Mathematics

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.

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, 21b, or MATH 22a, 22b 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.

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 15.

Faculty

Gerald Schwarz, Chair

Algebraic groups. Transformation groups.

Mark Adler

Analysis. Differential equations. Completely integrable systems.

Mehrzad Ajoodanian

Gauge theory. Low dimensional topology.

Fred Diamond

Number theory.

Ira Gessel

Combinatorics. Computer science.

Kiyoshi Igusa, Undergraduate Administrator

Differential topology. Homological algebra.

Michael Kleber

Combinatorics. Representation theory.

Dmitry Kleinbock

Dynamical systems. Ergodic theory. Number theory.

Jerome Levine

Differential topology. Knot theory and related algebra.

Bong Lian, Graduate Advising Head

Representation theory. Calabi-Yau geometry. String theory.

Alan Mayer

Classical algebraic geometry and related topics in mathematical physics.

Paul Monsky, Undergraduate Advising Head

Number theory. Arithmetic algebraic geometry. Commutative algebra.

Susan Parker, Elementary Mathematics Coordinator

Combinatorics. Elementary mathematics instruction.

Daniel Ruberman

Geometric topology and gauge theory.

Harry Tamvakis

Arithmetic algebraic geometry. Arakelov theory.

Pierre Van Moerbeke

Stochastic processes. Korteweg-deVries equation. Toda lattices.

Requirements for the Undergraduate Concentration

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

**B. **MATH 23b or exemption. See item E in Special Notes Relating to Undergraduates.

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

**D.** MATH 28a or 30a.

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

Honors

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

**E.** 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, C, and D above and the following:

**E.** MATH 8a (Introduction to Probability and Statistics) or 36a (Probability).

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

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

**H.** 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.

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. MATH 23b being a prerequisite for these courses, must be taken in addition to three of these courses. With permission of the Undergraduate Advising Head, courses taken in other Brandeis departments or taken at other universities may be substituted for mathematics courses required for the minor.

Special Notes Relating to Undergraduates

**A.** With permission of the Undergraduate Advising Head, 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 should normally take MATH 23b before taking upper level courses (i.e., those numbered above 23). For many students this means taking MATH 23b concurrently with MATH 15a, or MATH 20a, MATH 21 a or b, MATH 22 a or b. Students may also take MATH 23b concurrently with MATH 35a or b and MATH 36a or b since these do not have MATH 23b as a prerequisite. A student may be exempted from the requirement of taking MATH 23b by satisfactory performance on a placement exam. The placement exam will be given at the beginning of the fall semester and the end of the spring semester.

**F.** 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.

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

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

2. Offered once each year are MATH 8a, 21a and b, 23b, 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., 2000-01) and 1-0 indicates odd-even years (e.g., 2001-02). Slashes distinguish between fall and spring semesters:

**MATH 32a** Differential Geometry

**MATH 34a** Introduction to Topology

**MATH 37a** Differential Equations

**MATH 38b** Number Theory

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.

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.

Teaching Requirements

An important part of the doctoral program is participation, as teaching fellows, in undergraduate teaching. All graduate students are expected to teach a section of calculus or pre-calculus for at least four semesters, usually beginning in their second year of study. First-year graduate students are not expected to teach, but are required to act as graders, with close faculty supervision, for upper-level undergraduate courses and to participate in the department's evening drop-in tutoring program. Prior to beginning teaching, during the spring semester of their first year, every student must take part in our teaching apprenticeship program. Teaching fellows must also enroll every fall semester in the Teaching Practicum, in which their teaching will be evaluated and discussed.

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.

Courses of Instruction

(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 every second year.

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, manual and virtual, with a rich variety of mathematical objects like curves, surfaces, linkages, knots, and braids. Usually offered every second year.

Staff

MATH 5a Precalculus 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

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

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 in the spring.

Staff

MATH 10a Techniques of Calculus (a)

[ **sn** ]

Prerequisite: MATH 5a or placement by examination. Enrollment limited to 25 per section.

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)

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** ]

Prerequisites: 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. Adler and Staff (fall)

Mr. Levine and 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.

Mr. Ajoodanian (fall)

Mr. Mayer (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. Ruberman

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. Ajoodanian

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 every second year.

Mr. Lian

MATH 23b Introduction to Proofs

[ wi sn ]

Prerequisites: MATH 15a, 20a, 21a, 22a, or permission of the instructor. Enrollment limited to 25.

Emphasizes the analysis and writing of proofs. Various techniques of proof are introduced and illustrated with topics chosen from set theory, calculus, algebra, and geometry. Usually offered every year.

Mr. Gessel

MATH 28a Introduction to Algebraic Structures, Part I

[ **sn** ]

Prerequisite: MATH 23b and either MATH 15a, 21a or 22a.

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. Monsky

MATH 28b Introduction to Algebraic Structures, Part II

[ **sn** ]

Prerequisite: MATH 28a.

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

Mr. Monsky

MATH 30a Introduction to Algebra, Part I

[ **sn** ]

Prerequisite: MATH 23b and either MATH 21a and b, 22a 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. Diamond

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. Diamond

MATH 32a Differential Geometry

[ **sn** ]

Prerequisite: MATH 23b and either MATH 21b, 22b, 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 every second year.

Mr. Ruberman (spring)

MATH 34a Introduction to Topology

[ **sn** ]

Prerequisite: MATH 23b and either MATH 21a and b, 22a and b, or permission of the instructor.

An introduction to point set topology, covering spaces, and the fundamental group. Usually offered every second year.

