NEW YORK UNIVERSITY
THE UNDERGRADUATE PROGRAM
Mathematics Department
ACADEMIC YEAR 2008 – 2009
(Revised October 2008)
New York University
Mathematics Department
Warren Weaver Hall
251 Mercer Street
Room 724
New York, New York 10012
(212) 998-3005
Fax (212) 995-4121
This guide is meant to provide majors with a comprehensive picture of the offerings of the
Mathematics Department. It contains information on courses, degree requirements, special
programs, activities, prizes and awards. It supplements and repeats the information contained in
the New York University Bulletin for the College of Arts and Sciences.
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Table of Contents
1.
Welcome............................................................................................ ………p. 4
2.
Advising……. ........................................................................................
p. 4
3.
Degree Programs......................................................................................
p. 5
4.
Honors Program.................................................................... …………….. p. 7
5.
Transfer Students ....................................................................................... p. 8
6.
Courses Offered................................................................................ ………p. 9
7.
Suggested Course Program..................................................................
p. 10
8.
Activities.................................................................................................
p. 11
9.
Awards....................................................................................................
p. 11
10.
Work Opportunities in the Mathematics Department ................................
p. 12
11.
Courses Descriptions.................................... ……………………………. p. 12
12.
Graduate Courses Open to Undergraduates............................................
p. 18
13.
Departmental Faculty.............................................................................
p. 18
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1. Welcome To The Mathematics Department
The undergraduate division of the Department of Mathematics offers a wide variety of courses in
pure and applied mathematics. The faculty are members of the University’s Courant Institute of
Mathematical Sciences, noted for its advanced training and research programs, which emphasize
the applications of mathematics to technology and other branches of science.
In accordance with the Institute’s philosophy, emphasizing the applications of mathematics to
technology and other branches of science, the department participates in many interdisciplinary
programs. Joint programs are available in mathematics and (1) computer sciences, (2)
economics, (3) engineering and (4) minor in mathematics secondary school education. They lead
to the B.A. in four years, with the exception of the engineering option, which leads to a joint B.S.
degree from New York University and a B.E. degree from Stevens Institute of Technology.
Additionally, an accelerated B.A. and M.S. degree from New York University in five years is
offered. These programs are described in more detail below. Special courses in the
mathematical aspects of finance, biology, and medicine are also available.
The department provides an honors program in (1) mathematics and (2) mathematics and
computer science for outstanding students. Additionally, independent study courses are available
for students with special interests. Qualified undergraduates are granted an opportunity to enroll
in courses in the graduate division of the department. All students have access to the Institute’s
library which houses a large, up-to-date collection of books and technical journals in
mathematics and computer science. The library is located on the twelfth floor of the Courant
Institute.
2. Advising
Students intending to major in mathematics are urged to declare their major as early as possible
as and no later than the beginning of their junior year. Students can declare their major by
completing a major declaration form which is found on the undergraduate mathematics website:
math.nyu.edu/degree/undergrad/. The department office is located in rooms 724 and 725 of
Warren Weaver hall. All matters concerning undergraduate advisement in mathematics are
handled there. When necessary, the department administrator will arrange appointments with the
Director or Undergraduate Studies for advisement or academic counseling.
Each math major is assigned to a faculty advisor in the math department. All declared
mathematics majors must have their proposed course schedules approved each term by his/her
faculty advisor. The approved scheduled must be signed by their faculty advisor and delivered to
the math department administrative staff in room 724 or 725 for final clearance.
Students who believe that they are prepared to start their work in mathematics at an advanced
level, or who feel qualified to enter a course without the formal prerequisites, should arrange to
submit a transcript of prior college level coursework and a course description to the Director of
Undergraduate Studies. Students who have not taken college level coursework but feel that they
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are prepared to enroll in advanced courses should arrange to demonstrate proficiency by taking a
final examination in the prerequisite course for their desired course. Placement exams are offered
each semester.
3. Degree Programs
Major in Mathematics. The requirements for a mathematics major are twelve courses
numbered V63.0120 or higher. The only exception is Introduction to Abstract Algebra
(V63.0246) which counts for the math minor only. The courses must include: 1) Calculus I, II,
and III (V63.0121, V63.0122, V63.0123), 2) Linear Algebra (V63.0140), (3) Analysis I
(V63.0325) and Algebra I (V63.0343). The rest of the 12 required courses must include one of
the following: Analysis II, (V63.0326); Algebra II, (V63.0344), or Vector Analysis, (V63.0244).
In addition, the student must maintain a 2.5 math GPA.
The sequence Honor Calculus I and II, (V63.0221, V63.0222) is counted as two courses; it
covers material from Calculus II and III. Therefore, students enrolling in honors calculus are
expected to have a solid background in Calculus I.
Any two computer science courses numbered V22.0101 or higher may be credited toward the 12
course requirement. Students who complete the Pre-medical or Pre-dental Program and who
wish to major in mathematics may count any two of the following courses toward their 12 course
math requirement: General Physics I and II (V85.0011 and 0012) or Physics I, II (V85.0100 and
0101). However, if these physics courses are used towards the mathematics major, the computer
sciences courses will not apply towards the major. Courses taken under the pass/fail option
cannot be counted toward the major. A grade of C or better is required in all courses used to
fulfill the major requirement.
