Department of

Mathematics

Courses of Study:

Minor

Major (B.A.)

Combined B.A./M.S

Master of Arts

Doctor of Philosophy

Department website: http://www.math.brandeis.edu/

Objectives

Undergraduate Major

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 majors 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 Major

Students who enjoy mathematics are urged to consider concentrating in it; Brandeis offers a wide variety of mathematics courses, and majors will have the benefits of small classes and individual faculty attention. To become a major 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

**Bong Lian, Chair **Representation theory. Calabi-Yau geometry. String theory.

**Mark Adler **Analysis. Differential equations. Completely integrable systems.

**Fred Diamond, Graduate Advising Head **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.

**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.

**James Propp **Combinatorics.

**Daniel Ruberman **Geometric topology and gauge theory.

**Gerald Schwarz **Algebraic groups. Transformation groups.

**Harry Tamvakis **Arithmetic algebraic geometry. Arakelov theory.

**Pierre Van Moerbeke **Stochastic processes. Korteweg-deVries equation. Toda lattices.

Requirements for the Undergraduate Major

**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, 28b 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 major 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 major 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 major. 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. Most MATH courses numbered 27 or higher require MATH 23b as a prerequisite.

Students interested in analysis, physics, or applied mathematics are advised to choose additional courses from among MATH 35a, 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 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 major 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. 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.

**D.** A student may not receive credit for more than one of MATH 15a, 21a, and 22a; or MATH 20a, 21b, and 22b. Similarly, a student may not receive credit for all three of MATH 28a, 28b, and 30a.

**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 and MATH 36a 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 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, 36a and b, 40a and b, 45a.

3. In addition, the following semester courses are usually offered according to the following schedule where even-odd years (e.g., 2002-03) and odd-even years (e.g., 2003-04).

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: MATH 101a and b, MATH 110a, MATH 111a and b, and MATH 121a and b.

**C.** 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 a teaching fellow, in a structured program of undergraduate teaching. During the spring semester of their first year, every student takes part in our teaching apprenticeship program to learn basic classroom teaching skills. All graduate students are then expected to teach a section of calculus or pre-calculus for at least four semesters, usually beginning in their second year of study. Teaching fellows must also enroll every fall semester in the Teaching Practicum, in which their teaching is 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 **[

Mathematical reasoning; where it would be expected, and elsewhere. A variety of short topics involving games and puzzles, number theory, combinatorics, and topology. Usually offered every third year. Last offered in the spring of 2000.

Mr. Monsky

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

Staff

**MATH 5a Precalculus Mathematics **

Ms. Parker and Staff

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

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. Last offered in the spring of 2002.

Mr. Adler

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

Ms. Parker and Staff (fall)

Mr. Kleber (spring)

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

Mr. Ruberman and Staff (fall)

Ms. Parker and Staff (spring)

**MATH 15a Applied Linear Algebra **[

Messrs. Monsky and Propp (fall)

Mr. Propp (spring)

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

Mr. Levine (fall)

Mr. Propp (spring)

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

Mr. Kleinbock

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

Mr. Kleinbock

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

Mr. Lian

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

Mr. Levine

**MATH 23b Introduction to Proofs **

Mr. Gessel

**MATH 28a Introduction to Groups **[

Mr. Diamond

**MATH 28b Introduction to Rings and Fields **[

Mr. Ruberman

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

Mr. Igusa

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

Mr. Monsky

**MATH 32a Differential Geometry **[

Staff (spring)

**MATH 34a Introduction to Topology **[

Staff (fall)

**MATH 35a Advanced Calculus **[

Mr. Mayer

**MATH 36a Probability **[

Mr. Adler

**MATH 36b Mathematical Statistics **[

Mr. Igusa

**MATH 37a Differential Equations **[

Mr. Monsky

**MATH 38b Number Theory **[

Mr. Diamond

**MATH 39a Introduction to Combinatorics **[

Mr. Gessel

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

Mr. Tamvakis

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

Mr. Kleber

**MATH 45a Introduction to Complex Analysis **[

Mr. Mayer

**MATH 47a Introduction to Mathematical Research **[

Mr. Gessel

**MATH 98a Independent Research **

Staff

**MATH 98b Independent Research **

Staff

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

**MATH 101a Algebra I **[

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. Will be offered in the fall of 2002.

