(file last updated: [8/10/1998 - 15:26:34])

Objectives

**Undergraduate Concentration**

As our society becomes moretechnological, it is more affected by mathematics. Quite sophisticatedmathematics is now central to the natural sciences, to ecologicalissues, 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 toact in it.

Mathematics is, at the sametime, a subject of the greatest depth and beauty with a historyextending from antiquity. The department attempts to make thisdepth and beauty manifest. The undergraduate major program introducesstudents to some fundamental fields--algebra, real and complexanalysis, geometry, and topology--and to the habit of mathematicalthought. Mathematics concentrators may go on to graduate school,scientific research, or mathematics teaching, but many choosethe major for its inherent interest with unrelated career intentions.

**Graduate Program in Mathematics**

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

How to Become an UndergraduateConcentrator

Students who enjoy mathematicsare urged to consider concentrating in it; Brandeis offers a widevariety of mathematics courses, and concentrators will have thebenefits of small classes and individual faculty attention. Tobecome a concentrator a student should have completed either MATH15 and 20, MATH 21a, b, or MATH 22a, b by the end of the sophomoreyear--these courses are prerequisites to the higher level offerings.Therefore, it is important for students to start calculus andlinear algebra (MATH 10, 15, 20, 21, or 22) in the first year.

How to Be Admitted tothe Graduate Program

The general requirements foradmission to graduate work in mathematics are the same as thosefor the Graduate School as a whole. The department has availablea variety of fellowships and scholarships for well-qualified students.To be considered for such financial support the student shouldsubmit an application by February 15.

Faculty

**Daniel Ruberman, Chair**

Geometric topology and gaugetheory.

**Mark Adler**

Analysis. Differential equations.Completely integrable systems.

**David Buchsbaum**

Commutative algebra. Homologicalalgebra. Representation theory.

**Ira Gessel**

Combinatorics. Computer science.

**Markus Hunziker**

Lie groups. Algebraic geometry.

**Kiyoshi Igusa, UndergraduateAdministrator**

Algebraic K-theory.

**Jerome Levine**

Differential topology. Knottheory and related algebra.

**Bong Lian, UndergraduateAdvising Head**

Representation theory.

**Alan Mayer**

Classical algebraic geometryand related topics in mathematical physics.

**Paul Monsky, Graduate AdvisingHead**

Number theory. Arithmetic algebraicgeometry. Commutative algebra.

**Susan Parker, ElementaryMathematics Coordinator**

Combinatorics. Elementary mathematicsinstruction.

**Gerald Schwarz**

Algebraic groups. Transformationgroups.

**Pierre Van Moerbeke**

Stochastic processes. Korteweg-deVriesequation. Toda lattices.

**Kari Vilonen**

Representation theory. Topology.

Requirements for the UndergraduateConcentration

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

**B. **MATH23b 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 numbered27 or higher or cross-listed courses.

**Honors**

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

**E.**Six additional semester courses, either MATH courses numbered27 or higher or cross-listed courses, passed with at least a gradeof B. At least four of the courses used to satisfy the concentrationrequirement must be honors courses. The honors courses are MATH30a, 30b, 32a, 34a, 38b, 40a, 40b, 45a, and all MATH courses numbered100 or higher.

**Teacher Preparation Track**

Students who complete the Brandeisprogram for Massachusetts High School Teacher Certification (seesection on Education Program in this *Bulletin*) may earna bachelor's degree in mathematics by satisfying concentrationrequirements A, B, C, and D above and the following:

**E.**MATH 8a or 36a.

**F.**Two additional courses, either MATH courses numbered 27 or higheror 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 areeligible for the B.A./M.S. program in mathematics if they havecompleted MATH 101a,b; 110a, 111a,b, and 121 a,b with a gradeof B- or better, and demonstrated a reading knowledge of mathematicalFrench, 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 tothis program on forms available at the Graduate School office.This must be done no later than May 1 preceding his/her finalyear of study on campus.

Requirements for the UndergraduateMinor

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

**B.**Three additional semester courses, either MATH courses numbered27 or higher or cross-listed courses.

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

Special Notes Relatingto Undergraduates

**A.**With permission of the Undergraduate Mathematics Advisor, coursestaken in other Brandeis departments or taken at other universitiesmay be substituted for required mathematics courses. A grade ofC or better is required in courses satisfying the field of concentrationrequirements.