Joint Major in Mathematics and Computer Science. An interdisciplinary major offered
jointly by the Department of Mathematics and Computer Science, provides the opportunity to
study both computer science and relevant mathematics courses such as analysis, algebra,
probability, and statistics. Eight computer science courses are required for the major. Please
refer to the undergraduate computer science website for more information about the computer
science courses: http://cs.nyu.edu/web/Academic/Undergrad/ The math requirements are: ten
courses numbered V63.0120 or higher from the Mathematics Department, which must include
Analysis I (V63.0325) and Algebra I (V63.0343); Intro to Abstract Algebra (V63.0246) cannot
be counted if Algebra I (V63.0343) is applied toward the major. The remaining math courses
must include one of the following: Analysis II (V63.0326), Algebra II (V63.0344), or Vector
Analysis (V63.0244). A grade of C or better is required in all these courses. Please note that it is
strongly suggested for students to take Analysis I (V63.0325) before enrolling in Algebra I
(V63.0343).
Joint Major in Mathematics and Economics. An interdisciplinary major is offered for studies
jointly by the Departments of Mathematics and Economics, providing the opportunity to study
economics, computer science, and in mathematics, analysis, statistics and operations research.
The mathematics requirements are nine courses consisting of Calculus I, II, III, (V63.0121 - 123),
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Linear Algebra (V63.0140), Analysis I, II (V63.0325, 0326) plus three courses chosen below
Vector Analysis (V63.0224)
Probability & Statistics (V63.0235)
Combinatorics (V63.0240)
Function of Complex Variables (V63.0282)
Theory of Numbers (V63.0248)
Mathematics of Finance (V63.0250)
Theory of Probability (V63.0233)
Transformations & Geometries (V63.0270)
Ordinary Differential Equations (V63.0262)
Partial Differential Equations (V63.0363)
Chaos & Dynamical Systems (V63.0264)
Honors Linear Algebra (V63.0141)
Algebra I, II (V63.0343, 0344)
Numerical Analysis (V63.0252)
Mathematical Statistics (V63.0234)
The economics requirements are 9 courses in Economics including V31.0005, V31.0006,
V31.0011, V31.0013, V31.0020, V31.0266, plus any three economics elective courses, at least
two of which must be theory electives numbered V31.0300 to V31.0399. Please refer to the
undergraduate economics website for more information about the economics courses:
http://econ.as.nyu.edu/page/ugprog. Students in the joint mathematics/economics major may
only take the theory sequence. A grade of C or better is required in all of these courses.
Mathematics Minor. The requirement for a mathematics minor is four courses in the
department numbered V63.0120 or higher. Only two of these courses may apply simultaneously
to the requirements for any other major. At most two mathematics courses in the minor may be
transferred from other colleges. Courses taken under the pass/fail option are not counted toward
the minor. A grade of C or better is required in all courses applied to the minor.
Joint Mathematics and Computer Science Minor. The requirements are the four courses
V63.0121, V63.0122, V22.0101 and V22.0102. A grade of C or better is required. At most one
mathematics course in the joint minor may be transferred from other colleges.
Advanced Mathematical Methods Minor. The Advanced Mathematical Methods Minor (for
Stern undergraduates) provides students with mathematical tools to handle complex business
problems. The requirements are V63.0123, V63.0140, C22.0014, and V63.0252. Students who
have a more advanced mathematics background and have the equivalent of V63.0123or
V63.0140 can substitute one of the following more advanced courses V63.262, V63.263, or
C22.0015.
Mathematics Education Minor for Math Majors. The Mathematics Education Minor (for
Mathematics Majors) enables you to complete coursework as an undergraduate in CAS that will
reduce the number of credits required of a Steinhardt School of Education M.A. degree in
Teachers of Mathematics, Grades 7-12 that will fulfill the requirements for Initial teacher
certification. The master's degree will require 30 credits including student teaching internships
beyond the bachelor's degree.
Ideally you would want to declare the minor and begin taking these courses during your first
semester of junior year. Please be advised that most of these courses require a fieldwork
component consisting of classroom observation and participation. For course descriptions, please
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see the current Steinhardt School of Education's Undergraduate Bulletin at
www.steinhardt.nyu.edu/bulletin.
B.S. /B.E. Program in Engineering. The College of Arts and Science, in cooperation with
Stevens Institute of Technology offers a joint B.S./B.E. program in engineering. Students in the
program receive the B.S. degree in mathematics from New York University and the B.E. degree
from Stevens Institute in either computer, electrical or mechanical engineering. The joint
math/engineering students must complete the 12 course math requirement. Students are allowed
to take Differential Equation, MA221; and/or Complex Variables, MA234 at Stevens Institute,
which are equivalent to V63.0262 and V63.0282, respectively. Students are also allowed to
substitute Computer Science I, II, V22.0101, 0102, or Physics I, II, V85.0091, 0093 for a
maximum of two math classes. Further information is available from Joseph Hemmes, the
coordinator of the B.S./B.E. program, in the College Advising Center, Silver Center, 100
Washington Square East, Room 905, (212) 998-3133.
B.A./M.S. Program in Mathematics. New York University has introduced a program that
allows students to obtain a Mathematics Bachelor’s and Masters degree in five years. The
program is suited to students who have decided not to enter a Ph.D. program at this time but who
would like to increase their skills before entering the job market. Qualifying students are
accepted into a program toward the end of the sophomore year or during their junior year before
they reach 96 credits. Students must earn a minimum of a 3.50 GPA in order to qualify for
acceptance into the joint B.A. /M.A. For more information, please contact advisor Dr. Justin
Lorts the CAS Advising Center at (212) 998-8521. During the last few undergraduate semesters
students should accelerate by taking one quarter of their graduate courses during regular terms
and/or during the summer.
