### Courses of Study

### Sections

## Department of Mathematics

Last updated: December 18, 2018 at 09:26 a.m.

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. To major in Mathematics or Applied Mathematics, one needs to take more advanced courses. Starting from the academic year 2018-2019, the Department of Mathematics offers two degrees: Bachelor of Arts in Mathematics and Bachelor of Science in Applied Mathematics. This is a testament to the fact that mathematics is, at the same time, both a subject of the greatest inherent depth and beauty with a history extending from antiquity, and also a powerful tool for understanding our world.

**Bachelor of Arts in Mathematics**

The undergraduate major introduces students to some fundamental fields of mathematics—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.

**Bachelor of Science in Applied Mathematics**

Applications of mathematics to physics, biology, chemistry, economics and social sciences have proved particularly fruitful, and have led to the development of new mathematical tools and methods. The Applied Mathematics major will introduce students to the essential tools used in such applications. It will prepare students for professional careers in public institutions, research centers or private companies using quantitative methods (such as modeling, data analysis or optimization) to understand and solve complex real-world problems.

**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 coursework, 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.

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, and business. The Mathematics Department at Brandeis serves a diverse audience, consisting of students with 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

Note that the Mathematics Department at Brandeis offers a Minor in Mathematics, but not in Applied Mathematics.

**Bachelor of Arts in Mathematics or Bachelor of Science in Applied Mathematics**

**Knowledge**

Students completing the major in mathematics will:

- Understand the fundamental concepts of mathematical proof, logic, abstraction and generalization.
- Achieve a basic knowledge of the following areas of mathematics:

- Matrices, linear algebra, and multivariable calculus.
- Analysis in one and several variables, including properties of the real numbers and of limits.
- Axiomatically defined algebraic structures, such as groups, rings, fields, and vector spaces.

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, and differential geometry.

Students completing the major in applied mathematics will:

- Gain knowledge on the fundamental objects, frameworks and theorems in applied mathematics, including the fields of probability, mathematical modeling, numerical analysis and differential equations.
- Understand the main connections between the mathematical sciences and other scientific or humanistic disciplines.
- Acquire the principles specific to applications of mathematics and use then in developing models and analyzing them rigorously. Be able to formalize and abstract a concrete problem into mathematical models, and apply mathematical concepts and reasoning to solve problems arising in other sciences or in industry.

**Core Skills**

Mathematics majors will be able to read and write mathematical proofs, abstract general principles from examples, and distinguish correct from fallacious arguments. Majors will learn to apply general principles to specific cases, solve non-routine mathematical problems, and to apply mathematics to the real world.

Applied Mathematics majors will develop their ability to:

- Understand, modify or construct mathematical models of systems arising in natural or social sciences.
- Assess their relevance, accuracy and usefulness.
- Analyze formally these models and provide relevant information on the application domain.
- Clearly communicate the results of mathematical analysis to various audiences.

**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

**Graduate Program in Mathematics**

**Master of Arts in Mathematics**

**Knowledge**

- M.A. students are required to demonstrate a broad and deep knowledge of algebra, topology, geometry, and analysis. This is done by passing with a high mark the exams of the seven fundamental courses offered every year in those fields (Algebra I and II, Topology I and II, Real Analysis, Complex Analysis, and Geometric Analysis).
- A wide array of more advanced or more specialized elective courses is also offered, as well as reading courses, and M.A. students are required to take at least two of them.
- Seminars, colloquium and special lectures are also regularly given by scholars from all over the world, and allow the students to be exposed to current-research mathematics.

**Core Skills**

Students graduating with a Master's in Mathematics at Brandeis possess a rigorous foundation in modern mathematics.

**Outcome**

Students graduating with a Master's in Mathematics are ideally prepared to apply for a Ph.D. program in pure or applied mathematics, physics, and other sciences. They also have competencies in mathematics that are in high demand in many industries, or for certain jobs in the government.

**Doctor of Philosophy in Mathematics**

**Knowledge**

- Ph.D. students are required, before they begin to work on their dissertation, to demonstrate a broad and deep knowledge in algebra, topology, geometry, and analysis. This is done by passing with a high mark the exams of the seven courses offered every year in those fields (Algebra I and II, Topology I and II, Real Analysis, Complex Analysis, and Geometric Analysis).
- A wide array of more advanced or more specialized elective courses is also offered, and students are required to take a certain number of them, according to their taste and to the needs of their progress towards their dissertation.
- Many reading courses, where one or a small group of students read a research paper or a mathematical book under the guidance of a professor, are offered, often on demand. They allow the students to acquire progressively the knowledge necessary to enter current research.
- Seminars, colloquium, and special lectures are also regularly given by scholars from all over the world, and allow the students to learn more current-research mathematics.