**B.**Students who intend to take mathematics courses numbered 10 orhigher should take the departmental placement exam. On the basisof the exam, recommendations are made placing students out ofthe first year of calculus or into MATH 5a, 10a, or 10b. Studentsreceiving a score of 5 on the advanced placement MATH AB examor a score of 4 or more on the MATH BC exam place out of the first-yearcalculus sequence. Students receiving a score of 4 on the MATHAB exam or a score of 3 on the MATH BC exam place out of firstsemester calculus. Such students must take the departmental placementexam if they wish to place out of second semester calculus. Questionsabout placement should be directed to the Elementary MathematicsCoordinator, or the undergraduate advising head.

**C.**The usual calculus sequence is MATH 10a, 10b, and 20a. Studentsmay precede this with MATH 5a. Many students also take MATH 15a(Applied Linear Algebra), which has MATH 5a (or placement outof 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 stronginterest in mathematics and science are encouraged to take MATH21a,b or 22a,b in place of MATH 15a and 20a. Similarly, a studentmay not receive credit for both MATH 28a and 30a, or for bothMATH 28b and 30b.

**E.**Students should normally take MATH 23b before taking upper levelcourses (i.e., those numbered above 23). For most students thismeans taking MATH 23b in the spring semester, concurrently withMATH 21b, MATH 22b, MATH 15a, or MATH 20a. A student may be exemptedfrom the requirement of taking Math 23b by satisfactory performanceon a placement exam. The placement exam is given in early November,in time for preenrollment.

**F.**Students interested in graduate school or a more intensive studyof mathematics are urged to include all of the following coursesin their program:

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

2. MATH 30a and b.

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

4. MATH 45a.

5. A course numbered 100 orhigher.

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

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

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

3. In addition, the followingsemester courses are usually offered according to the followingschedule where 0-1 indicates even-odd years (e.g., 1998-99) and1-0 indicates odd-even years (e.g., 1999-2000). Slashes distinguishbetween fall and spring semesters:

**MATH 32a** > Differential Geometry

**MATH 34a** > Introduction to Topology

**MATH 37a** > Differential Equations

Requirements for the Degreeof 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 Degreeof Doctor of Philosophy

**Program of Study**

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

**Residence Requirement**

The minimum residence requirementis three years.

**Language Requirement**

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

**Qualifying Examination**

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

**Dissertation and Defense**

The doctoral degree will beawarded only after the submission and acceptance of an approveddissertation and the successful defense of that dissertation.

Courses of Instruction

(1-99) Primarily for UndergraduateStudents

**MATH 1a Introduction toMathematical Concepts**

[ **sn** ]

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

Staff

**MATH 2a Order and Chaos**

[ **cl ^{31}**

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

The "new science"of chaos, the study of deterministic but nonrepetitive behaviorthat is extremely sensitive to small changes in the initial conditionsof a system, is an exciting development in 20th-century science.We develop the mathematical background to understand how suchremarkable and complex behavior arises from apparently simpledescriptions of physical systems. Usually offered every thirdyear. 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 computergraphics and other hands-on activities to convey the spirit ofmodern mathematics. We will concentrate on experimentation, manualand virtual, with a rich variety of mathematical objects likecurves, surfaces, linkages, knots, and braids. Usually offeredin even years.

Staff

**MATH 5a Precalculus Mathematics**

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

Brief review of algebra followedby the study of functions. Emphasis on graphing functions andon trigonometric functions. The course's goal is to prepare studentsfor MATH 10 or 15a. The decision to take this course should beguided by the results of the mathematics placement exam. Severalsections will be offered. Usually offered every semester.

Ms. Parker and staff

**MATH 8a Introduction toProbability and Statistics**

[ **cl ^{31}**

Discrete probability spaces,random variables, expectation, variance, approximation by thenormal curve, sample mean and variance, and confidence intervals.Does not require calculus, only high school algebra and graphingof functions. Usually offered in the fall.

Mr. Adler

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

[ **sn** ]

*Prerequisite: MATH 5a orplacement 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 ontechniques and applications. Several sections will be offered.Usually offered every semester.

Ms. Parker and staff (fall)

Mr. Vilonen and staff (spring)

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

[ **sn** ]

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

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

Mr. Levine and staff (fall)

Ms. Parker and staff (spring)

**MATH 15a Applied LinearAlgebra**

[ **sn** ]

*Prerequisite: MATH 5a andpermission of the instructor, placement by examination, or anymathematics course numbered 10 or above. Students may not takemore than one of MATH 15a, 21a, and 22a for credit.*

Matrices, determinants, linearequations, vector spaces, eigenvalues, quadratic forms, linearprogramming. Emphasis on techniques and applications. Usuallyoffered every semester.