4. Honors Program
The honors program is designed for students with a strong commitment to mathematics. It is
recommended for students who intend to pursue graduate study in mathematics. The course
requirements are: Analysis I, II (V63.0325, 0326) and Algebra I, II (V63.0343, 0344), both
usually taken during the junior year; and Honors, I-II, (V63.0393, 0394), usually taken in the
senior year. Under special circumstances, with the permission of the department, certain graduate
courses may be substituted for Honors I-II. Potential honors students should register for Honors
Calculus I, II, (V63.0221-0222). Students must also complete a senior research project. Students
should register for two semesters of independent study (V63.0997, 0998) under faculty
supervision. Students should seek approval of their research project from the Director of
Undergraduate Studies. The research project can also be completed through the mathematics
summer research program (S.U.R.E.). Students are required to present their research at the
undergraduate research forum in the fall semester of their senior year. The requirements for
admission into the honors program are: (1) maintain a grade point average of 3.65 or higher in
the major sequence (including Honors requirements); (2) maintain a general grade point average
of 3.65 or better, and (3) approval of the director of the honors program. Interested students can
consult with the mathematics Honors Advisor, Professor Eli Hameiri at hameiri@cims.nyu.edu.
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For further general requirements, please see under College of Arts and Sciences Honors and
Awards.
Joint Honors in Mathematics and Computer Sciences This is an interdisciplinary major
offered by the Department of Mathematics and the Department of Computer Sciences. The
mathematics requirements are V63.0120, V63.0121, V63.0122, V63.0123, V63.0140, V63.0325,
V63.0326, V63.0343, V63.344, V63.0393, V63.0394 (V63.0393 and V63.0394 may be
substituted for two graduate classes with math faculty approval). The computer science
requirements are V22.0101, V22.0102, V22.0201, V22.0202, V22.0310, V22.0421, V22.0453,
and three computer science courses listed at the V22.0400 level. Four courses, numbered
V22.0101 to V22.0499, must be completed with HONORS credit, one of which must be
V22.0300-level or above. Students should complete a guided research, sponsored by either
department and be presented at the Dean’s Undergraduate Research Conference which takes
places in late April. Students are expected to dedicate 10-20 hours per week toward their
research.
To be graduated with Latin honors, a student must have completed at least 64 points in courses in
the College with passing grades. All graded courses taken while enrolled in the College, or in
other divisions of the University, will be used to compute the honors average. Pass grades are
not counted, nor are grades received in courses at other institutions. The student must have a
good record of conduct and maintain a minimum general average as follows: cum laude, 3.50;
magna cum laude, 3.70; summa cum laude, 3.90.
5. Transfer Students
Transfer students who have taken mathematics courses at other colleges should consult the
Director of Undergraduate Studies as soon as possible to determine proper placement. They
should bring copies of their transcripts along with a catalog descriptions/syllabus of all
mathematics courses taken elsewhere. Transfer courses that are to be counted toward major or
optional minor requirements must be approved by the department. Official transcripts must also
be sent to the University registrar’s office.
To graduate with a major in mathematics, transfer students must complete at least 12 eligible
mathematics courses, each with grade of C or better. At least half the courses comprising the
major must be taken in the College of Arts and Science at New York University during two or
more semesters. For the minor in mathematics, at least two of the four required mathematics
courses must be taken in residence at the College of Arts and Science at New York University,
and can not be applied towards major or minor requirements in other departments.
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6. Courses Offered in Academic Year 2008 – 2009
FALL
Discrete Math
Calculus I
Calculus II
Calculus III
Linear Algebra
Intensive Linear Algebra I
Theory of Probability
Theory of Numbers
Mathematics of Finance
Math in Medicine and
Biology
Ordinary Differential
Equations
Transformations and
Geometry
Analysis I
Algebra I
Topology
Special Topics I
Independent Study
V63.0120
V63.0121
V63.0122
V63.0123
V63.0140
V63.0141
V63.0233
V63.0248
V63.0250
V63.0255
SPRING
Discrete Math
Calculus I
Calculus II
Calculus III
Linear Algebra
Intensive Linear Algebra II
Vector Analysis
Mathematical Statistics
Probability & Statistics
Combinatorics
V63.0120
V63.0121
V63.0122
V63.0123
V63.0140
V63.0142
V63.0224
V63.0234
V63.0235
V63.0240
V63.0262 Intro to Abstract Algebra
V63.0246
V63.0270 Intro to Math Modeling
V63.0251
V63.0325 Numerical Analysis
V63.0343 Computers in Medicine and
Biology
V63.0375 Ordinary Differential Equations
V63.0393 Partial Differential Equations
V63.0997 Chaos and Dynamical Systems
Functions of a Complex Variable
Analysis I
Analysis II
Algebra I
Algebra II
Special Topics II
Special Topics in Fourier
Analysis
Independent Study
V63.0252
V63.0256
V63.0262
V63.0263
V63.0264
V63.0282
V63.0325
V63.0326
V63.0343
V63.0344
V63.0394
V63.0395
V63.0998
NOTES:
• Combinatorics (V63.0240) is offered every spring.
• Transformations and Geometry is offered every fall.
• Special Topics (V63.0395, 96) Topics and prerequisites are announced and posted in advance.
• Honors (V63.0393, 94) Topics are announced and posted in advance.
• The math department rarely offers Logic courses due to low enrollment. However, if students
wish to take a Logic course offered by the Philosophy Department, they should consult the
Director of Undergraduate Studies to seek approval for the course to count towards the math
major.
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7. Suggested Course Programs
The programs shown below are suggestions only. Many students will take more mathematics
electives than are listed in these programs.
A. Possible program for math majors:
YEAR
FALL
First Year
V63.0121 (Calculus I)
Second Year V63.0123 (Calculus III)
Third Year
Fourth Year
V63.0325 (Analysis I)
Math Elective
V63.0343 (Algebra I)
Math Elective
SPRING
V63.0122 (Calculus II)
V63.0140 (Linear Algebra)
Math Elective
V63.0326 (Analysis II) or V63.0343 (Algebra I)
Math Elective
Math Elective
V63.0326 (Analysis II) or (V63.0344) Algebra II
NOTE: By the end of the seventh semester students should complete Analysis I, Algebra I, and
one or more proofs courses.
