Mathematics

S = Objectives

**Undergraduate Concentration**

As our society becomes more
technological, it is more affected by mathematics. Quite sophisticated
mathematics is now central to the natural sciences, to ecological
issues, to economics, and to our commercial and technical life.
A student who takes such general level courses as Math 5, 8, 10,
15, or 20 will better understand the world and be prepared to
act in it.

Mathematics is, at the same
time, a subject of the greatest depth and beauty with a history
extending from antiquity. The department attempts to make this
depth and beauty manifest. The undergraduate major program introduces
students to some fundamental fields--algebra, real and complex
analysis, geometry, and topology--and to the habit of mathematical
thought. Mathematics concentrators may go on to graduate school,
scientific research, or mathematics teaching, but many choose
the major for its inherent interest with unrelated career intentions.

**Graduate Program in Mathematics**

The graduate program in mathematics is designed primarily to lead to the Doctor of Philosophy degree. The formal course work gives the student a broad foundation for work in modern pure mathematics. An essential part of the program consists of seminars on a variety of topics of current interest in which mathematicians from Greater Boston often participate. In addition, the Brandeis-Harvard-MIT-Northeastern Mathematics Colloquium gives the student an opportunity to hear the current work of eminent mathematicians from all over the world.

S = How to Become an Undergraduate Concentrator

Students who enjoy mathematics are urged to consider concentrating in it; Brandeis offers a wide variety of mathematics courses, and concentrators will have the benefits of small classes and individual faculty attention. To become a concentrator a student should have completed either MATH 15 and 20, MATH 21a, b, or MATH 22a, b by the end of the sophomore year--these courses are prerequisites to the higher level offerings. Therefore, it is important for students to start calculus and linear algebra (MATH 10, 15, 20, 21, or 22) in the first year.

G = How to Be Admitted to the Graduate Program

The general requirements for admission to graduate work in mathematics are the same as those for the Graduate School as a whole. The department has available a variety of fellowships and scholarships for well-qualified students. To be considered for such financial support the student should submit an application by February 1.

S = Faculty

**Ira Gessel, Chair**

Combinatorics. Computer science.

**Mark Adler**

Analysis. Differential equations.
Completely integrable systems.

**David Buchsbaum**

Commutative algebra. Homological
algebra. Representation theory.

**David Eisenbud, Graduate
Advising Head**

Commutative algebra. Algebraic
geometry.

**Markus Hunziker**

Lie groups. Algebraic geometry.

**Kiyoshi Igusa**

Algebraic K-theory.

**Jerome Levine**

Differential topology. Knot
theory and related algebra.

**Bong Lian, Undergraduate
Advising Head**

Representation theory.

**Alan Mayer**

Classical algebraic geometry
and related topics in mathematical physics.

**Paul Monsky**

Number theory. Arithmetic algebraic
geometry.

**Susan Parker, Elementary
Mathematics Coordinator**

Combinatorics. Elementary mathematics
instruction.

**Daniel Ruberman, Undergraduate
Administrator**

Geometric topology, gauge theory,
and low dimensional manifolds.

**Gerald Schwarz**

Algebraic groups. Transformation
groups.

**Pierre Van Moerbeke**

Stochastic processes. Korteweg-deVries
equation. Toda lattices.

**Kari Vilonen**

Topology. Representation theory.

S = Requirements for the Undergraduate Concentration

**A.**
MATH 21a, 22a, or 15a; MATH 21b, 22b, or 20a.

**B.**
MATH 35a, 40a, or 45a.

**C.**
MATH 28a or 30a.

**D.**
Four additional semester courses, either MATH courses numbered
27 or higher or cross-listed courses.

**Honors**

A degree with honors requires
items A, B, and C above as well as:

**D.**
Six additional semester courses, either MATH courses numbered
27 or higher or cross-listed courses, passed with at least a grade
of B. At least four of the courses used to satisfy the concentration
requirement must be honors courses. The honors courses are MATH
30a, 30b, 32a, 34a, 38b, 40a, 40b, 45a, and all MATH courses numbered
100 or higher.

