### Courses of Study

### Sections

## Department of Mathematics

Last updated: July 19, 2013 at 1:38 p.m.

**Undergraduate Major**

As our society becomes more technological, it is increasingly 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 5a, 8a, 10a, 10b, 15a, or 20a 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 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, finance, actuarial science, or mathematics teaching, but many choose the major for its inherent interest.

**Postbaccalaureate Program in Mathematics
** The mathematics department offers a postbaccalaureate program for students with a bachelor’s degree in a different field who wish to prepare for graduate school or a career requiring enhanced mathematical skills.

**Graduate Program in Mathematics
** The graduate program in mathematics includes course work, seminar participation, teaching and research experience, and is designed to lead to a broad understanding of the subject. The program offers the Master of Arts and Doctor of Philosophy degrees.

Entering students may be admitted to either the master's or the doctoral program.The courses offered by the department, participation in seminars, and exposure to a cutting-edge research environment provide the students with a broad foundation for work in modern pure mathematics and prepare them for careers as mathematicians in academia, industry, or government.

Mathematics is both a subject of great depth and beauty, with a history extending from antiquity, and a powerful tool for understanding our world. Students may study mathematics for several reasons: for its own intrinsic interest, for its applications to other fields such as economics, computer science, and physical and life sciences, and for the analytical skills that it provides for such fields of study as law, medicine, or business. The mathematics major at Brandeis serves a diverse audience, consisting of students whose motivations cover all of these reasons.

**Learning goals for non-majors**:

Non-majors who take mathematics courses include pre-medical students, education minors, many science and economics majors, and mathematics minors. Although their mathematical goals may vary depending on their interests, the following are among the most important:

- improved analytical reasoning skills
- enhanced basic computational skills
- familiarity with basic mathematical terms and their physical meanings
- the ability to model real-world problems mathematically
- an appreciation for the power of mathematical thinking

**Learning goals for majors**:

**Knowledge**: Students completing the major in mathematics will understand the funda- mental concepts of

- mathematical proof
- abstraction and generalization
- the rules and uses of logic and will achieve a basic knowledge of the following areas of mathematics:
- analysis in one and several variables, including properties of the real numbers and of limits
- matrices and linear algebra
- axiomatically defined algebraic structures, such as groups, rings, fields, and vector spaces

Moreover, mathematics majors will know the basic ideas of some, but not necessarily all, of the following areas:

- differential equations
- probability and statistics
- number theory
- combinatorics
- real and complex analysis
- topology
- differential geometry

**Core Skills**: Mathematics majors will be able to

- formulate mathematical statements precisely
- read and write mathematical proofs
- communicate mathematical ideas orally and in writing
- distinguish correct from fallacious arguments
- abstract general principles from examples, and apply general principles to specific cases
- solve non-routine mathematical problems
- apply mathematics to real-world problems
- extend their knowledge of mathematics through independent reading

**Upon Graduation**: Mathematics majors with appropriate backgrounds and preparation may

- pursue graduate study and a scholarly career in mathematics
- work as actuaries
- teach mathematics at the K-12 level
- work in fields such as computer science, operations research, economics, finance, biology, physics, or other sciences
- attend medical, law, or business school

**Dmitry Kleinbock, Chair
** Dynamical systems. Ergodic theory. Number theory.

**Mark Adler, Undergraduate Advising Head (on leave spring 2014)
** Analysis. Differential equations. Completely integrable systems.

**Joël Bellaïche (On leave spring 2014)
** Number theory.

**Olivier Bernardi**

Combinatorics.

**Ruth Charney**

Geometric group theory. Topology.

**Luke Cherveny**

Differential and algebraic geometry. Mirror symmetry. Gromov-Witten theory.

**Ira Gessel
** Combinatorics.

**Kiyoshi Igusa, Graduate Advising Head**

Differential topology. Homological algebra.

**Bong Lian, Undergraduate Advising Head
** Representation theory. Calabi-Yau geometry. String theory.

**Alan Mayer (on leave fall 2013)
** Classical algebraic geometry and related topics in mathematical physics.

**Susan Parker, Elementary Mathematics Coordinator
** Combinatorics. Elementary mathematics instruction.

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

**Rebecca Torrey**

Number theory.

**A.**MATH 22a or 15a; MATH 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, but Math 35a, 36a, 36b, 37a, and 39a do not.