B. Possible program for math majors who intend to go into secondary school education:
YEAR
First Year
Second Year
Third Year
Fourth Year
FALL
SPRING
V63.0121 (Calculus I)
V63.0122 (Calculus II)
V63.0123 (Calculus III)
V63.0140 (Linear Algebra)
V63.0233 (Probability)
Math Elective
Math Elective
Math Elective
V63.0270 (Transform & Geom.) Math Elective
V63.0325 (Analysis I)
Math Elective
Recommended electives are V63.0120 (Discrete Mathematics), V63.0255 (Mathematics in
Medicine and Biology), V63.0248 (Theory of Numbers), V63.0264 (Chaos & Dynamical
Systems), V63.0262 (Ordinary Differential Equations), and V63.0282 (Function of Complex
Variables).
C. Possible Honors Program. (Especially recommended for students who intend to go to
graduate school for advanced work in mathematics).
YEAR
First Year
Second Year
Third Year
Fourth Year
FALL
V63.0221 (Honor Cal. I)
V63.0140 (Linear Algebra)
V63.0325 (Analysis I)
V63.0343 (Algebra I)
V63.0393 (Honors I)
Elective
SPRING
V63.0222 (Honor Cal. II)
V63.0262 (Ordinary Diff. Equations)
V63.0326 (Analysis II)
V63.0344 (Algebra II)
V63.0394 (Honors II)
Elective
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Students with advanced standing should begin their freshman year at the appropriate level.
Highly qualified students are encouraged to take graduate mathematics courses provided they
satisfy the prerequisites and obtain permission from both undergraduate and graduate
departmental advisors.
8. Activities
Mathematics Club. There is an active math club open to all students interested in the study of
mathematics. An organizational meeting is held shortly after classes begin in the fall to plan for
the coming academic year. Activities include talks by faculty and guest speakers on a variety of
topics including mathematics and career opportunities as well as attending conferences. The club
is under the supervision of Professor Adi Rangan.
William Lowell Putnam Competition. The mathematics Department participates in the annual
William Lowell Putnam Competition, a mathematics competition open to all undergraduate
mathematics students in the U.S.A. and Canada. Interested students should contact the
department as early as possible in the school year since the contest takes place in early
December. A series of preparation sessions is held under the supervision of math faculty.
Peer Mentor Program. The math department has an active peer mentor program for math
majors. The program is designed to assist new math majors in making the transition to the math
major and life at NYU. If you’re interested in becoming a mentor or mentee please send an email
to Jackie Klein at jklein@cims.nyu.edu.
9. Awards
Prizes. Each year the Sidney Goldwater Roth Prize is presented to the graduating senior who has
been of greatest service to the department and his or her fellow students, and who shows the
greatest promise in mathematics.
Four awards, two senior and two junior, are awarded for excellence in mathematics and/or for
service to the department and fellow students.
The Hollis Cooley Memorial Prize is awarded for excellence and exceptional promise in
mathematics.
The Perley Lenwood Thorne Medal, endowed by the faculty to honor Professor Thorne at the
time of his retirement in 1949, is awarded for outstanding scholarship in mathematics.
Scholarships and Grants. New York University sponsors and administers a wide variety of
financial aid programs, including its own scholarship and grant funds and some New York State
and Federal funds. For additional information, see the College of Arts and Science Bulletin.
11
10. Work Opportunities in the Mathematics Department
Tutoring. The department offers paid undergraduate tutoring positions for advanced
mathematics majors. Tutors provide free tutoring for undergraduate students taking lower level
mathematics courses. Preferred applicants have completed and received at least a B+ in Calculus
I, II, III and Linear Algebra, and Analysis I. Interested students should complete an application in
room 705 Warren Weaver Hall.
Grading. Paid grader positions are available for advanced mathematics majors. Students grade
homework problems for undergraduate mathematics courses. Preferred applicants have
completed and received at least a B in Calculus I, II, III, Linear Algebra, and Discrete
Mathematics. Interested students should complete an application in room 705 Warren Weaver
Hall.
11. Courses Descriptions
Listed below are descriptions of all mathematics courses that satisfy the major and minor
requirements. Some of the courses are given only in the fall, others only in the spring, a few are
given once every other year, and some only on request. However, any course may be scheduled
if there is sufficient demand.
Discrete Mathematics V63.0120 (offered each term)
Prerequisite: Passing V63.0121 with a grade of C or better, or permission of the department. 4
points. A first course in discrete mathematics. Sets, algorithms, induction. Combinatorics.
Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.
Calculus Tracks: Two tracks are currently available: The standard track (V63.0121 - 0123) and
the honor track V63.0221 -0222). The honors track assumes that the student knows the material
from Calculus I since the track covers Calculus II and III. The courses V63.0221 - 0222 count as
the equivalent of two mathematics courses. For more information about Honors Calculus please
visit the Honors Calculus website: http://math.nyu.edu/degree/undergrad/honors_calculus.html.
While it is possible in some circumstance to switch tracks, it is neither advised nor encouraged.
Thus, a student who intends to take the full calculus sequence should be prepared to continue on
the same track to complete the sequence. Hand calculators are optional in all sections.
Calculus I V63.0121 (offered each term)
Prerequisite: Passing Departmental placement exam, Advanced Placements scores: AB 4 or
better or BC 3 or better, SAT math score of 750 or ACT math score of 34 or Algebra & Calculus
(V63.0009) with a grade of C or better, or passing the calculus placement exam. 4 points.
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.
Calculus II V63.0122 (offered each term)
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Prerequisite: Passing V63.0121 with a grade of C or better, or the equivalent. 4 points.
Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and
parametric equations. Infinite series, including power series.
