• Course Code: 01:640:432
• Semester(s) Offered: Spring (frequently)
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 311. Material from prior courses (Math 250 and Calc III) will also be essential.

General Information

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. Differential geometry is a vast subject. A comprehensive introduction would require prerequisites in several related subjects, and would take at least two or three semesters of courses. In this elementary introductory course we develop much of the language and many of the basic concepts of differential geometry in the simpler context of curves and surfaces in ordinary 3 dimensional Euclidean space. Our aim is to build both a solid mathematical understanding of the fundamental notions of differential geometry and sufficient visual and geometric intuition of the subject. We hope that this course is of interest to students from a variety of math, science and engineering backgrounds, and that after completing this course, the students will be in a position to (i) apply their knowledge and skills in this course to their related subjects, (ii) be ready to study more advanced topics such as global properties of curves and surfaces, geometry of abstract manifolds, tensor analysis, and general relativity.

Prerequisites:

The officially listed prerequisite is 01:640:311. But equally essential prerequisites from prior courses are Multivariable Calculus and Linear Algebra. Most notions of differential geometry are formulated with the help of Multivariable Calculus and Linear Algebra.
01:640:311, which itself requires Multivariable Calculus and Linear Algebra as prerequisites, is an important prerequisite because it helps students build mathematical maturity and gain the ability to understand, formulate and present precise mathematical concepts and proofs.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Sample Syllabus

Syllabus, Spring 2008. (for the DoCarmo text). Now obsolete.

This course is taught in the Spring semester.

Schedule of Sections

Previous semesters

• Spring 2009. Prof. Nussbaum
• Spring 2008. Prof. Han

• Course Code: 01:640:435
• Semester(s) Offered: Fall (frequently)
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250 and Calc III and Math 300

General Information (Catalog listing)

01:640:435 Geometry (3)
Various geometries, including projective and non-Euclidean geometries, and geometric axiom systems.
Prerequisites: CALC3, 01:640:250, and 300, or permission of department.

Prerequisites

Most notions of differential geometry are formulated with the help of Multivariable Calculus and Linear Algebra. Math 300 is an important prerequisite because it helps students build mathematical maturity and gain the ability to understand, formulate and present precise mathematical concepts and proofs. Additional exposure to proofs in courses like Math 311 and Math 350 is also helpful.

Textbook

For current textbook please refer to our Master Textbook List page

Sample Syllabus

Prof. Tunnell (2019)

This course is taught during the Fall semester.

Schedule of Sections:

01:640:435 Schedule of Sections

Previous semesters:

• Fall 2008. Prof. Han
• Fall 2007, Sec. 01. Prof. Carbone, Sec. 02. Prof. Gonzalez
• Fall 2006, Sec. 1. Prof. Han
• Fall 2006, Sec. 02. Prof. X. Huang
• Fall 2005, Section 1, Prof. Han
• Fall 2004, Section 1, Prof. Han
• Fall 2002, Section 1, Prof. Han

• Course Code: 01:640:437
• Semester(s) Offered: Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250 and Calc III

Course Description

This course will present an overview of the development of mathematics from ancient civilizations to the beginning of the 19th century. Selected topics from the history of mathematics in China, India, Europe, and the Islamic world, including number systems, Euclidean geometry, algebra, combinatorics, and calculus. Recurrent themes include calculation of areas and volumes, finding zeros of polynomials, progressive enlargement of number systems, and changing concepts of rigorous proof.

Prerequisites: Math 250 and Calc III (Math 251 or Math 291)

This course may include oral or written presentations of selected topics by students. However, it does not satisfy writing requirements for the SAS core curriculum.

Textbook

For current textbook, please refer to our Master Textbook List page

Previous Semesters

• Spring 2026: Kiessling
• Spring 2025: Retakh
• Fall 2021: Zeilberger

• Course Code: 01:640:441
• Semester(s) Offered: Fall
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Calc IV and Analysis (Math 311 or Math 411)

General Information (Catalog listing)

01:640:441-442 Introductory Topology I,II (3,3)
Math 441: Introduction to topology with emphasis on the foundations of analysis; Euclidean spaces, metric spaces, topological spaces, and their properties; applications to analysis. Math 442: Basic concepts of algebraic topology, including the fundamental group, plane curves, homotopy, and a brief introduction to homology.
Prerequisites: CALC4 and Math 311 (Analysis I) or permission of department.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus

See the 2004 syllabus for a general idea, but be prepared for considerable variation in emphasis from one year to the next.