Students interested in analysis, physics, or applied mathematics are advised to choose additional courses from among MATH 35a, 36a, 36b, 37a, and 115a. Students interested in algebra or computer science are advised to consider MATH 28a, 28b, 100a, 100b, and 108b.

**C.** No grade below a C- will be given credit toward the minor.

**D.** No course taken pass/fail may count towards the the minor requirements.

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

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

**C.** MATH 35a, 110a, or 115a.

**D.** MATH 28a, 28b, or 100a.

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

No grade below a C- will be given credit toward the major.

No course taken pass/fail may count towards the the major requirements.

**Honors**

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

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 all MATH courses numbered 100 or higher.**Teacher Preparation Track**

Students who complete the Brandeis program for Massachusetts High School Teacher Licensure (see the Education Program section 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 Licensure Program.

**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.

**B.** Students planning to take MATH 10a or 10b or to place into MATH 15a or 20a should take the Calculus Placement Exam. This online exam can be found, along with instructions for scoring and interpreting the results, http://www.brandeis.edu/registrar/newstudent/testing.html. Students planning to take MATH 22a must take the MATH 22a Placement Exam, which can be found at the same place.

Students with AP Mathematics credit should consult the chart in this *Bulletin* to see which Brandeis mathematics courses are equivalent to their AP credit. Note: Students who want to use their AP score to place into an upper level course must still take the Calculus Placement Exam or the MATH 22a Placement Exam to make sure that their preparation is sufficient. 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 sequence 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 22a,b in place of MATH 15a and 20a.

**D.** A student may not receive credit for more than one of MATH 15a and 22a; or MATH 20a and 22b; or ECON 184b and 185a. Similarly, a student may not receive credit for all three of MATH 28a, 28b, and 100a.

**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 or MATH 22a or b. Students may also take MATH 23b concurrently with MATH 35a and MATH 36a as 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 an exemption exam. The exemption exam will be given at the beginning of the fall 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 22a and b.

2. MATH 100a and b.

3. MATH 35a or 110a and b.

4. MATH 115a.

5. Other courses numbered 100 or higher.

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

1. Offered every semester are MATH 5a, 10a and b, 15a, 20a, and 23b.

2. Offered once each year are MATH 8a, 35a, 36a and b, 37a, 100a, 110a, 115a.

3. In addition, the following semester courses are usually offered every second year according to the following schedule:

a. even-odd years (e.g., 2010-2011): MATH 3a, 28a, 47a, 100b, and 102a.

b. odd-even years (e.g., 2009-2010): MATH 28b, 39a, 104a, 108b, 110b, and 126a.

**H.** The number of cross-listed courses used to satisfy the requirements for the major, the honors, or teacher preparation track must not exceed two; for the minor, the limit is one.

**A.** Two core courses: MATH 15a and MATH 20a.

**B.** Elective courses: At least four additional MATH courses. Students who have taken linear algebra and/or multivariable calculus prior to entering the program may substitute additional electives for these two courses. At most one cross-listed course may be used to fulfill the elective requirement.

**Course Requirement**

The Master's program requires 8 courses including a reading course leading up to the minor examination. This takes the form either of a paper or lecture of expository nature outlining the material learned in the course. Master's students take the same first-year courses as PhD students. The only additional course requirement is MATH 140a, usually taken in the first semester of the second year plus one other MATH course numbered 130 or higher which may be a reading course.

**Residence Requirement**

The minimum residence requirement is one year. The program may take an additional one or two semesters to complete as an Extended Master's student.

**Language Requirement**

One language examination is required for the Master's degree; qualifying examinations in Mathematics are not required.

**Program of Study**

The normal first year of study consists of MATH 131a and b, 141a and b, and 151a and b. With the permission of the graduate adviser, 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 140a 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 adviser and begin work on a thesis. This should be accompanied by advanced courses and seminars. In addition, all first-year Ph.D students are required to take CONT 300b (Responsible Conduct of Science), offered in the spring.

**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 the 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 precalculus for at least three semesters, usually beginning in the 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 academic residence requirement is three years.

**Language Requirement
** Proficiency in reading one of French, German, or Russian.

**Qualifying Examination
** The qualifying examination consists of two parts: a major examination and a minor examination. Both are normally completed by the end of 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 toward the PhD. The minor examination will be more limited in scope and less advanced in content. 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
3a
Mathematics for Elementary and Middle School Teachers
**

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*This is an experiential learning course. *

An in-depth exploration of the fundamental ideas underlying the mathematics taught in elementary and middle school. Emphasis is on problem solving, experimenting with mathematical ideas, and articulating mathematical reasoning. Usually offered every second year.