Calculus III V63.0123 (offered each term).
Prerequisite: Passing V63.0122 with a grade of C or better, or the equivalent. 4 points.
Functions of several optimization and 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.
Linear Algebra V63.0140 (offered each term)
Prerequisite: Passing V63.0121 with a grade of C or better, or the equivalent. 4 points. Systems
of linear equations, Gaussian elimination, matrices, determinants, Cramer’s rule. Vectors, vector
spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic
forms.
Honors Linear Algebra I V63.0141. Identical to G63.2110. (offered fall term) NOTE: The
graduate course is offered in spring under a different name.
Prerequisite: A grade of B or better in V63.0325 and/or V63.0343 or the equivalent. 4 points.
Linear spaces, subspaces, and quotient spaces; linear dependence and independence; basis and
dimensions. Linear transformation and matrices; dual spaces and transposition. Solving linear
equations. Theory of Determinants. Quadratic forms and their relation to local extrema of
multivariable functions.
Honors Linear Algebra II V63.0142 Identical to G63.2120. (offered spring term)
Prerequisite: V63.0141. 4 points. Spectral theory, eigenvalues and eigenvectors; nilpotent
operations and Jordan canonical form. Inner products orthogonality, theory of bilinear forms.
Self-adjoint mappings, matrix inequalities, polar decomposition. Nomed linear spaces and the
exponential map. Implicit function theorem and introduction to Lie groups.
Calculus for Biology and the Life Sciences I V63.0143. Identical to V23.0101 (offered fall
term). Prerequisite: V63.0009. 4 points. Equivalent to V63.0121, this course is for calculus
students who plan to enter the medical or life sciences. Examples and problems are drawn from
a wide selection of topics in biology, including physiology, ecology, genetics, bioinformatics,
probability, biostatistics, enzymology and neurophysiology.
Honors Calculus I, II V63.0221-0222 (offered fall and spring terms, respectively)
Prerequisite for Honors Calculus I: A score of 5 on the Advanced Placement Calculus AB exam
or 3 on BC, a B+ or better in Calculus I (V63.0121), placement exam, or permission of the
instructor. Prerequisite for Honors Calculus II: Honors I or permission of the instructor. 5 points
each term. Covers the same material as V63.0122, .0123, but at a faster pace. Appropriate for
science, mathematics, and computer science majors. V63.0221 covers differential and integral
calculus of one variable, with applications, and the elementary transcendental functions.
V63.0222 includes techniques of integration, infinite series, and the calculus several variables
with applications.
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Vector Analysis V63.0224 Identical to G63.1002. (offered spring term)
Prerequisite: Passing V63.0122, V63.0123 and V63.0140 with a grade of C+ or better. 4 points.
Functions of several variables. Partial derivatives, chain rule, change of variables. Lagrange
multipliers. Inverse and implicit function theorems. Divergence, gradient and curl; Theorems of
Gauss, Green, and Stokes with applications to fluids, gravity, electromagnetism and the like.
Introduction to differential forms. Degree and fixed points of mappings with applications.
Additional topics depending on the interests of the class, as time permits.
Theory of Probability V63.0233 (offered fall term)
Prerequisite: V63.0123 with a grade of C or better and/or the equivalent. 4 points. 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.
Mathematical Statistics V63.0234 (offered spring term)
Prerequisite: V63.0233 with a grade of C or better and/or the equivalent. Not open to students
who have taken V63.0235. 4 points. 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.
Probability and Statistics V63.0235 (offered spring term)
Prerequisite: V63.0122 with a grade of C or better and/or the equivalent. Not open to math
majors and/or students who have taken V63.0233. 4 points. A combination of V63.0233 and
V63.0234 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.
Combinatorics V63.0240 (offered spring of even year)
Prerequisite: V63.0122 with a grade of C or better and/or the equivalent. 4 points. Techniques
for counting and enumeration including generating functions, the principle of inclusion and
exclusion, and Polyacounting. Graph theory. Modern algorithms and data structures for graphtheoretic problems.
Introduction to Cryptography V63.0243 Identical to V22.0480 (offered spring term)
Prerequisite: V22.0310 with a grade of C or better or permission of the instructor. 4 points.
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,
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EIGamal); digital signatures (RSA, Fiat-Shamir); authentication applications (identification,
zero-knowledge) and others time permitting.
Logic V63.0245 (offered in Philosophy department)
Prerequisite: V63.0122 with a grade of C or better and/or the equivalent. 4 points. Propositional
calculus, quantification theory, properties of axiomatic systems, Henkin’s theorem. Introduction
to set theory, interpretations, models. Computability and its applications to the incompleteness
theorem.
Abstract Algebra V63.0246 (offered spring term)
Prerequisite: V63.0122 and V63.0140 with a grade of C or better. 4 points. 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, V63.0243, 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.
Theory of Numbers V63.0248 (offered fall term)
Prerequisite: V63.0122 with a grade of C or better and/or the equivalent. 4 points. Divisibility
theory and prime numbers. Linear and quadratic congruences. The classical number-theoretic
functions. Continued fractions. Diophantine equations. RSA cryptography algorithim.
Mathematics of Finance V63.0250 (offered fall term)
Prerequisite: V63.0123 and one of the followings V63.0233/V63.235/V31.0018/V31.0120
(Theory of Probability, Probability & Statistics, Statistics or Analytical Statistics) with a grade of
C or better and/or the permission of the instructor. 4 points. Introduction to the mathematics of
finance. Topics include: Linear programming with application. 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.
Introduction to Mathematical Modeling V63.0251 (offered spring term)
Prerequisites: V63.0123 with a grade of C or better or permission of the instructor. 4 points.
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.
Numerical Analysis V63.0252 (offered spring term)
Prerequisites: V63.0123 and V63.0140 with a grade of C or better or permission of the instructor.