**Teacher Preparation Track**

Students who complete the Brandeis
program for Massachusetts High School Teacher Certification (see
section on Education Program in this *Bulletin*) may earn
a bachelor's degree in mathematics by satisfying concentration
requirements A, B, and C above and the following:

**D.**
MATH 8a or 36a.

**E.**
Two additional courses, either MATH courses numbered 27 or higher
or cross-listed courses.

**F.**
A computer science course numbered 10 or higher.

**G.**
Completion of the High School Teacher Certification Program.

**Combined B.A./M.S. Program**

Undergraduate students are eligible for the B.A./M.S. program in mathematics if they have completed MATH 101a,b; 110a, 111a,b and 121 a,b with a grade of B- or better, and demonstrated a reading knowledge of mathematical French, German, or Russian. No more than three of these courses, however, may be counted towards the concentration. In addition, students must fulfill a minimum of three years' residence on campus. A student must make formal written application for admission to this program on forms available at the Graduate School office. This must be done no later than May 1 preceding his/her final year of study on campus.

S = Requirements for the Undergraduate Minor

**A.**
MATH 21a, 22a or 15a; MATH 21b, 22b, or 20a.

**B.**
Three additional semester courses, either MATH courses numbered
27 or higher or cross-listed courses.

Students interested in analysis, physics, or applied mathematics are advised to choose additional courses from among MATH 35a, 35b, 36a, 36b, 37a, and 45a. Students interested in algebra or computer science are advised to consider MATH 28a, 28b, 30a, 30b, and 38b. With permission of the Undergraduate Mathematics Advisor, courses taken in other Brandeis departments or taken at other universities may be substituted for mathematics courses required for the minor.

S = Special Notes Relating to Undergraduates

**A.**
With permission of the Undergraduate Mathematics Advisor, courses
taken in other Brandeis departments or taken at other universities
may be substituted for required mathematics courses. A grade of
C or better is required in courses satisfying the field of concentration
requirements.

**B.**
Students who intend to take mathematics courses numbered 10 or
higher should take the departmental placement exam. On the basis
of the exam, recommendations are made placing students out of
the first year of calculus or into MATH 5a, 10a, or 10b. Students
receiving a score of 5 on the advanced placement MATH AB exam
or a score of 4 or more on the MATH BC exam place out of the first-year
calculus sequence. Students receiving a score of 4 on the MATH
AB exam or a score of 3 on the MATH BC exam place out of first
semester calculus. Such students must take the departmental placement
exam if they wish to place out of second semester calculus. Questions
about placement should be directed to the Elementary Mathematics
Coordinator, or the undergraduate advising head.

**C.**
The usual calculus sequence is MATH 10a, 10b, and 20a. Students
may precede this with MATH 5a. Many students also take MATH 15a
(Applied Linear Algebra), which has MATH 5a (or placement out
of MATH 5a) as a prerequisite.

**D.**
A student may not receive credit for more than one of MATH 15a,
21a, and 22a; or MATH 20a, 21b, and 22b. Students with a strong
interest in mathematics and science are encouraged to take MATH
21a,b or 22a,b in place of MATH 15a and 20a. Similarly, a student
may not receive credit for both MATH 28a and 30a, or for both
MATH 28b and 30b.

**E.**
Students interested in graduate school or a more intensive study
of mathematics are urged to include all of the following courses
in their program:

1. MATH 21a and b or 22a and
b.

2. MATH 30a and b.

3. MATH 35a and b or 40a and
b.

4. MATH 45a.

5. A course numbered 100 or
higher.

**F.**
The following schedule determines course offerings in mathematics:

1. Offered every semester are
MATH 5a, 10a and b, 15a, 20a.

2. Offered once each year are
MATH 8a, 21a and 21b, 28a and b, 30a and b, 35a and b, 36a and
b, 40a and b, 45a.

3. In addition, the following
semester courses are usually offered according to the following
schedule where 0-1 indicates even-odd years (e.g., 1996-97) and
1-0 indicates odd-even years (e.g., 1997-98). Slashes distinguish
between fall and spring semesters:

**MATH 32a**
Differential Geometry

**MATH 34a**
Introduction to Topology

**MATH 37a**
Differential Equations

**MATH 38b**
Number Theory

G = Requirements for the Degree of Master of Arts

**A.**
One year's residence as a full-time student.

**B.**
Successful completion of an approved schedule of courses.

**C.**
Satisfactory performance in examinations in algebra, analysis,
topology, and geometric analysis.

**D.**
Proficiency in reading French, German, or Russian.

G = Requirements for the Degree of Doctor of Philosophy

**Program of Study**

The normal first year of study
consists of MATH 101a and b, 111a and b, and 121a and b. With
the permission of the graduate advisor, a student with superior
preparation may omit one or more of these courses and elect higher
level courses instead. In this case the student must take an examination
in the equivalent material during the first two weeks of the course.
The second year's work will normally consist of MATH 110a and
higher level courses in addition to preparation for the qualifying
examinations described below and participation in the second-year
seminar. Upon completion of the qualifying examinations, the student
will choose a dissertation advisor and begin work on a thesis.
This should be accompanied by advanced courses and seminars.

**Residence Requirement**

The minimum residence requirement
is three years.

**Language Requirement**

Proficiency in reading one
of French, German, or Russian, and one other language (besides
English) determined with the consent of the advisor.

**Qualifying Examination**

The qualifying examination
consists of two parts: a major examination and a minor examination.
Both are normally taken in the latter part of the second year
but may occasionally be postponed until early in the third year.
For the major examination, the student will choose a limited area
of mathematics (e.g., differential topology, several complex variables,
or ring theory) and a major examiner from among the faculty. Together
they will plan a program of study and a subsequent examination
in that material. The aim of this study is to prepare the student
for research towards the Ph.D. The minor examination will be more
limited in scope and less advanced in content. The procedures
are similar to those for the major examination, but its subject
matter should be significantly different from that of the major
examination.

**Dissertation and Defense**

The doctoral degree will be awarded only after the submission and acceptance of an approved dissertation and the successful defense of that dissertation.

S = Courses of Instruction

S = (1-99) Primarily for Undergraduate Students

**MATH 1a Introduction to
Mathematical Concepts**

[ **sn** ]

Is mathematics an art or a science? What would the world be without mathematics? What is mathematical insight and how does it happen? Why do we love mathematics? Why do we hate mathematics? These and other questions will be discussed, along with examples of mathematical reasoning in a variety of contexts. Usually offered in even years.

Staff

**MATH 2a Order and Chaos**

[ **cl ^{31}**

*Prerequisite: High school
algebra. Enrollment limited to 20. This course may not be taken
for credit by students who have received credit for CHSC 7a.*

The "new science" of chaos, the study of deterministic but nonrepetitive behavior that is extremely sensitive to small changes in the initial conditions of a system, is an exciting development in 20th-century science. We develop the mathematical background to understand how such remarkable and complex behavior arises from apparently simple descriptions of physical systems. Usually offered every third year. Last offered in the spring of 1995.

Staff

**MATH 4a Looking Into Mathematics:
A Visual Invitation to Mathematical Thinking**

[ **sn** ]

*Enrollment limited to 12.*

We use interactive computer graphics and other hands-on activities to convey the spirit of modern mathematics. We will concentrate on experimentation, both manual and virtual, with a rich variety of mathematical objects like curves, surfaces, linkages, knots, and braids. Usually offered in even years.

Staff

**MATH 5a Precalculus Mathematics**

*Does not meet any of the
options of the University Studies requirement in science and mathematics.
Does not satisfy the School of Science requirement. Enrollment
limited to 20 per section.*

Brief review of algebra followed by the study of functions. Emphasis on graphing functions and on trigonometric functions. The course's goal is to prepare students for MATH 10 or 15a. The decision to take this course should be guided by the results of the mathematics placement exam. Several sections will be offered. Usually offered every semester.