This course is offered during the Fall semester.

Schedule of Sections:

01:640:441 Schedule of Sections

Previous semesters:

• Fall 2019: Prof. Weibel
• Fall 2018: Prof. Tarasca
• Fall 2017: Prof. Nussbaum
• Fall 2016: Prof. Rong
• Fall 2010. Prof. Saks
• Fall 2008. Prof. Speer
• Fall 2007. Prof. Bahri; Followed in Spring 2008 by Math 442, taught by Prof. Bahri.
• Spring 2007. Prof. Chanillo.
• Fall 2005. Prof. Sahi
• Fall 2004. Prof. Luo

• Course Code: 01:640:442
• Semester(s) Offered: Spring (frequently)
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 441

General Information (Catalog listing)

01:640:441-442 Introductory Topology I,II (3,3)
Math 441: Introduction to topology with emphasis on the foundations of analysis; Euclidean spaces, metric spaces, topological spaces, and their properties; applications to analysis. Math 442: Basic concepts of algebraic topology, including the fundamental group, plane curves, homotopy, and a brief introduction to homology.
Prerequisites: CALC4 and Math 311 (Analysis I) or permission of department.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus

See the syllabus from 2020 for a general idea, but be prepared for considerable variation in emphasis from one year to the next.

Schedule of Sections:

01:640:442 Schedule of Sections

Previous semesters:

• Spring 2020. Prof. Weibel
• Fall 2010. Prof. Saks
• Fall 2008. Prof. Speer
• Fall 2007. Prof. Bahri; Followed in Spring 2008 by Math 442, taught by Prof. Bahri.
• Spring 2007. Prof. Chanillo.
• Fall 2005. Prof. Sahi
• Fall 2004. Prof. Luo

• Course Code: 01:640:451
• Semester(s) Offered: Fall
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Special permission only. Typically students have already taken Math 350 or Math 351.

General Information (Catalog listing)

01:640:451-452 Abstract Algebra I,II (3,3)
Rigorous study of abstract algebraic systems including groups, rings, and fields.
Prerequisites: Permission of department and instructor. For students preparing for graduate study in the mathematical sciences.

This course forms part of the Honors Track sequence. Special permission is required for admission to Math 451. Requests for admission are evaluated based on the student's prior achievements, level of interest, and potential for success in the course.

Required Textbook

Michael Artin; Algebra (second edition); Prentice-Hall, 2011 (560 pp.); (ISBN: 0-13-241377-9; ISBN13: 9780132413770).
See also the  Master Textbook List page.

Short Description

Groups: axiomatic formulation, subgroups, isomorphisms, cosets, quotient groups
Symmetry: actions of groups on sets and vector spaces
Theory of groups: groups of finite order, Sylow theorems, free groups, generators and relations
Representations of groups: representing groups via linear transformations

Topics, in approximate order

Groups and subgroups
Group homomorphisms and isomorphisms
Cosets and Lagrange’s theorem
Conjugation and normal subgroups
Quotient groups
Group actions
Groups of isometries
Groups of permutations
The orbit-stabilizer theorem
Groups of finite order
Sylow theorems
Free groups, generators and relations
Group representations
Characters
The regular representation

As time allows, with the instructor’s discretion, some additional topics may also be covered, such as the following:
Finite groups of Lie type, finite simple groups, classifying abelian groups, linear groups.

Sample Syllabus

Spring 2013

Schedule of Sections

Previous semesters

• Course Code: 01:640:452
• Semester(s) Offered: Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 451

General Information (Catalog listing)

01:640:451-452 Abstract Algebra I,II (3,3)
Rigorous study of abstract algebraic systems including groups, rings, and fields.
Prerequisites: Permission of department and instructor. For students preparing for graduate study in the mathematical sciences.

Math 452 is part of the Honors Track sequence. It is intended to be taken immediately after Math 451. Students enrolled in Math 451 during the Fall semester should be able to preregister for Math 452 in the Spring without assistance. All other students should contact the Honors Advisor <>.

Required Textbook

Michael Artin; Algebra (second edition); Prentice-Hall, 2011 (560 pp.); (ISBN: 0-13-241377-9; ISBN13: 9780132413770).
See also the Master Textbook List page.