4 points. In numerical analysis one explores how mathematical problems can analyzed and
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solved with a computer. As such, numerical analysis has very broad applications in mathematics,
physics, engineering, finance, and the life sciences. This course introduces the subject for
mathematics majors. Theory and practical examples using Matlab are combined in the studying
of topics ranging from simple root-finding procedures to differential equations and the finite
element method.
Mathematics in Medicine and Biology V63.0255 Identical to G23.1501. (offered fall term)
Prerequisite: V63.0121 and V23.0011 or permission of the instructor. 4 points. 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.
Computers in Medicine and Biology V63.0256 Identical to G23.1502 (offered spring term)
Prerequisites: V63.0255 or permission of the instructor. Familiarity with a programming
language such as Pascal, FORTRAN, or BASIC is recommended. 4 points. Introduces the
students of biology or mathematics to the use of the computer as a tool for modeling
physiological phenomena. Each student constructs two computer models selected from the
following list: circulation, gas exchange in the lung, control of cell volume, and the renal
countercurrent mechanism. The student uses the models to conduct simulated physiological
experiments.
Ordinary Differential Equations V63.0262 (offered each term)
Prerequisites: V63.0123 and V63.0140 with a grade of C or better or the equivalent. 4 points.
First and second order equations. Series solutions. Laplace transforms. Introduction to partial
differential equations and Fourier series.
Partial Differential Equations V63.0263 (offered spring term)
Prerequisite: V63.0262 with a grade of C or better or the equivalent. 4 points. 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.
Chaos and Dynamical Systems V63.0264 (offered fall term)
Prerequisite: V63.0121 with a grade of B or better or the equivalent. 4 points. Topics to include
fixed points of one dimensional maps, linear operators and linear approximations; stability and
bifurcation; logistic maps, Cantor sets, fractal sets, symbolic dynamics, conjugacy of maps.
Introduction to dynamics in two dimensions. Introduction for students with little preparation to
the recent discovery that , in certain regimes, fully deterministic mechanics can produce chaotic
behavior.
Transformations and Geometries V63.0270 (offered fall term)
Prerequisite: V63.0123 with a grade of C or better or the equivalent. 4 points. An axiomatic and
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algebraic study of Euclidean, non-Euclidean, affine, and conformal and projective geometries.
Special attention to group theoretic methods and the use of complex variables.
Functions of Complex Variable V63.0282 (offered spring term)
Prerequisites: V63.0123 and V63.0140 plus one higher level course with the grade of C or better.
4 points. 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.
Analysis I V63.0325 (offered each term)
Prerequisite: V63.0122, V63.0123 and V63.0140 or the equivalent. 4 points. 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.
Analysis II V63.0326 (offered spring term)
Prerequisite: V63.0325 or permission of the department. 4 points. Functions of several
variables. Limits and continuity. Partial derivatives. The implicit function theorem.
Transformation of multiple integrals. The Riemann integral and its extensions.
Algebra I V63.0343 (offered each term)
Prerequisites: V63.0123 and V63.0140 with a grade of C or better and/or the equivalent. It is
strongly recommended to take V63.0325 before registering for this course. 4 points. Groups,
homomorphisms, automorphisms, permutation groups. Rings, ideals and quotient rings,
Euclidean rings, polynomial rings.
Algebra II V63.0344 (offered spring term)
Prerequisite: V63.0343 with a grade of C or better. 4 points. Extension fields, roots of
polynomials. Construction with straight-edge and compass. Elements of Galois theory.
Topology V63.0375 (offered fall term)
Prerequisite: V63.0325 or permission of the department. 4 points. Set-theoretic preliminaries.
Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy
groups.
Differential Geometry V63.0377 (offered spring upon request)
Prerequisite: V63.0326 or permission of the department. 4 points. The differential properties of
curves and surfaces. Introduction to differential manifolds and Riemannian geometry.
Honors I-II V63.0393, 0394. Topic varies from year to year. In 2007-8 it is identical to
G63.2450/G63.2460 (offered each term)
Prerequisite: Approval of the director of the honors program. 4 points each term. Complex
numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations;
construction and geometry of the elementary functions; Green’s theorem, Cauchy’s theorem;
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Jordan curve theorem, Cauchy’s formula; Taylor’s theorem, Laurent expansion; analytic
continuation; isolated singularities, Liouville’s theorem; Abel’s convergence theorem an the
Poisson integral formula.
Special Topics I –II V63.0395, 0396 (offered upon request)
Prerequisite: Permission of the department. 4 points each term. Covers topics not offered
regularly; experimental courses and courses offered on student demand. Detailed course
descriptions are available during preregistration.
Independent Study V63.0997, 0998 (offered each term)
Prerequisite: Permission of the department. 2 or 4 points each term. 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.
12. Graduate Courses Open to Undergraduates
Qualified students may take the courses listed below in the Graduate School of Arts and Science
provided they first obtain permission from both undergraduate and graduate departmental
advisors. If these courses are offered toward fulfillment of the requirements for the baccalaureate
degree, no advanced credit is allowed for them in the graduate school. These courses are all
three credit courses.
Numerical Methods G63.2010, 2020
Linear Algebra G63.2111 (for students who have not taken G63.0142)
Scientific Computing G63.2043
Algebra G63.2130, 2140
Number Theory G63.2210, 2220
Topology G63.2310, 2320
Real Variables G63.2430, 2440
Ordinary Differential Equations G63.2470
Introduction to Applied mathematics G63.2701, 2702
Game Theory, Linear Programming G63.2731, 2742
Mathematical Topics in Biology G63.2859, 2851
Basic Probability G63.2911, 2912
Differential Geometry I-II G63.2350-2360
13. Departmental Faculty
Marco M. Avellaneda. Professor. Licenciado en Ciencias 1981, Buenos Aires; Ph.D. 1985,
Minnesota. Research interests: applied mathematics, mathematical modeling in finance,
probability.