Mr. Igusa (fall)

MATH 35a Advanced Calculus, Part I

[ **sn** ]

Prerequisites: MATH 15a, 21a, or 22a and MATH 20a, 21b or 22b

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. Gessel

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.

Staff

MATH 36a Probability

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

Prerequisite: MATH 20a, 21b or 22b.

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. Kleinbock

MATH 36b Mathematical Statistics

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

Prerequisite: MATH 36a or permission of the instructor.

Probability distributions, estimators, hypothesis testing, data analysis. Theorems will be proved and applied to real data. Topics include maximum likelihood estimators, the information inequality, chi-square test, and analysis of variance. Usually offered every year.

Staff

MATH 37a Differential Equations

[ **sn** ]

Prerequisite: MATH 15a, 21a or 22a and MATH 20a, 21b or 22b.

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 every second year.

Staff

MATH 38b Number Theory

[ **sn** ]

Prerequisite: MATH 23b and either MATH 21a, 22a, or permission of the instructor.

Congruences, finite fields, the Gaussian integers, and other rings of numbers. Quadratic reciprocity. Such topics as quadratic forms or elliptic curves will be covered as time permits. Usually offered every second year.

Staff

MATH 39a Introduction to Combinatorics

[ **sn **]

Prerequisites: COSI 29a or MATH 23b

Topics include graph theory (trees, planarity, coloring, Eulerian and Hamiltonian cycles), combinatorial optimization (network flows, matching theory), enumeration (permutations and combinations, generating functions, inclusion-exclusion), and extremal combinatorics (pigeonhole principle, Ramsey's theorem). Usually offered every second year. Will be offered in the fall of 2002.

Mr. Gessel

MATH 40a Introduction to Real Analysis, Part I

[ **sn** ]

Prerequisites: MATH 23b and either MATH 21a and b, 22a 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.

Staff

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** ]

Prerequisites: MATH 15a, 21a or 22a and MATH 20a, 21b or 22b and MATH 23b or permission of the instructor.

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

Mr. Monsky

MATH 47a Introduction to Mathematical Research

[** sn **]

Signature of the instructor required.

Students work on research projects that involve generating data, making conjectures and proving theorems, and present their results orally and in writing. Introduces applications of computers in mathematical research: symbolic computation, typesetting, and literature search. The special topic for spring 2000 was Fibonacci numbers. Usually offered every third year. Last offered in the spring of 2000.

Staff

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

(100-199) For Both Undergraduate and Graduate Students *Undergraduates should consult with the instructor regarding the required background.*

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.

Staff

MATH 101b Algebra II

[ **sn** ]

Continuation of MATH 101a. Usually offered every year.

Mr. Igusa

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.

Staff

MATH 110b Introduction to Lie Groups and Differential Geometry

[ **sn** ]

The correspondence between Lie groups and Lie algebras. Usually offered every second year.

Mr. Ajoodanian

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. schwarz

MATH 121a Topology I

[ **sn** ]

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

Mr. Ruberman

MATH 121b Topology II

[ **sn** ]

Continuation of MATH 121a. Manifolds and orientation, cup and cap products, Poincaré 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 every second year.

Mr. Gessel

(200 and above) Primarily for Graduate Students

MATH 200a Second-Year Seminar

Usually offered every year.

Mr. Lian

MATH 201a Topics in Algebra

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

Staff

MATH 201b Topics in Algebra

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

Staff

MATH 202a Algebraic Geometry I

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

Mr. Schwarz

MATH 202b Algebraic Geometry II

Continuation of MATH 202a. Usually offered every year.

Staff

MATH 203a Number Theory

Topics include basic algebraic number theory (number fields, ramification theory, class groups, Dirichlet unit theorem), zeta and L-functions (Riemann Zeta-function, Dirichlet L-functions, primes in arithmetic progressions, prime number theorem). Other possible topics are 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 every second year.

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. Diamond

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. Kleinbock

MATH 221a Topology III

Vector bundles and characteristic classes. Elementary homotopy theory and obstruction theory. Cobordism and transversality; other topics as time permits. Usually offered every year.

Mr. Ajoodanian

MATH 221b Topology IV

Topics in topology and geometry. In recent years, topics have included knot theory, symplectic and contact topology, gauge theory, and three dimensional topology. Usually offered every year.

Mr. Ruberman

MATH 224a Advanced Topics in Lie Groups and Representation Theory

(Formerly MATH 324a)

Usually offered every second year.

Staff

MATH 224b Advanced Topics in Lie Groups and Representation Theory

(Formerly MATH 324b)

Usually offered every second year.

Staff

MATH 250a Riemann Surfaces

An introductory course on Riemann surfaces. Usually offered every second year.

Staff

MATH 299a and b Readings in Mathematics

Staff

MATH 301a Advanced Topics in Algebra

Staff

MATH 302a Topics in Algebraic Geometry

Staff

MATH 302b Topics in Algebraic Geometry

Staff

MATH 311a Advanced Topics in Analysis

Staff

MATH 311b Advanced Topics in Analysis

Staff

MATH 321a Topics in Topology

Staff

MATH 321b Topics in Topology

A continuation of MATH 321a.

Staff

MATH 326a Topics in Mathematics

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

Staff

MATH 326b Topics in Mathematics

A continuation of MATH 326a.

Staff

MATH 399a and b Readings in Mathematics

Staff

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

MATH 401d Research

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

Staff

Cross-Listed Courses

Introduction to Combinatorics

Mathematical Logic

Courses of Related Interest

Philosophy of Mathematics