**Core Skills**

Students graduating with a Ph.D. in Mathematics at Brandeis:

- Have learned to read and understand research papers, both in English and another language of their choice;
- Have learned how to present mathematical materials, in particular their own results, in seminars and other expositions destined to fellow graduate students and researchers;
- Have participated, as a teaching fellow, in a structured program of undergraduate teaching, giving them the skills and experience necessary to teach successfully mathematics at various undergraduate levels;
- Have attained research expertise and completed a significant body of original research that advances a specific field of study in mathematics;
- Have written and defended a Ph.D. dissertation.

**Outcome**

Students graduating with a Ph.D. have been trained to be effective teachers and cutting-edge researchers. They may work in academia, either in a research-oriented institution or in a teaching-oriented one, in many industries, or in the government.

Students who enjoy mathematics are urged to consider majoring in either Mathematics or Applied Mathematics. Note that a student can declare a Major in Mathematics or a Major in Applied Mathematics but not both. Brandeis offers a wide variety of mathematics courses, and majors will have the benefits of small classes and individual faculty attention. For either of the majors 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.

**Joël Bellaïche, Chair
** Number theory.

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

**Olivier Bernardi, Graduate Advising Head (fall) (on leave spring 2019)**

Combinatorics.

**Corey Bregman**

Topology. Geometric group theory.

**Ruth Charney, Undergraduate Advising Head (spring)**

Geometric group theory. Topology.

**Thomas Fai**

Applied Mathematics.

**Hansol Hong**

Symplectic geometry, Lagrangian Floer theory, Homological mirror symmetry.

**An Huang**

Algebraic Geometry and Graph Theory.

**Kiyoshi Igusa**

Differential topology. Homological algebra.

**Dmitry Kleinbock, Graduate Advising Head (spring)
** Dynamical systems. Ergodic theory. Number theory.

**Bong Lian, Undergraduate Advising Head (fall) (on leave spring 2019)
** Representation theory. Calabi-Yau geometry. String theory.

**Konstantin Matveev**

Probability and Algebraic Combinatorics.

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

**Keith Merrill**

Ergodic theory. Dynamical systems. Number theory.

**Omer Offen
** Number theory and Representation theory.

**Gail Peretti**

Statistics.

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

**Rebecca Torrey**

Number theory.

**Jonathan Touboul**

Mathematical Neuroscience.

**John Wilmes**

Algorithms and Combinatorics.

**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 in Mathematics. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used. Most Math courses numbered 27 or higher require Math 23b as a prerequisite, but Math 35a, 36a, 36b, 37a, and 39a do not.

**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. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used.

**Bachelor of Arts in Mathematics**

Nine semester courses are required, including the following:

**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 in Mathematics.

**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. Only cross-listed courses in Mathematics and not in Applied Mathematics may be used.

**Bachelor of Arts in Mathematics with Honors**

A degree in Mathematics with honors requires satisfactory completion of requirements A-E, as well as:

**I.** Two additional semester courses, either MATH courses numbered 27 or higher or cross-listed courses in Mathematics.

**J.** 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 except MATH 121a, 122a and 123a. Cross-listed courses are not considered honors courses.

**K.** All courses used to satisfy the major requirements must be passed with a grade of B 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 in Mathematics. Only cross-listed courses in Mathematics and not in Applied Mathemaics may be used.

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

**H.** Completion of the High School Teacher Licensure Program.

**Bachelor of Science in Applied Mathematics**

At least twelve semester courses are required, including the following:

**A.** Five core courses: MATH 15a or MATH 22a, MATH 20a or MATH 22b, MATH 36a, MATH 36b, and MATH 40a. Note that MATH 23b (or exemption) is recommended but not required.