Gerard Ben Arous. Professor. Ph.D. 1981, University of Paris. Research interests: probability
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theory, stochastic processes, partial differential equations.
Simeon M. Berman. Professor. B.A. 1956, City College; M.A. 1958, Ph.D. 1961, Columbia.
Research interests: stochastic processes, probability theory, applications.
Fedor A. Bogomolov. Professor. Diploma 1970, Moscow University; Ph.D. 1974, Steklov
Institute of Mathematics. Research interests: algebraic geometry and related problems in
algebra, topology, number theory.
Oliver Bühler. Associate Professor. Ph.D. 1996, Cambridge University. Research interests:
geophysical fluid dynamics, interactions between waves and vortices, acoustics, statistical
mechanics.
David Cai. Professor. B.S. 1984, Peking University; M.S. 1989, Ph.D. 1994, Northwestern.
Research interests: nonlinear stochastic behavior in physical and biological systems.
Sylvain E. Cappell. Professor. B.A. 1966, Columbia; Ph.D. 1969, Princeton. Research interests:
algebraic and geometric topology, symplectic and algebraic geometry.
Jeff Cheeger. Professor. B.A. 1964, Harvard College; M.S. 1966, Ph.D. 1967, Princeton.
Research interests: differential geometry and its connections to analysis and topology.
Yu Chen. Associate Professor. B.S. 1982, Tsinghua University; M.S. 1988, Ph.D. 1991, Yale.
Research interests: numerical scattering theory, ill-posed problems, scientific computing.
Francesca Chiaromonte. Associate Professor. Laurea 1990, University of La Sapienza; Ph.D.
1996, University of Minnesota. Research interests: multivariate analysis and regression, Markov
modeling, analysis and modeling of large-scale genomic data.
Percy A. Deift. Professor. B.S. 1967, M.S. 1970, Natal, Durban; M.S. 1971, Rhodes South
Africa; Ph.D. 1976, Princeton. Research interests: spectral theory and inverse spectral theory,
integrable systems, Riemann-Hilbert problems.
Paul R. Garabedian. Professor. B.A. 1946, Brown; M.A. 1947, Ph.D. 1948, Harvard. Research
interests: complex analysis, computational fluid dynamics, plasma physics.
Jonathan B. Goodman. Professor. B.S. 1977, Massachusetts Institute of Technology; Ph.D.
1982, Stanford. Research interests: fluid dynamics, computational physics, computational
finance.
Leslie Greengard. Professor. B.A. 1979, Wesleyan; M.D. 1987, Ph.D. 1987, Yale. Research
interests: applied and computational math, partial differential equations, computational
chemistry, mathematical biology.
Frederick P. Greenleaf. Professor. B.S. 1955 Pennsylvania State; M.A. 1961, Ph.D. 1964, Yale.
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Research interests: noncommutative harmonic analysis, Lie groups and group representations,
invariant partial differential operators.
Mikhael Gromov. Professor. Maitrise 1965, Doctorat 3e Cycle 1969, D.Sc. 1973, University of
Leningrad. Research interests: Riemannian manifolds, symplectic manifolds, infinite groups,
math models of biomolecular systems.
Sinan Gunturk. Associate Professor. B.S. 1996, Bogazici University; Ph.D. 2000, Princeton.
Research interests: harmonic analysis, information theory, signal processing.
Eliezer Hameiri. Professor. B.A. 1970, M.A. 1972, Tel Aviv; Ph.D. 1976, New York. Research
interests: applied mathematics, magnetohydrodynamics, plasma physics.
Fengbo Hang. Associate Professor. B.S. 1993, Tsinghua University; M. S. 1996, Peking
University; Ph.D. 2001, New York University. Research interests: Geometric analysis and
nonlinear partial differential equations.
Helmut Hofer. Professor. Diplom 1979, Ph.D. 1981, University of Zurich. Research interests:
symplectic geometry, dynamical systems, partial differential equations.
David Holland. Professor. B.S. 1984, B.A. 1993, M.S. 1986, Memorial University
(Newfoundland); Ph.D. 1993, McGill. Research interests: ocean-ice studies, climate theory and
modeling.
Richard Kleeman. Professor. B.S. 1980, Australian National University, Ph.D. 1986, Adelaide
University. Research interests: climate dynamics, El Nino, predictability of weather and climate
dynamical systems.
Bruce Kleiner Professor. Research interests: Geometric analysis, geometric group theory and
geometric evolution equations.
Robert V. Kohn. Professor. A.B. 1974, Harvard; M.S. 1975, Warwick (England); Ph.D. 1979,
Princeton. Research interests: nonlinear partial differential equations, materials science,
mathematical finance.
Matthew Leingang. Clinical Associate Professor. Ph.D. 2000, Harvard; Research interests:
Mathematics Education, Web Pedagogies, differential geometry.
Fang-Hua Lin. Professor. B.S. 1981, Zhejing; Ph.D. 1985, Minnesota. Research interests:
partial differential equations, geometric measure theory.
Andrew Majda. Professor. B.S. 1970, M.S. 1971, Ph.D. 1973, Stanford. Research interests:
modern applied mathematics, atmosphere/ocean science, partial differential equations.
Nader Masmoudi. Professor. Ph.D. 1999, University of Paris Dauphine. Research interests:
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nonlinear parallel differential equations.
Henry P. McKean. Professor. B.A. 1952, Dartmouth; Ph.D. 1955, Princeton. Research
interests: probability, partial differential equations, complex function theory.