Ms. Torrey (spring)

**
MATH
5a
Precalculus Mathematics
**

*Does not satisfy the School of Science requirement. Students may not take MATH 5a if they have received a satisfactory grade in any math class numbered 10 or higher.*

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

Staff (fall and spring)

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

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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 year.

Mr. Adler (fall)

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

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*Prerequisite: A satisfactory grade of C- or higher in MATH 5a or placement by examination. Students may not take MATH 10a if they have received a satisfactory grade in MATH 10b or MATH 20a.*

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

Ms. Parker (fall) Ms. Torrey (spring)

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

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*Prerequisite: A satisfactory grade of C- or higher in MATH 10a or placement by examination. Continuation of 10a. Students may not take MATH 10a and MATH 10b simultaneously. Students may not take MATH 10b if they have received a satisfactory grade in MATH 20a.*

Introduction to integral calculus of one variable with emphasis on techniques and applications. Usually offered every semester in multiple sections.

Ms. Torrey (fall) Ms. Parker (spring)

**
MATH
15a
Applied Linear Algebra
**

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*Prerequisites: MATH 5a and permission of the instructor, placement by examination, or any mathematics course numbered 10 or above. Students may take MATH 15a or 22a for credit, but not both.*

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

Ms. Torrey (fall), Mr. Sultan (spring)

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

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*Prerequisites: MATH 10a and b or placement by examination. Students may take MATH 20a or 22b for credit, but not both. Students may not take MATH 10a or 10b concurrently with MATH 20a.*

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. Joyner (fall and spring)

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

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*Prerequisite: MATH 22 placement exam and permission of the instructor. Students may take MATH 15a or 22a for credit, but not both.*

MATH 22a and b cover linear algebra and calculus of several variables. The material is similar to that of MATH 15a and MATH 20b, but with a more theoretical emphasis and with more attention to proofs. Usually offered every year.

Mr. Lian (fall)

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

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*Prerequisite: MATH 22a or permission of the instructor. Students may take MATH 20a or 22b for credit, but not both.*

See MATH 22a for course description. Usually offered every year.

Mr. Ruberman (spring)

**
MATH
23b
Introduction to Proofs
**

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*Prerequisites: MATH 15a, 20a, or 22a, or permission of the instructor.*

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 semester.

Mr. Igusa (fall), Ms. Charney (spring)

**
MATH
28a
Introduction to Groups
**

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

Groups. Lagrange's theorem. Modulo n addition and multiplication. Matrix groups and permutation groups. Homomorphisms, normal subgroups, cosets, and factor groups. Usually offered every second year.

Mr. Sultan (fall)

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

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

Fields. Z/p and other finite fields. Commutative rings. Polynomial rings and subrings of C. Euclidean rings. The quotient ring A/(f). Polynomials over Z. Usually offered every second year.

Mr. Levine (spring)

**
MATH
35a
Advanced Calculus
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b.*

Infinite series: convergence tests, power series, and Fourier series. Improper integrals: convergence tests, the gamma function, Fourier and Laplace transforms. Complex numbers. Usually offered every year.

Mr. Bernardi

**
MATH
36a
Probability
**

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*Prerequisite: MATH 20a 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.

Staff

**
MATH
36b
Mathematical Statistics
**

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*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
**

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*Prerequisites: MATH 15a or 22a and MATH 20a 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 year.

Mr. Mayer (spring)

**
MATH
39a
Introduction to Combinatorics
**

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*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.

Mr. Lipset (fall)

**
MATH
47a
Introduction to Mathematical Research
**

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*Prerequisite: MATH 23b or permission of the instructor. This is an experiential learning course.*

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. Usually offered every second year.

Staff

**
MATH
98a
Independent Research
**

Usually offered every year.

Staff

**
MATH
98b
Independent Research
**

Usually offered every year.

Staff

#### (100-199) For Both Undergraduate and Graduate Students

Undergraduate students should consult with the instructor regarding the required background for each course.

**
MATH
100a
Introduction to Algebra, Part I
**

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*Prerequisite: MATH 23b and MATH 22a, or permission of the instructor. May not be taken for credit by students who took MATH 30a in prior years.*

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

Mr. Lipsett (fall)

**
MATH
100b
Introduction to Algebra, Part II
**

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*Prerequisite: MATH 100a or permission of the instructor. May not be taken for credit by students who took MATH 30b in prior years.*

A continuation of MATH 100a, culminating in Galois theory. Usually offered every second year.

