# Courses for Mathematics

**MATH W 1003x or y College Algebra and Analytic
Geometry**

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.

*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.*

*3 points*

**MATH V 1101x or y Calculus I**

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) **

*Prerequisites: see Courses for First-Year Students. Functions, limits,
derivatives, introduction to integrals. BC: Fulfillment of 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. BC: Fulfillment of
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 or the equivalent. BC: Fulfillment of
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: MATH V1102, V1201, or the equivalent. BC: Fulfillment of General
Education Requirement: Quantitative and Deductive Reasoning
(QUA)..
3 points*

**MATH V 1208y Honors Mathematics 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. BC: Fulfillment of General Education Requirement: Quantitative and
Deductive Reasoning (QUA)..*

*4 points*

**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.*

*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. BC: Fulfillment of 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. BC: Fulfillment of 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. BC: Fulfillment of General Education Requirement:
Quantitative and Deductive Reasoning (QUA)..*

*3 points*

**MATH V 3020y Number Theory and Cryptography**

Congruences. Primitive roots. Quadratic residues. Contemporary applications.

*Prerequisites: one year of calculus. BC: Fulfillment of General Education
Requirement: Quantitative and Deductive Reasoning (QUA)..*

*3 points*

**MATH V 3025x 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.*

*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 V1102-MATH V1201 or the equivalent. Corequisites:
MATH V2010. BC: Fulfillment of 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 BC: Fulfillment of
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 V2030) 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. BC: Fulfillment
of 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. BC: Fulfillment of 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. BC: Fulfillment of General Education Requirement: Quantitative
and Deductive Reasoning (QUA)..*

*3 points*

**MATH W 4041x or y-W4042 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.

*BC: Fulfillment of General Education Requirement: Quantitative and
Deductive Reasoning (QUA)..*

*3 points*

**MATH W 4043x 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. BC: Fulfillment of General
Education Requirement: Quantitative and Deductive Reasoning
(QUA)..*

*3 points*

**MATH W 4044y 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 4046x Introduction to Category Theory**

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

*Prerequisites: MATH W4041 Not offered in 2015-2016.*

*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. BC:
Fulfillment of General Education Requirement: Quantitative and Deductive
Reasoning (QUA)..*

*3 points*

**MATH W 4052y 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, Math W4041 and Math W4051. Not offered in 2015-2016.*

*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 V2010, MATH W4041, MATH W4051*

*3 points*

**MATH W 4061x or y-W4062 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. BC: Fulfillment of 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 and y 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. BC: Fulfillment of
General Education Requirement: Quantitative and Deductive Reasoning
(QUA)..*

*3 points*

**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. - O. Savin

*Prerequisites: MATH W4051 or W4061 and V2010.*

*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*

## Engineering Courses

**MATH V 1207x Honors Mathematics A**

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.*

*4 points*

**MATH V 1208y Honors Mathematics 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.*

*4 points*

**MATH V 2000x or y 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. CC/GS: Partial Fulfillment of
Science Requirement. BC: Fulfillment of General Education Requirement:
Quantitative and Deductive Reasoning (QUA).

*3 points*

**MATH BC 2001x Perspectives in Mathematics**

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.

*Prerequisites: Some calculus or permission of the instructor.*

*1 point*

**MATH V 2030x 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 V1102-MATH V1201 or the equivalent.*

*3 points*

**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
clasificiation of flows in the plane (poincare-Bendixson Therem);
conservative and dissipative systems; linear and nonlinear stability analysis
of equilibria and periodic solutions; stble and unstable manifoleds;
bifurcations, e.g. Andronov-Hopf; sensitive depeneence and chaotic dynamics;
slected applications. - <.>

*Prerequisites: APMA E2101 (or MATH E1210)and APMA E3101*

*3 points*

**APMA E 4101y 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 4150x Applied Function Analysis**

Introduction to modern tools in functional analysis that are used in the
analysis of deterministic and stochastic partial differential equations and
in the analysis of numerical methods: metric and normed spaces. Banach
space of continuous functions, measurable spaces, the contraction mapping
theorem, Banach and Hilbert spaces bounded linear operators on Hilbert spaces
and their spectral decomposition, and time permitting distributions and
Fourier transforms.

*Prerequisites: Advanced calculus and a course in basic analysis, or
instructor's approval.*

*3 points*

**APMA E 4200x Partial Differential Equations**

Techniques of solution of partial differential equations. Separtion of the
variables. Orthogonality and characteristic functions, nonhomogeneous
boundary value problems. Solutions in orthogonal curvilinear coordinate
systems. Applications of Fourier integrals, Fourier and Laplace transforms.
Problems from the fields of vibrations, heat conduction, electricity, fluid
dynamics, and wave propagation are considered.

*Prerequisites: A course in ordinary differential equations*

*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*

## Cross-Listed Courses

### Computer Science

W3203 Discrete Mathematics: Introduction to Combinatorics and Graph Theory

W3251 Computational Linear Algebra

W4203 Graph Theory