Mr. Lian and Mr. Monsky (fall)

Mr. Igusa (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 22bfor credit.*

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

Staff (fall)

Mr. Mayer (spring)

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

[ **sn** ]

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

MATH 21a and 21b cover calculusof several variables for those with a serious interest in mathematics.The course starts with an introduction to linear algebra and thendiscusses various important topics in vector calculus, includingdirectional 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 orpermission of the instructor. Students may not take more thanone of MATH 20a, 21b, and 22b for credit.*

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

Mr. Schwarz

**MATH 22a Linear Algebraand Intermediate Calculus, Part I**

[ **sn** ]

*Prerequisite: MATH 10a,bor placement by examination. Students intending to take the courseshould consult with the instructor or the undergraduate administrator.Students may not take more than one of MATH 15a, 21a, or 22a forcredit.*

MATH 22a and 22b cover linearalgebra and calculus of several variables. The material is similarto that of MATH 21a and MATH 21b, but with a more theoreticalemphasis and with more attention to proofs. Usually offered everyyear.

Mr. Lian

**MATH 22b Linear Algebraand Intermediate Calculus, Part II**

[ **sn** ]

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

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

Staff

**MATH 23b Introduction toProofs**

**[ wi sn ]**

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

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

Mr. Hunziker

**MATH 28a Introduction toAlgebraic Structures, Part I**

[ **sn** ]

*Prerequisite: MATH 15a or20a or 21a.*

MATH 28a and b give an introductionto 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 toAlgebraic Structures, Part II**

[ **sn** ]

*Prerequisite: MATH 28a.*

See MATH 28a for course description.Usually offered every year, will not be offered 1998-99.

Staff

**MATH 30a Introduction toAlgebra, Part I**

[ **sn** ]

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

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

Mr. Buchsbaum

**MATH 30b Introduction toAlgebra, Part II**

[ **sn** ]

*Prerequisite: MATH 30a orpermission of the instructor.*

Groups and Galois theory. Usuallyoffered every year.

Mr. Buchsbaum

**MATH 32a Differential Geometry**

[ **sn** ]

*Prerequisite: MATH 21b orpermission of the instructor.*

Classical differential geometryof curves and surfaces, which, time permitting, will be followedby and motivate a brief introduction to differential manifolds.Usually offered in odd years.

Staff

**MATH 34a Introduction toTopology**

[ **sn** ]

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

An introduction to point settopology, covering spaces, and the fundamental group. Usuallyoffered in odd years.

Staff

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

[ **sn** ]

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

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

Mr. Mayer

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

[ **sn** ]

*Prerequisite: MATH 35a orpermission of the instructor.*

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

Mr. Mayer

**MATH 36a Probability**

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

*Prerequisite: MATH 20a or21b.*

Sample spaces and probabilitymeasures, elementary combinatorial examples. Random variables;expectations, variance, characteristic, and distribution functions.Independence and correlation. Chebychev's inequality and the weaklaw of large numbers. Central limit theorem. Markov and Poissonprocesses. Usually offered every year.

Mr. Igusa

**MATH 36b Mathematical Statistics**

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

*Prerequisite: MATH 36a orpermission of the instructor.*

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

Mr. Adler

**MATH 37a Differential Equations**

[ **sn** ]

*Prerequisite: MATH 15a or21a, MATH 20a or 21b.*

A first course in ordinarydifferential equations. Study of general techniques, with a viewto solving specific problems such as the brachistochrone problem,the hanging chain problem, the motion of the planets, the vibratingstring, Gauss's hypergeometric equation, the Volterra predator-preymodel, isoperimetric problems, and the Abel mechanical problem.Usually offered in even years.

Mr. Hunziker

**MATH 38b Number Theory**

[ **sn** ]

*Prerequisite: MATH 21a orpermission 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 coveredas time permits. Usually offered in even years.

Mr. Monsky

**MATH 40a Introduction toReal Analysis, Part I**

[ **sn** ]

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

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

Mr. Vilonen

**MATH 40b Introduction toReal Analysis, Part II**

[ **sn** ]

*Prerequisite: MATH 40a orpermission of the instructor.*

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

Mr. Levine

**MATH 45a Introduction toComplex Analysis**

[ **sn** ]

*Prerequisite: MATH 15a or21a, MATH 20a or 21b.*

An introduction to functionsof a complex variable. Topics include analytic functions, lineintegrals, power series, residues, conformal mappings. Usuallyoffered every year.

Staff

**MATH 98a Independent Research**

*Signature of the instructorrequired.*

Usually offered every year.

Staff

**MATH 98b Independent Research**

*Signature of the instructorrequired.*

Usually offered every year.

