Department of Mathematics

Last updated: July 10, 2014 at 1:57 p.m.

Objectives

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 be prepared to engage with the modern world.

Mathematics is, at the same time, a subject of the greatest depth and beauty with a history extending from antiquity, and a powerful tool for understanding our world. The department attempts to make this 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 offers the Master of Arts and Doctor of Philosophy degrees. The Master's program gives students a rigorous foundation in graduate-level mathematics. The doctoral program, in addition to course work, includes seminar participation, teaching and research experience, and is designed to lead to a broad understanding of the subject.

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.

Learning Goals

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 fundamental 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

How to Become a Major

Students who enjoy mathematics are urged to consider majoring 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 15a and 20a, or MATH 22a and b by the end of the sophomore year—these courses are prerequisites to the higher-level offerings. Therefore, it is important for students to start calculus and linear algebra (MATH 10a, 10b, 15a, 20a, or 22a and 22b) in the first year.

How to Be Admitted to the Postbaccalaureate and Graduate Programs

The general requirements for admission are the same as those for the Graduate School as a whole. The department has a variety of fellowships and scholarships available for well-qualified PhD students. The application deadline for PhD students is January 15. Admission of postbaccalaureate and M.A. students is rolling until the class is filled, beginning January 15. PhD and MA applications must contain three letters of recommendation; certificate applications may contain one or two letters of recommendation. Graduate Record Exam (GRE) general and subject tests are recommended, but not required.

Faculty

Daniel Ruberman, Chair
Geometric topology and gauge theory.

Mark Adler
Analysis. Differential equations. Completely integrable systems.

Joël Bellaïche (on leave fall 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, Undergraduate Advising Head
Combinatorics.

Kiyoshi Igusa, Graduate Advising Head
Differential topology. Homological algebra.

Dmitry Kleinbock (on leave academic year 2014-2015)
Dynamical systems. Ergodic theory. Number theory.

Bong Lian
Representation theory. Calabi-Yau geometry. String theory.

Alan Mayer
Classical algebraic geometry and related topics in mathematical physics.

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

Rebecca Torrey
Number theory.

Carl Wang Erickson
Number theory.

Requirements for the Minor

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 minor requirements.

E. No more than one cross-listed course may be used to satisfy the requirements for the minor.

Requirements for the Major

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.

F. No grade below a C- will be given credit toward the major, honors, or the teacher preparation track.

G. No course taken pass/fail may count towards the major, honors, or the teacher preparation track requirements.

H. No more than two cross-listed courses may be used to satisfy the requirements for the major, honors, or the teacher preparation track.

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.

Combined BA/MA Program

Undergraduate students are eligible for the BA/MA program in mathematics if they have completed MATH 131a and b; 140a; 141a and b; and 151 a and b; plus one other MATH course numbered 130 or higher, which may be a reading course with a grade of B- or better; and have demonstrated a reading knowledge of mathematical French, German, or Russian. 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.

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. 

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.

Requirements for the Postbaccalaureate Program in Mathematics

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.

Requirements for the Degree of Master of Arts

Course Requirement
The Master's program requires 8 courses. 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. Qualifying examinations in Mathematics are not required.

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.

Requirements for the Degree of Doctor of Philosophy

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 PhD 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 four 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. Usually preparation to the exam takes the form of a reading course, and an exam will consist of a paper or lecture of expository nature outlining the material learned in the course.

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 Explorations in Math: A Course for Educators
[ sn ]
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.
Ms. Parker (fall and spring)

MATH 8a Introduction to Probability and Statistics
[ qr sn ]
Discrete probability spaces, random variables, expectation, variance, approximation by the normal curve, sample mean and variance, and confidence intervals. Does not require calculus; only high school algebra and graphing of functions. Usually offered every year.
Mr. Bernardi (fall)

MATH 10a Techniques of Calculus (a)
[ sn ]
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)
[ sn ]
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
[ sn ]
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), Ms. Ray (spring)