Short Description

Review of vector spaces, linear transformations and matrices, including Jordan canonical form.
Factoring in commutative rings: principal ideal domains, unique factorization domains; examples.
Modules; free modules; diagonalizing integer matrices; application to the structure of finitelygenerated abelian groups.
Modules over polynomial rings; application to rational canonical form for linear operators.

Topics, in approximate order

Review of vector spaces, linear transformations, matrices and Jordan canonical form
Rings
Commutative rings
Homomorphisms and ideals
Quotient rings, product rings
Rings of fractions
Maximal ideals
Factoring in commutative rings
Principal ideal domains
Unique factorization domains
Modules
Free modules
Noetherian rings
Structure of abelian groups
Application to linear operators, rational canonical form

 As time allows, with the instructor’s discretion, some additional topics may also be covered, such as the following:
Quadratic number fields, fields and Galois theory, non-commutative rings, algebras.

Sample Syllabus

Spring 2013

Schedule of Sections

Previous semesters

• Course Code: 01:640:454
• Semester(s) Offered: Fall, Summer
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250 and Calc II

General Information (Catalog listing)

01:640:454 Combinatorics (3)
Existence and enumeration of designs and patterns such as codes, graphs, and block designs, and extremal problems related to such objects. Emphasis on applications to computer, biological, physical, and social problems.
Prerequisites: CALC2 and 01:640:250.

Textbook

Textbook varies

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus set by Instructor

Sample syllabus: Fall 2021 Prof. Weibel

Taught in Summer and Fall semesters.

Schedule of Sections:

01:640:454 Schedule of Sections

Previous semesters:

• Fall 2011: Prof. Weibel (Roberts/Tesman)
• Fall 2010: Prof. Beck.
• Fall 2009: Prof. Beck.
• Summer 2009: Paul Raff (Roberts/Tesman)
• Fall 2008: Prof. Beck (Roberts/Tesman)
• Summer 2008, Paul Ellis (Roberts/Tesman)
• Fall 2007, Prof. Bumby (Roberts/Tesman)
• Summer 2007, Michael Weingart (Roberts/Tesman)
• Fall 2006, Prof. Beck. Using Brualdi
• Fall 2005, Prof. Cherlin. Using Roberts/Tesman

• Course Code: 01:640:461
• Semester(s) Offered: Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 300 and Calc III

General Information (Catalog listing)

Intuitive and formal development of the sentential and predicate calculus. Special emphasis given to questions of consistency, completeness, and independence. Formal systems; incompleteness and undecidability; theorems of Gödel. Exploration of which properties of structures can be defined in the first-order language.
Prerequisite: CALC3 and either 01:640:300 or permission of department

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Sample Syllabus

Variable: Spring 2007

This course is taught each Spring semester.

Schedule of Sections

Archives

For Instructors

Previous semesters:

• Spring 2009. A. Deloro
• Spring 2008. Sam Coskey
• Spring 2007. Prof. Thomas

• Course Code: 01:640:477
• Semester(s) Offered: Fall, Spring, Summer
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Calc III

• General Information
• Textbook
• Special Accommodations
• Academic Integrity
• Sample Syllabus
• Related Sites
• Sections Taught this Semester
• Previous Semesters

General Information

Prerequisites: CALC III -- third-semester, multiple-variable calculus, which is Math 251 at Rutgers-NB -- is an unwaivable prerequisite. A working knowledge of multiple integrals and partial derivatives is essential for the course.

Restrictions on Credit: A student can receive credit for at most one of the courses

• 01:640:477
• 01:198:206
• 01:960:381
• 14:332:226

(despite the fact that Math 477 covers considerably more material than Comp Sci 206).

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Special Accommodations

Students with disabilities requesting accommodations must follow the procedures outlined at https://ods.rutgers.edu/students/applying-for-services

Academic Integrity

All Rutgers students are expected to be familiar with and abide by the academic integrity policy. Violations of the policy are taken very seriously.

Sample Syllabus

Prof. Salamon (2022)
Prof. Speer (2010).

Related Sites

• Mathematical Finance at Rutgers
• Actuarial Club
• Probability Theory (Wikipedia)

Schedule of Sections:

01:640:477 Schedule of Sections

Previous semesters