Mr. Bellaïche (spring)

**
MATH
102a
Differential Geometry
**

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*Prerequisites: MATH 23b and either MATH 22b or permission of the instructor. May not be taken for credit by students who took MATH 32a in prior years.*

Introduces the classical geometry of curves and surfaces. Topics include the Frenet equations and global properties of curves, local surface theory, including the fundamental forms and the Gauss map, intrinsic geometry of surfaces, Gauss's fundamental theorem and the Gauss-Bonnet Theorem. Usually offered every second year.

Mr. Ruberman (fall)

**
MATH
104a
Introduction to Topology
**

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*Prerequisites: MATH 23b and either MATH 22a and b or permission of the instructor. May not be taken for credit by students who took MATH 34a in prior years.*

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

Mr. Cherveny (fall)

**
MATH
108b
Number Theory
**

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*Prerequisites: MATH 23b and either MATH 22a or permission of the instructor. May not be taken for credit by students who took MATH 38b in prior years.*

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
110a
Introduction to Real Analysis, Part I
**

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*Prerequisites: MATH 23b and MATH 22a and b or permission of the instructor. May not be taken for credit by students who took MATH 40a in prior years.*

MATH 110a and b give a rigorous introduction to metric space topology, continuity, derivatives, and Riemann and Lebesgue integrals. Usually offered every year.

Mr. Ruberman (fall)

**
MATH
110b
Introduction to Real Analysis, Part II
**

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*Prerequisite: MATH 110a or permission of the instructor. May not be taken for credit by students who took MATH 40b in prior years.*

See MATH 110a for course description. Usually offered every second year.

Mr. Cherveny (spring)

**
MATH
115a
Introduction to Complex Analysis
**

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*Prerequisites: MATH 15a or 22a and MATH 20a or 22b, and MATH 23b or permission of the instructor. May not be taken for credit by students who took MATH 45a in prior years.*

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 (spring)

**
MATH
126a
Introduction to Stochastic Processes and Models
**

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*Prerequisites: MATH 15a, 20a, and 36a. May not be taken for credit by students who took MATH 56a in prior years.*

Basic definitions and properties of finite and infinite Markov chains in discrete and continuous time, recurrent and transient states, convergence to equilibrium, Martingales, Wiener processes and stochastic integrals with applications to biology, economics, and physics. Usually offered every second year.

Mr. Adler and Mr. Miller

**
MATH
131a
Algebra I
**

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*May not be taken for credit by students who took MATH 101a in prior years.*

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. Bernardi (fall)

**
MATH
131b
Algebra II
**

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*May not be taken for credit by students who took MATH 101b in prior years.*

Continuation of MATH 131a. Usually offered every year.

Mr. Joyner (spring)

**
MATH
139a
Differential Topology
**

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*May not be taken for credit by students who took MATH 109a in prior years.*

Introduction to the topology of smooth manifolds. Inverse/implicit function theorems, Morse theory, vector fields, Euler characteristics, intersections and transversally. Other topics may include classification of surfaces, Lefschetz fixed point theorem, and elementary knot theory. Usually offered every third year.

Staff

**
MATH
140a
Geometric Analysis
**

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*May not be taken for credit by students who took MATH 110a in prior years.*

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

Mr. Mayer (fall)

**
MATH
140b
Differential Geometry
**

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*May not be taken for credit by students who took MATH 110b in prior years.*

Riemannian metrics, parallel transport, geodesics, curvature. Introduction to Lie groups and Lie algebras, vector bundles and principal bundles. Usually offered every second year.

Mr. Adler (spring)

**
MATH
141a
Real Analysis
**

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*May not be taken for credit by students who took MATH 111a in prior years.*

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

Mr. Kleinbock (fall)

**
MATH
141b
Complex Analysis
**

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*May not be taken for credit by students who took MATH 111b in prior years.*

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. Adler (spring)

**
MATH
151a
Topology I
**

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*May not be taken for credit by students who took MATH 121a in prior years.*

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

Mr. Sultan (fall)

**
MATH
151b
Topology II
**

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*May not be taken for credit by students who took MATH 121b in prior years.*

Continuation of MATH 151a. Manifolds and orientation, cup and cap products, Poincaré duality. Other topics as time permits. Usually offered every year.