Ms. Parker and staff

**MATH 8a Introduction to
Probability and Statistics**

[ **cl ^{31}**

Discrete probability spaces, random variables, expectation, variance, approximation by the normal curve, sample mean and variance, and confidence intervals. Does not require calculus, only high school algebra and graphing of functions. Usually offered every spring.

Mr. van Moerbeke

**MATH 10a Techniques of Calculus
(a)**

[ **sn** ]

*Prerequisite: MATH 5a or
placement by examination. Enrollment limited to 25 per section.
May not be taken for credit by students who have taken MATH (PHYS)
13a,b.*

Introduction to differential (and some integral) calculus of one variable, with emphasis on techniques and applications. Several sections will be offered. Usually offered every semester.

Ms. Parker and staff (fall)

Mr. Monsky and staff (spring)

**MATH 10b Techniques of Calculus
(b)**

[ **sn** ]

*Prerequisite: MATH 10a or
placement by examination. Enrollment limited to 25 per section.
Continuation of 10a.*

Introduction to integral calculus of one variable with emphasis on techniques and applications. Several sections will be offered. Usually offered every semester.

Mr. Igusa and staff (fall)

Ms. Parker and staff (spring)

**MATH 15a Applied Linear
Algebra**

[ **sn** ]

*Prerequisite: MATH 5a and
permission of the instructor, placement by examination, or any
mathematics course numbered 10 or above. Students may not take
more than one of MATH 15a, 21a, and 22a for credit.*

Matrices, determinants, linear equations, vector spaces, eigenvalues, quadratic forms, linear programming. Emphasis on techniques and applications. Usually offered every semester.

Mr. Levine and Mr. Mayer (fall)

Staff (spring)

**MATH 20a Techniques of Calculus:
Calculus of Several Variables**

[ **sn** ]

*Prerequisite: MATH 10a,b.
Students may not take more than one of MATH 20a, 21b, and 22b
for credit.*

Among the topics treated are vectors and vector-valued functions, partial derivatives and multiple integrals, extremum problems, line and surface integrals, Green's and Stokes's theorems. Emphasis on techniques and applications. Usually offered every semester.

Staff (fall)

Mr. Ruberman (spring)

**MATH 21a Intermediate Calculus:
Linear Algebra and Calculus of Several Variables, Part I**

[ **sn** ]

*Prerequisite: MATH 10a,b
or placement by examination. Students intending to take the course
should consult the instructor or the undergraduate administrator.
Students may not take more than one of MATH 15a, 21a, and 22a
for credit.*

MATH 21a and 21b cover calculus of several variables for those with a serious interest in mathematics. The course starts with an introduction to linear algebra and then discusses various important topics in vector calculus, including directional derivatives, Jacobian matrices, multiple integrals, line integrals and surface integrals, and differential equations. Usually offered every year.

Mr. Schwarz

**MATH 21b Intermediate Calculus:
Linear Algebra and Calculus of Several Variables, Part II**

[ **sn** ]

*Prerequisite: MATH 21a or
permission of the instructor. Students may not take more than
one of MATH 20a, 21b, and 22b for credit.*

See MATH 21a for special notes and course description. Usually offered every year.

Mr. Schwarz

**MATH 22a Linear Algebra
and Intermediate Calculus, Part I**

[ **sn** ]

*Prerequisite: MATH 10a,b
or placement by examination. Students intending to take the course
should consult with the instructor or the undergraduate administrator.
Students may not take more than one of MATH 15a, 21a, or 22a for
credit.*

MATH 22a and 22b cover linear algebra and calculus of several variables. The material is similar to that of MATH 21a and MATH 21b, but with a more theoretical emphasis and with more attention to proofs. Usually offered every year.

Mr. Lian

**MATH 22b Linear Algebra
and Intermediate Calculus, Part II**

[ **sn** ]

*Prerequisite: MATH 22a or
permission of the instructor. Students may not take more than
one of MATH 20a, 21b, or 22b for credit.*

See MATH 22a for course description. Usually offered in even years.