David W. McLaughlin. Professor. B.S. 1966, Creighton. M. S. 1969, Ph.D. 1971, Indiana
University. Research interests: applied mathematics, nonlinear wave equations, visual neural
science.
Assaf Naor. Associate Professor. B. S. 1996, M.S. 1998, Ph.D. 2002, Hebrew University.
Research interests: analysis, probability; applications to combinatorics, mathematical physics,
and theoretical computer science.
Charles M. Newman. Professor. B.S. 1966, Massachusetts Institute of Technology; M.A. 1968,
Ph.D. 1971, Princeton. Research interests: probability theory, statistical physics, stochastic
models.
Albert B. J. Novikoff. Professor. B.A. 1946, Brown; Ph.D. 1954, Stanford. Research interests:
analysis, history of mathematics, pedagogy.
Olivier Pauluis. Assistant Professor. B.S./M.S. 1995, Université Catholique de Louvain; Ph.D.
2000, Princeton. Research interests: climate and the general circulation of the atmosphere,
moist convection, tropical meteorology, numerical modeling.
Jerome K. Percus. Professor. B.S. 1947, M.A. 1948, Ph.D. 1954, Columbia. Research
interests: chemical physics, mathematical biology.
Charles S. Peskin. Professor. B.A. 1968, Harvard; Ph.D. 1972, Yeshiva. Research interests:
applications of mathematics and computing to problems in medicine and biology, cardiac fluid
dynamics, molecular machinery within biological cells, mathematical/computational
neuroscience.
Aaditya Rangan. Assistant Professor. B.A. 1999, Dartmouth; Ph.D. 2003 California (Berkley)
Research interests: Large-scale scientific modeling of physical biological, and neurobiological
phenomena.
Weiqing Ren. Associate Professor. B.S., 1994, University of Nanjing; Ph.D. 2002, NYU.
Research interests: applied mathematics, scientific computing, multiscale modeling of fluids.
John Rinzel. Professor. B.S. 1967, University of Florida; M.S. 1968, Ph.D. 1973, New York.
Research interests: computational neuroscience, nonlinear dynamics of neurons and neural
circuits, sensory processing.
Sylvia Serfaty. Professor. M.S. 1995, Ecole Normale Superieure; Ph.D. 1999, University of
Paris-Orsay. Research interests: partial differential equations, nonlinear analysis applied to
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physics.
Jalal M. I. Shatah. Professor. B.S. 1979, Texas (Austin); Ph.D. 1983, Brown. Research
interests: partial differential equations, analysis.
Scott Sheffield. Associate Professor. A. B., A. M. 1998 Harvard; Ph.D. 2003 Stanford.
Research interests: Probability and mathematical physics.
Michael J. Shelley. Professor. B.A. 1981, Colorado; M.S. 1984, Ph.D. 1985, Arizona. Research
interests: applied math and modeling, visual neuroscience, fluid dynamics, computational
physics and neuroscience.
K. Shafer Smith. Associate Professor. B.S. 1992, Indiana; Ph.D. 1999, UC Santa Cruz.
Research interests: geophysical fluid dynamics, physical oceanography and climate.
Joel H. Spencer. Professor. B.S. 1965, Massachusetts Institute of Technology; Ph.D. 1970,
Harvard. Research interests: discrete mathematics, theoretical computer science.
Daniel Stein. Professor. Sc.B. 1975, Brown University; M.A. 1977, Ph.D. 1979, Princeton
University. Research interests: theoretical condensed matter physics, statistical mechanics, and
mathematical physics.
Esteban G. Tabak. Professor. Bach. 1988, University of Buenos Aires; Ph.D. 1992,
Massachusetts Institute of Technology. Research interests: dynamics of the atmosphere and
ocean, energy transfer in systems with many degrees of freedom.
Anna-Karin Tornberg. Associate Professor. Ph.D. in Numerical Analysis, 2000, The Royal
Institute of Technology (KTH), Stockholm, Sweden. Research interests: numerical analysis;
computational fluid dynamics; moving boundary problems.
Daniel A. Tranchina. Professor. B.A. 1975, SUNY (Binghamton); Ph.D. 1981, Rockefeller.
Research interests: mathematical modeling in neuroscience.
Yuri Tschinkel. Professor. M.A. 1990, Moscow State University; Ph.D. 1992, Massachusetts
Institute of Technology. Research interests: algebraic geometry, number theory, automorphic
forms.
Eric Vanden-Eijnden. Professor. B. S., M. S., 1992, Ph.D. 1997, Universite Libre de Bruxelles.
Research interests: stochastic partial differential equations, statistical mechanics, turbulence
theory.
S. R. Srinivasa Varadhan. Professor. B.S. 1959, M.A. 1960, Madras; Ph.D. 1963, Indian
Statistical Institute. Research interests: probability theory, stochastic processes, partial
differential equations.
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Akshay Venkatesh. Professor. B.Sc. 1997 University of Western Australia; Ph.D. 2002,
Princeton University. Research interests: analytic number theory, algebraic geometry.
Harold Weitzner. Professor. B.A. 1954, California; M.A. 1955, Ph.D. 1958, Harvard. Research
interests: plasma physics, fluid dynamics, differential equations.
Olof B. Widlund. Professor. M.S. 1960, Ph.D. 1964, Royal Institute of Technology
(Stockholm); D. Phil. 1966, Upsala. Research interests: numerical analysis, partial differential
equations, parallel computing.
Lai-Sang Young. Professor. B.A. 1973, Wisconsin; M.A. 1976, Ph.D. 1978, California
(Berkeley). Research interests: dynamical systems and ergodic theory.
Jun Zhang. Associate Professor. B.S. 1985, Wuhan University (China); Ph.D. 1994, University
of Copenhagen. Research interests: fluid dynamics, biophysics, complex systems.
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