Mr. Sultan (spring)

**
MATH
180a
Combinatorics
**

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*May not be taken for credit by students who took MATH 150a in prior years.*

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.

Staff

**
MATH
180b
Topics in Combinatorics
**

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*May not be taken for credit by students who took MATH 150b in prior years.*

Possible topics include symmetric functions, graph theory, extremal combinatorics, combinatorial optimization, coding theory. Usually offered every second year.

Mr. Bernardi (spring)

#### (200 and above) Primarily for Graduate Students

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

**
MATH
200a
Second-Year Seminar
**

A course for second-year students in the PhD program designed to provide exposure to current research and practice in giving seminar talks. Students read recent journal articles and preprints and present the material. Usually offered every year.

Mr. Kleinbock (spring)

**
MATH
201a
Topics in Algebra
**

Introduction to a field of algebra. Possible topics include representation theory, vertex algebras, algebraic groups. Usually offered every year.

Mr. Aldi (fall)

**
MATH
202a
Algebraic Geometry I
**

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

Staff

**
MATH
202b
Algebraic Geometry II
**

Continuation of MATH 202a. Usually offered every second year.

Staff

**
MATH
203a
Number Theory
**

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). Usually offered every second year.

Mr. Barnet-Lamb (spring)

**
MATH
203b
Topics in Number Theory
**

Possible topics include class field theory, cyclotomic fields, modular forms, analytic number theory, ergodic number theory. Usually offered every 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.

Ms. Parker (fall)

**
MATH
205b
Commutative Algebra
**

Associated primes, primary decomposition. Filtrations, completions, graded rings. Dimension theory, Hilbert functions. Regular sequences, depth, regular local rings. Other topics as time permits. Usually offered every second year.

Mr. Bellaïche (fall)

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

Possible topics include complex manifolds, elliptic operators, index theory, random matrix theory, integrable systems, dynamical systems, ergodic theory. Usually offered every year.

Mr. Kleinbock (spring)

**
MATH
212b
Functional Analysis
**

Banach and Hilbert spaces, linear operators, operator topologies, Banach algebras. Convexity and fixed point theorems, integration on locally compact groups. Spectral theory. Other topics as time permits. Usually offered every second year.

Staff

**
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. Levine (fall)

**
MATH
221b
Topics in Topology
**

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 (spring)

**
MATH
223a
Lie Algebras
**

Theorems of Engel and Lie. Semisimple Lie algebras, Cartan's criterion. Universal enveloping algebras, PBW theorem, Serre's construction. Representation theory. Other topics as time permits. Usually offered every second year.

Mr. Igusa (fall)

**
MATH
224b
Lie Groups
**

Basic theory of Lie groups and Lie algebras. Homogeneous spaces. Haar measure. Compact Lie groups, representation theory, Peter-Weyl theorem, differential slice theorem. Complex reductive groups. Other topics as time permits. Usually offered every second year.

Staff

**
MATH
250a
Complex Algebraic Geometry I
**

Riemann surfaces, Riemann-Roch theorems, Jacobians. Complex manifolds, Hodge decomposition theorem, cohomology of sheaves, Serre duality. Vector bundles and Chern classes. Other topics as time permits. Usually offered every second year.

Mr. Lian (fall)

**
MATH
250b
Complex Algebraic Geometry II
**

Continuation of MATH 250a. Usually offered every second year.

Mr. Mayer (spring)

**
MATH
299a
Readings in Mathematics
**

Staff

**
MATH
311a
Further Topics in Analysis
**

Staff

**
MATH
399a
Readings in Mathematics
**

Staff

**
MATH
401d
Research
**

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

Staff

#### Cross-Listed in Mathematics

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BIOL
51a
Biostatistics
**

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*Prerequisite: MATH 10a.*

A basic introduction to methods of statistics and mathematical analysis applied to problems in the life sciences. Topics include statistical analysis of experimental data, mathematical description of chemical reactions, and mathematical models in neuroscience, population biology, and epidemiology. Usually offered every year.

Staff

**
BIOL
135b
The Principles of Biological Modeling
**

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*Prerequisite: MATH 10a or 10b.*

With examples from neuroscience, cell biology, ecology, evolution, and physiology, dynamical concepts of significance throughout the biological world are discussed. Simple computational and mathematical models are used to demonstrate important roles of the exponential function, feedback, stability, oscillations, and randomness. Usually offered every second year.