Staff

**MATH 28a Introduction to
Algebraic Structures, Part I**

[ **sn** ]

*Prerequisite: MATH 15a or
20a or 21a.*

MATH 28a and b give an introduction to the major algebraic systems. The main topics are integers, groups, rings, integral domains, fields, real and complex numbers, and polynomials. Usually offered every year.

Mr. Buchsbaum

**MATH 28b Introduction to
Algebraic Structures, Part II**

[ **sn** ]

*Prerequisite: MATH 28a.*

See MATH 28a for course description. Usually offered every year.

Mr. Vilonen

**MATH 30a Introduction to
Algebra, Part I**

[ **sn** ]

*Prerequisite: MATH 21a and
b, or permission of the instructor.*

An introduction to the basic notions of modern algebra--rings, fields, and linear algebra. Usually offered every year.

Mr. Monsky

**MATH 30b Introduction to
Algebra, Part II**

[ **sn** ]

*Prerequisite: MATH 30a or
permission of the instructor.*

Groups and Galois theory. Usually offered every year.

Mr. Monsky

**MATH 32a Differential Geometry**

[ **sn** ]

*Prerequisite: MATH 21b or
permission of the instructor.*

Classical differential geometry of curves and surfaces, which, time permitting, will be followed by and motivate a brief introduction to differential manifolds. Usually offered in odd years.

Mr. Lian

**MATH 34a Introduction to
Topology**

[ **sn** ]

*Prerequisite: Math 21a and
b or permission of the instructor.*

An introduction to point set topology, covering spaces, and the fundamental group. Usually offered in odd years.

Mr. Vilonen

**MATH 35a Advanced Calculus,
Part I**

[ **sn** ]

*Prerequisites: MATH 15a
or 21a, MATH 20a or 21b.*

Solutions of linear first order and autonomous differential equations. Power series solutions. Fourier series and the Fourier and Laplace integrals. Applications. Usually offered every year.

Mr. Adler

**MATH 35b Advanced Calculus,
Part II**

[ **sn** ]

*Prerequisite: MATH 35a or
permission of the instructor.*

Laplace, heat, and wave equations. Solutions by separation of variables and Fourier analysis. Special functions and eigenvalue problems. Green's functions. Characteristics and wave propagation (some elementary material about linear operators and distributions may be covered). Usually offered every year.

Mr. Levine

**MATH 36a Probability**

[ **qr** **sn** ]

*Prerequisite: MATH 20a or
21b.*

Sample spaces and probability measures, elementary combinatorial examples. Random variables; expectations, variance, characteristic, and distribution functions. Independence and correlation. Chebychev's inequality and the weak law of large numbers. Central limit theorem. Markov and Poisson processes. Usually offered every year.

Mr. Eisenbud

**MATH 36b Mathematical Statistics**

[ **qr** **sn** ]

*Prerequisite: MATH 36a or
permission of the instructor.*

Basic notions of statistics. Distributions. Bayesian methods. Analysis of variance. Topics include order statistics, sequential analysis, limit theorems. Usually offered every year.

Mr. Adler

**MATH 37a Differential Equations**

[ **sn** ]

*Prerequisite: MATH 15a or
21a, MATH 20a or 21b.*

A first course in ordinary differential equations. Study of general techniques, with a view to solving specific problems such as the brachistochrone problem, the hanging chain problem, the motion of the planets, the vibrating string, Gauss's hypergeometric equation, the Volterra predator-prey model, isoperimetric problems, and the Abel mechanical problem. Usually offered in even years.

Staff

**MATH 38b Number Theory**

[ **sn** ]

*Prerequisite: MATH 21a or
permission of the instructor.*

Congruences, Z/p, quadratic reciprocity. Other topics as time allows. Usually offered in even years.

Staff

**MATH 40a Introduction to
Real Analysis, Part I**

[ **sn** ]

*Prerequisite: MATH 21a and
b, or permission of the instructor.*

MATH 40a and 40b give a rigorous introduction to metric space topology, continuity, derivatives, and Riemann and Lebesgue integrals. Usually offered every year.