MATH 20a Techniques of Calculus: Calculus of Several Variables
[ sn ]
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. Soroush (fall), Mr. Merrill (spring)

MATH 22a Linear Algebra and Intermediate Calculus, Part I
[ sn ]
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
[ sn ]
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. Bellaïche (spring)

MATH 23b Introduction to Proofs
[ sn wi ]
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.
Ms. Charney (fall), Mr. Gessell (spring)

MATH 28a Introduction to Groups
[ sn ]
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.
Staff

MATH 28b Introduction to Rings and Fields
[ sn ]
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. Lipsett (fall)

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. Mayer

MATH 36a Probability
[ qr sn ]
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.
Mr. Merrill (fall)

MATH 36b Mathematical Statistics
[ qr sn ]
Prerequisite: MATH 36a or permission of the instructor.
Probability distributions, estimators, hypothesis testing, data analysis. Theorems will be proved and applied to real data. Topics include maximum likelihood estimators, the information inequality, chi-square test, and analysis of variance. Usually offered every year.
Mr. Adler (spring)

MATH 37a Differential Equations
[ sn ]
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. Adler (fall)

MATH 39a Introduction to Combinatorics
[ sn ]
Prerequisites: COSI 29a or MATH 23b.
Topics include graph theory (trees, planarity, coloring, Eulerian and Hamiltonian cycles), combinatorial optimization (network flows, matching theory), enumeration (permutations and combinations, generating functions, inclusion-exclusion), and extremal combinatorics (pigeonhole principle, Ramsey's theorem). Usually offered every second year.
Mr. Bernardi (fall)

MATH 47a Introduction to Mathematical Research
[ sn wi ]
Prerequisite: MATH 23b or permission of the instructor.
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
[ sn ]
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. Lippsett (fall)

MATH 100b Introduction to Algebra, Part II
[ sn ]
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.
Ms. Ray (spring)

MATH 102a Differential Geometry
[ sn ]
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 (spring)

MATH 104a Introduction to Topology
[ sn ]
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. Igusa (fall)

MATH 108b Number Theory
[ sn ]
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
[ sn ]
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. Wang Erickson (fall)

MATH 110b Introduction to Real Analysis, Part II
[ sn ]
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.
Staff

MATH 115a Introduction to Complex Analysis
[ sn ]
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
[ sn ]
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.
Staff

MATH 131a Algebra I
[ sn ]
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. Bellaïche (fall)

MATH 131b Algebra II
[ sn ]
May not be taken for credit by students who took MATH 101b in prior years.
Continuation of MATH 131a. Usually offered every year.
Mr. Bernardi (spring)

MATH 140a Geometric Analysis
[ sn ]
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
[ sn ]
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.
Staff

MATH 141a Real Analysis
[ sn ]
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. Adler (fall)

MATH 141b Complex Analysis
[ sn ]
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. Gessell (spring)

MATH 151a Topology I
[ sn ]
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.
Ms. Ray (fall)

MATH 151b Topology II
[ sn ]
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.
Ms. Charney (spring)

MATH 180a Combinatorics
[ sn ]
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.
Mr. Gessel (spring)

MATH 180b Topics in Combinatorics
[ sn ]
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.
Staff

(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. Igusa (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. Wang Erickson (spring)

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. Bellaïche (fall)

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.
Staff

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.
Staff

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.
Mr. Adler (fall)

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.
Ms. Charney (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.
Staff

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. Cherveny (spring)

MATH 299a Readings in Mathematics
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

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 calculus requirement, to enroll in this course. 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, Mr. Pei 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

NPHY 115a Dynamical Systems
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Prerequisites: MATH 10b and MATH 15a or PHYS 20a or equivalent.
Covers analytic, computational and graphical methods for solving systems of coupled nonlinear ordinary differential equations. We study bifurcations, limit cycles, coupled oscillators and noise, with examples from physics, chemistry, population biology and many models of neurons. Usually offered every third year.
Mr. Miller

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. Lawrence

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.
Ms. Chakraborty

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