Staff

(100-199) For Both Undergraduateand Graduate Students

**MATH 101a Algebra I**

[ **sn** ]

Groups, rings, modules, Galoistheory, affine rings, and rings of algebraic numbers. Multilinearalgebra. 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. Monsky

**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 everyyear.

Mr. Adler

**MATH 110b Introduction toLie Groups and Differential Geometry**

[ **sn** ]

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

Mr. Ruberman

**MATH 111a Real Analysis**

[ **sn** ]

Measure and integration. Lpspaces, Banach spaces, Hilbert spaces. Radon-Nikodym, Riesz representationand Fubini theorems. Fourier transforms. Usually offered everyyear.

Mr. Mayer

**MATH 111b Complex Analysis**

[ **sn** ]

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

Mr. Adler

**MATH 121a Topology I**

[ **sn** ]

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

Mr. Levine

**MATH 121b Topology II**

[ **sn** ]

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

Mr. Igusa

**MATH 150a CombinatoricsI**

[ **sn** ]

Emphasis on enumerative combinatorics.Generating functions and their applications to counting graphs,paths, permutations, and partitions. Bijective counting, combinatorialidentities, Lagrange inversion and Möbius inversion. Usuallyoffered every year, will not be offered 1998-99.

Staff

**MATH 150b CombinatoricsII**

[ **sn** ]

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

Staff

(200 and above) Primarilyfor Graduate Students

**MATH 200a Second-Year Seminar**

Usually offered every year.

Mr. Vilonen

**MATH 201a Topics in Algebra**

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

Mr. Buchsbaum

**MATH 201b Topics in Algebra**

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

Staff

**MATH 202a Algebraic GeometryI**

Varieties and schemes. Cohomologytheory. Curves and surfaces. Usually offered every year.

Mr. Hunziker

**MATH 202b Algebraic GeometryII**

Continuation of MATH 202a.Usually offered every year.

Mr. Hunziker

**MATH 203a Number Theory**

Topics include basic algebraicnumber theory (number fields, ramification theory, class groups,Dirichlet unit theorem), zeta and L-functions (Riemann Zeta-function,Dirichlet L-functions, primes in arithmetic progressions, primenumber theorem). Other possible topics are class field theory,modular functions and modular forms, cyclotomic fields, and automorphicforms on adele groups. Usually offered every year.

Staff

**MATH 203b Number Theory**

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

Staff

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

Teaching elementary mathematicscourses is a subtle and difficult art, involving many skills besidesthose that make mathematicians good at proving theorems. Thiscourse focuses on the development and support of teaching skills.The main feature is individual observation of the graduate studentby the practicum teacher, who provides written criticism of, andconsultation on, classroom teaching practices. Usually offeredevery year.

Mr. Gessel

**MATH 211a Topics in DifferentialGeometry and Analysis**

Usually offered every year.

Staff

**MATH 211b Topics in DifferentialGeometry and Analysis**

Usually offered every year.

Staff

**MATH 221a Topology III**

Vector bundles and characteristicclasses. Elementary homotopy theory and obstruction theory. Cobordismand transversality; other topics as time permits. Usually offeredevery year.

Mr. Igusa

**MATH 221b Topology IV**

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

Mr. Levine

**MATH 224a Advanced Topicsin Lie Groups and Representation Theory**

(Formerly MATH 324a)

Usually offered in odd years.

Mr. Vilonen

**MATH 224b Advanced Topicsin Lie Groups and Representation Theory**

(Formerly MATH 324b)

Usually offered in even years.

Mr. Schwarz

**MATH 250a Riemann Surfaces**

An introductory course on Riemannsurfaces. Usually offered in odd years.

Mr. Van Moerbeke

**MATH 299a and b Readingsin Mathematics**

Usually offered every year.

Staff

**MATH 301a Advanced Topicsin Algebra**

Usually offered in even years.

Staff

**MATH 302a Topics in AlgebraicGeometry**

Usually offered in even years.

Mr. Vilonen

**MATH 302b Topics in AlgebraicGeometry**

Usually offered in even years.

**MATH 311a Advanced Topicsin Analysis**

Usually offered every year.

Staff

**MATH 311b Advanced Topicsin 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 topicchosen 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 Readingsin Mathematics**

Usually offered every year.

Staff

All graduate courses will haveorganizational meetings the first week of classes.

**MATH 401d Research**

Independent research for thePh.D. degree. Specific sections for individual faculty membersas requested.

Staff

Cross-Listed Courses

Introduction to Combinatorics

Mathematical Logic

Courses of Related Interest

Philosophy of Mathematics