Mr. Ruberman

**MATH 40b Introduction to
Real Analysis, Part II**

[ **sn** ]

*Prerequisite: MATH 40a or
permission of the instructor.*

See MATH 40a for course description. Usually offered every year.

Staff

**MATH 45a Introduction to
Complex Analysis**

[ **sn** ]

*Prerequisite: MATH 15a or
21a, MATH 20a or 21b.*

An introduction to functions of a complex variable. Topics include analytic functions, line integrals, power series, residues, conformal mappings. Usually offered every year.

Mr. Mayer

**MATH 98a Independent Research
**

*Signature of the instructor
required.*

Usually offered every year.

Staff

**MATH 98b Independent Research**

*Signature of the instructor
required.*

Usually offered every year.

Staff

G = (100-199) For Both Undergraduate and Graduate Students

**MATH 101a Algebra I**

[ **sn** ]

Groups, rings, modules, Galois theory, affine rings, and rings of algebraic numbers. Multilinear algebra. The Wedderburn theorems. Other topics as time permits. Usually offered every year.

Mr. Mayer

**Math 101b Algebra II**

[ **sn** ]

Continuation of MATH 101a. Usually offered every year.

Mr. Mayer

**MATH 110a Geometric Analysis**

[ **sn** ]

Manifolds, tensor bundles, vector fields, and differential forms. Frobenius theorem. Integration, Stokes's theorem, and deRham's theorem. Usually offered every year.

Mr. Ruberman

**MATH 110b Introduction to
Lie Groups and Differential Geometry**

[ **sn** ]

The correspondence between Lie groups and Lie algebras. Usually offered in odd years.

Mr. Adler

**MATH 111a Real Analysis**

[ **sn** ]

Measure and integration. Lp Spaces, Banach Spaces, Hilbert Spaces. Radon-Nikodym, Riesz representation and Fubini theorems. Fourier transforms. Usually offered every year.

Mr. Adler

**MATH 111b Complex Analysis**

[ **sn** ]

The Cauchy integral theorem, calculus of residues, and maximum modulus principle. Harmonic functions. The Riemann mapping theorem and conformal mappings. Other topics as time permits. Usually offered every year.

Mr. Vilonen

**MATH 121a Topology I**

[ **sn** ]

Fundamental group, covering spaces. Simplicial complexes, homology and cohomology theory, with applications. Usually offered every year.

Mr. Igusa

**MATH 121b Topology II**

[ **sn** ]

Continuation of MATH 121a. Manifolds and orientation, cup and cap products, Poincare duality. Other topics as time permits. Usually offered every year.

Mr. Levine

**MATH 150a Combinatorics
I**

[ **sn** ]

Emphasis on enumerative combinatorics. Generating functions and their applications to counting graphs, paths, permutations, and partitions. Bijective counting, combinatorial identities, Lagrange inversion and Möbius inversion. Usually offered every year.

Mr. Gessel

**MATH 150b Combinatorics
II**

[ **sn** ]

Representations of finite groups, with emphasis on symmetric groups. Symmetric functions, Pólya's theory of enumeration under group action, and combinatorial species. Usually offered in even years.

Staff

G = (200 and above) Primarily for Graduate Students

**MATH 200a Second-Year Seminar**

Usually offered every year.

Mr. Eisenbud

**MATH 201a Topics in Algebra**

Introduction to a field of algebra. Topic changes each year. Usually offered every year.

Mr. Vilonen

**MATH 201b Topics in Algebra**

Introduction to a field of algebra. Topic changes each year. Usually offered every year.

Mr. Schwarz

**MATH 202a Algebraic Geometry
I**

Varieties and schemes. Cohomology Theory. Curves and surfaces. Usually offered every year.

Mr. Eisenbud

**MATH 202b Algebraic Geometry
II**

Continuation of MATH 202a. Usually offered every year.