Mr. Propp

**MATH 101b Algebra II **[

Continuation of MATH 101a. Usually offered every year. Last offered in the spring of 2002.

Mr. Mayer

**MATH 110a Geometric Analysis **[

Manifolds, tensor bundles, vector fields, and differential forms. Frobenius theorem. Integration, Stokes's theorem, and deRham's theorem. Usually offered every year. Will be offered in the fall of 2002.

Mr. Adler

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

The correspondence between Lie groups and Lie algebras. Usually offered every second year. Last offered in the spring of 2002.

Mr. Ruberman

**MATH 111a Real Analysis **[

Measure and integration. Lp spaces, Banach spaces, Hilbert spaces. Radon-Nikodym, Riesz representation, and Fubini theorems. Fourier transforms. Usually offered every year. Will be offered in the fall of 2002.

Mr. Mayer

**MATH 111b Complex Analysis **[

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. Last offered in the spring of 2002.

Mr. Tamvakis

**MATH 121a Topology I **[

Fundamental group, covering spaces. Cell complexes, homology and cohomology theory, with applications. Usually offered every year. Will be offered in the fall of 2002.

Mr. Igusa

**MATH 121b Topology II **[

Continuation of MATH 121a. Manifolds and orientation, cup and cap products, Poincaré duality. Other topics as time permits. Usually offered every year. Last offered in the spring of 2002.

Mr. Igusa

**MATH 150a Combinatorics I **[

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 second year. Last offered in the fall of 2001.

Staff

**MATH 150b Combinatorics II **[

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 third year. Last offered in the fall of 2000.

Staff

(200 and above) Primarily for Graduate Students

**MATH 200a Second-Year Seminar **Usually offered every year. Last offered in the spring of 2002.

Mr. Diamond

**MATH 201a Topics in Algebra **Introduction to a field of algebra. Topic changes each year. Usually offered every third year. Last offered in the fall of 2000.

Mr. Kleber

**MATH 201b Topics in Algebra **Introduction to a field of algebra. Topic changes each year. Usually offered every year. Last offered in the spring of 2002.

Mr. Lian

**MATH 202a Algebraic Geometry I **Varieties and schemes. Cohomology theory. Curves and surfaces. Usually offered every second year. Last offered in the fall of 2001.

Mr. Tamvakis

**MATH 202b Algebraic Geometry II **Continuation of MATH 202a. Usually offered every year. Last offered in the spring of 2002.

Mr. Tamvakis

**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. Will be offered in the fall of 2002.

Mr. Diamond

**MATH 203b Number Theory **Continuation of MATH 203a. Usually offered every second year. Last offered in the spring of 2001.

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 second year. Last offered in the fall of 2001.

Mr. Kleber

**MATH 211a Topics in Differential Geometry and Analysis **Usually offered every second year. Last offered in the fall of 2001.

Mr. Ruberman

**MATH 211b Topics in Differential Geometry and Analysis **Usually offered every second year. Last offered in the spring of 2001.

Mr. Adler

**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 second year. Last offered in the fall of 2001.

Mr. Levine

**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. Last offered in the spring of 2002.

Mr. Levine

**MATH 224a Advanced Topics in Lie Groups and Representation Theory **Usually offered every fourth year. Last offered in the fall of 1998.

Staff

**MATH 224b Advanced Topics in Lie Groups and Representation Theory **Usually offered every second year. Last offered in the spring of 2001.

Staff

**MATH 250a Riemann Surfaces **An introductory course on Riemann surfaces. Usually offered every third year. Last offered in the spring of 2000.

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

**COSI 188a **Introduction to Combinatorics

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