Mr. Miller

**
COSI
30a
Introduction to the Theory of Computation
**

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*Prerequisite: COSI 29a.*

Formal treatment of models of computation: finite automata and regular languages, pushdown automata and context-free languages, Turing machines, and recursive enumerability. Church's thesis and the invariance thesis. Halting problem and undecidability, Rice's theorem, recursion theorem. Usually offered every year.

Mr. Mairson

**
COSI
190a
Introduction to Programming Language Theory
**

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*Prerequisite: COSI 21b or familiarity with a functional programming language, set theory and logic.*

An introduction to the mathematical semantics of functional programming languages. Principles of denotational semantics; lambda calculus and its programming idiom; Church-Rosser theorem and Böhm's theorem; simply typed lambda calculus and its model theory: completeness for the full type frame, Statman's 1-section theorem and completeness of beta-eta reasoning; PCF and full abstraction with parallel operations; linear logic, proofnets, context semantics and geometry of interaction, game semantics, and full abstraction. Usually offered every second year.

Mr. Mairson

**
ECON
184b
Econometrics
**

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*Prerequisites: ECON 83a. Corequisite: ECON 80a or permission of instructor. Students must earn C- or higher in MATH 10a, or otherwise satisfy the math requirement, to enroll in this course. See "Special Notes Relating to Undergraduates" in the economics section of the Bulletin for more information. This course may not be taken for credit by students who have previously taken or are currently enrolled in ECON 185a or ECON 311a.*

An introduction to the theory of econometric regression and forecasting models, with applications to the analysis of business and economic data. Usually offered every year.

Ms. Brainerd and Mr. Pettenuzzo

**
ECON
185a
Econometrics with Linear Algebra
**

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*Prerequisites: ECON 80a, 82b, 83a and MATH 15a. A working knowledge of linear algebra is required. Does not count toward the major in economics if the student has taken ECON 184b or an equivalent course.*

Students are first exposed to the necessary background in advanced probability theory and statistics. Then statistical theory for the linear regression model, its most important variants, and extensions to nonlinear methods including Generalized Method of Moments (GMM) and Maximum Likelihood Estimation (MLE) are covered. Theoretical analysis is accompanied by the study of empirical economic examples. Usually offered every second year.

Staff

**
PHIL
106b
Mathematical Logic
**

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Covers in detail several of the following proofs: the Gödel Incompleteness Results, Tarski's Undefinability of Truth Theorem, Church's Theorem on the Undecidability of Predicate Logic, and Elementary Recursive Function Theory. Usually offered every year.

Mr. Berger

**
PHYS
100a
Classical Mechanics
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*Prerequisites: PHYS 20a or permission of the instructor.*

Lagrangian dynamics, Hamiltonian mechanics, planetary motion, general theory of small vibrations. Introduction to continuum mechanics. Usually offered every second year.

Mr. Fell

**
PHYS
110a
Mathematical Physics
**

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*Prerequisite: PHYS 30a, PHYS 31a (formerly PHYS 30b), or permission of the instructor.*

A selection of mathematical concepts and techniques useful for formulating and analyzing physical theories. Topics may include: complex analysis, Fourier and other integral transforms, special functions, ordinary and partial differential equations (including their theory and methods for solving them), group and representation theory, and differential geometry. Usually offered every second year.

Staff

**
QBIO
110a
Numerical Modeling of Biological Systems
**

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*Prerequisite: MATH 10a and b or equivalent.*

Modern scientific computation applied to problems in molecular and cell biology. Covers techniques such as numerical integration of differential equations, molecular dynamics and Monte Carlo simulations. Applications range from enzymes and molecular motors to cells. Usually offered every second year.

Mr. Hagan

#### Courses of Related Interest

Note: the following courses do not count as credit toward the major or the minor in mathematics.

**
PHIL
138b
Philosophy of Mathematics
**

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*Prerequisite: A course in logic or permission of the instructor. May not be repeated for credit by students who have taken PHIL 38b in previous years.*

Basic issues in the foundations of mathematics will be explored through close study of selections from Frege, Russell, Carnap, and others, as well as from contemporary philosophers. Questions addressed include: What are the natural numbers? Do they exist in the same sense as tables and chairs? How can "finite beings" grasp infinity? What is the relationship between arithmetic and geometry? The classic foundational "programs," logicism, formalism, and intuitionism, are explored. Usually offered every second year.

Mr. Berger or Mr. Yourgrau