Mr. Eisenbud

**MATH 203a Number Theory**

Topics include basic algebraic number theory (number fields, ramification theory, class groups, Dirichlet unit theorem), zeta and L-functions (Riemann-function, Dirichlet L-functions, primes in arithmetic progressions, prime number theorem), class field theory, modular functions and modular forms, cyclotomic fields, and automorphic forms on Adele groups. Usually offered every year.

Mr. Monsky

**MATH 203b Number Theory**

Continuation of MATH 203a. Usually offered in even years.

Staff

**MATH 204a T.A. Practicum**

Teaching elementary mathematics courses is a subtle and difficult art, involving many skills besides those that make mathematicians good at proving theorems. This course focuses on the development and support of teaching skills. The main feature is individual observation of the graduate student by the practicum teacher, who provides written criticism of, and consultation on, classroom teaching practices. Usually offered every year.

Mr. Buchsbaum

**MATH 211a Topics in Differential
Geometry and Analysis**

Usually offered every year.

Mr. Lian

**MATH 211b Topics in Differential
Geometry and Analysis**

Usually offered every year.

Mr. Lian

**MATH 221a Topology III**

Elementary homotopy theory, fibrations, obstruction theory, and spectral sequences. Usually offered every year.

Mr. Levine

**MATH 221b Topology IV**

Differential topology: transversality and characteristic classes. Geometric definitions of cobordism, computation via homotopy theory. Other topics as time permits. Usually offered every year.

Mr. Ruberman

**MATH 224a Advanced Topics
in Lie Groups and Representation Theory**

(Formerly MATH 324a)

Usually offered in odd years.

Mr. Schwarz

**MATH 224b Advanced Topics
in Lie Groups and Representation Theory**

(Formerly MATH 324b)

Usually offered in even years.

Staff

**MATH 250a Riemann Surfaces**

An introductory course on Riemann surfaces. Usually offered in odd years.

Mr. Van Moerbeke

**MATH 291d Fellowship of
the Ring ó Seminar in Commutative Algebra**

Research seminar; not normally taken for credit. Usually offered every year.

Staff

**MATH 293d Topology Seminar**

Research seminar; not normally taken for credit. Usually offered every year.

Staff

**MATH 294d Differential Geometry
Seminar**

Research seminar; not normally taken for credit. Usually offered every year.

Staff

**MATH 295d Algebraic Geometry
Seminar**

Research seminar; not normally taken for credit. Usually offered every year.

Staff

**MATH 297d Number Theory
Seminar**

Research seminar; not normally taken for credit. Usually offered every year.

Staff

**MATH 299a and b Readings
in Mathematics**

Usually offered every year.

Staff

**MATH 301a Advanced Topics
in Algebra**

Usually offered in even years.

Staff

**MATH 302a Topics in Algebraic
Geometry**

Usually offered in even years.

Mr. Vilonen

**MATH 302b Topics in Algebraic
Geometry**

Usually offered in even years.

**MATH 311a Advanced Topics
in Analysis**

Usually offered every year.

Staff

**MATH 311b Advanced Topics
in Analysis**

Usually offered every year.

Staff

**MATH 321a Topics in Topology**

Usually offered in even years.

Mr. Levine

**MATH 321b Topics in Topology**

A continuation of MATH 321a. Usually offered in even years.

Staff

**MATH 326a Topics in Mathematics**

An advanced course on a topic chosen each year by the department. Usually offered every year.

Staff

**MATH 326b Topics in Mathematics**

A continuation of MATH 326a. Usually offered every year.

Staff

**MATH 399a and b Readings
in Mathematics**

Usually offered every year.

Staff

L =

All graduate courses will have organizational meetings the first week of classes.

L =

**MATH 401d Research**

Independent research for the Ph.D. degree. Specific sections for individual faculty members as requested.

Staff

S = Cross-Listed Courses

**COSI 188a**

Introduction to Combinatorics

**PHIL 106b**

Mathematical Logic

S = Courses of Related Interest

**PHIL 38b**

Philosophy of Mathematics