU4052 Introduction to Knot Theory. 3 points.

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 linear algebra, or equivalent.

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.

MATH GU4053 Introduction to Algebraic Topology. 3 points.

Prerequisites: MATH UN2010 and MATH GU4041 and MATH GU4051

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.

MATH GU4061 Introduction To Modern Analysis I. 3 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

MATH GU4062 Introduction To Modern Analysis II. 3 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

MATH GU4065 Honors Complex Variables. 3 points.

Prerequisites: (MATH UN1207 and MATH UN1208) or MATH GU4061

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.

MATH W4071 Introduction to the Mathematics of Finance. 3 points.

CC/GS: Partial Fulfillment of Science Requirement, BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

Prerequisites: MATH V1202, MATH V3027, STAT W4150, SEIOW4150, or their equivalents.

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.

MATH GU4081 Introduction to Differentiable Manifolds. 3 points.

Prerequisites: (MATH GU4051 or MATH GU4061) and MATH UN2010

Concept of a differentiable manifold. Tangent spaces and vector fields. The inverse function theorem. Transversality and Sard's theorem.  Intersection theory. Orientations. Poincare-Hopf theorem. Differential forms and Stokes' theorem.

MATH GU4155 Probability Theory. 3 points.

Prerequisites: MATH GU4061 or MATH UN3007

A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains.

MATH W4391 Intro to Quantum Mechanics: An Introduction for Mathematicians and Physicists I. 3 points.

CC/GS: Partial Fulfillment of Science Requirement
Not offered during 2019-20 academic year.

Prerequisites: MATH V1202 or the equivalent and MATH V2010.

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.

MATH W4392 Quantum Mechanics: An Introduction for Mathematicians and Physicists II. 3 points.

Not offered during 2019-20 academic year.

Prerequisites: MATH V1202 or the equivalent, MATH V2010, and MATH W4391.

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.

COMS S3251 Computational Linear Algebra. 3 points.

Not offered during 2019-20 academic year.

Prerequisites: two terms of calculus.

Computational linear algebra, solution of linear systems, sparse linear systems, least squares, eigenvalue problems, and numerical solution of other multivariate problems as time permits.

COMS W4203 Graph Theory. 3 points.

Lect: 3.

Prerequisites: (COMS W3203)

General introduction to graph theory. Isomorphism testing, algebraic specification, symmetries, spanning trees, traversability, planarity, drawings on higher-order surfaces, colorings, extremal graphs, random graphs, graphical measurement, directed graphs, Burnside-Polya counting, voltage graph theory.

COMS W3203 Discrete Mathematics: Introduction to Combinatorics and Graph Theory. 3 points.

Lect: 3.

Prerequisites: Any introductory course in computer programming.

Logic and formal proofs, sequences and summation, mathematical induction, binomial coefficients, elements of finite probability, recurrence relations, equivalence relations and partial orderings, and topics in graph theory (including isomorphism, traversability, planarity, and colorings).