**B.** MATH 35a, MATH 37a, and MATH 121a.

**C.** One of the following: MATH 122a, COSI 177a or COSI 178a.

**D.** One of the following: MATH 123a or NBIO 136b.

**E.** One of the following: MATH 39a, MATH 115a, MATH 126a, BCHM 145a, COSI 123a, NPHY 115a or PHYS 110a.

**F.** **At least two courses must be taken from another department in the School of Science or Economics from the following list: BCHM 102a, BCHM 104a, COSI 130a, COSI 180a, ECON 181b, ECON 184b, PHYS 100a, PHYS 163b. The non-math classes used to satisfy the requirements A-E may double count towards this requirement.**

**G.** No grade below a C- will be given credit toward the Bachelor of Science degree.

**H.** No course taken pass/fail may count towards the Bachelor of Science degree.

**Bachelor of Science in Applied Mathematics with Honors**

A degree in Applied Mathematics with honors requires satisfactory completion of all of the above requirements, as well as:

**I.**Passing all the courses used to satisfy the BS with at least a grade of B.

**J.** Completion and defense of a senior honors thesis. Students considering this option should enroll in MATH 99a and MATH 99b (Independent Research).

Undergraduate students are eligible for the BA/MA program in mathematics if they have completed MATH 131a and b; 140a; 141a and b; and 151a and b; plus one other MATH course (or readings course) numbered 130 or higher, with a grade of B- or better. In addition, students must fulfill a minimum of three years' residence on campus. A student must make an appointment with the Undergraduate Advising Head in the Department of Mathematics in order to add the BA/MA to his/her program. This must be done no later than May 1 preceding his/her final year of study on campus.

**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 more than one of MATH 28a and 100a or MATH 28b or 100b.

**E.** Students should normally take MATH 23b before taking upper-level pure mathematics courses (i.e., those which require 23b as a prerequisite). 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:

- MATH 22a and b.
- MATH 100a and b.
- MATH 35a or 110a and b.
- MATH 115a.
- Other courses numbered 100 or higher.

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

- Offered every semester are MATH 5a, 8a, 10a and b, 15a, 20a, and 23b.
- Offered once each year are MATH 22a and b, 35a, 36a and b, 37a, 40a, 100a, 110a, 115a, 121a, 122a, 123a.
- In addition, the following semester courses are usually offered every second year according to the following schedule:

a. even-odd years (e.g., 2018-2019): MATH 3a, 28a, 100b, 102a, and 108b.

b. odd-even years (e.g., 2019-2020): MATH 28b, 39a, 104a, 110b, and 126a.

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

**B.** A grade below a C will not count towards the post-baccalaureate program.

**C.** 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: MATH 131a and b, 141a and b, and 151a and b, 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.

**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. Students in their second and third years are required to take at least three lecture courses per semester. Students may count a reading course towards the total each semester, but Math 204A T.A. Practicum does not count. 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 PhD students are required to take the Division of Science Responsible Conduct of Research (RCR) workshop, offered in the spring. Students in their first year of study may wait until their second year to fulfill this requirement.

**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.

#### (1-99) Primarily for Undergraduate Students

**
MATH
3a
Explorations in Math: A Course for Educators
**

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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 year.

Staff

**
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.

Rebecca Torrey

**
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 semester.

Gail Peretti

**
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.

Keith Merrill (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.

Keith Merrill (fall), Rebecca Torrey (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.

Keith Merrill (fall) Omer Offen (spring)

**
MATH
20a
Multi-variable Calculus
**

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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.

Daniel Ruberman (fall) Staff (spring)

**
MATH
22a
Honors Linear Algebra and Multi-variable 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.

Bong Lian (fall)

**
MATH
22b
Honors Linear Algebra and Multi-variable 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.

Dmitry Kleinbock (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.

Corey Bregman (fall), Ruth Charney and Konstantin Matveev (spring)

**
MATH
28a
Introduction to Groups
**

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*Prerequisites: MATH 23b and either MATH 15a or 22a, or permission of the instructor. Students may take MATH 28a or 100a for credit, but not both.*

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
**

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*Prerequisites: MATH 23b and either MATH 15a, 22a, or permission of the instructor. Students may take MATH 28b or 100b for credit, but not both.*

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.

Kiyoshi Igusa (fall)

**
MATH
35a
Advanced Calculus and Fourier Analysis
**

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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.

Olivier Bernardi (fall)

**
MATH
36a
Probability
**

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*Prerequisite: MATH 20a or 22b.*

Sample spaces and probability measures, elementary combinatorial examples. Conditional probability. Random variables, expectations, variance, distribution and density 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.

Mark Adler (fall)

**
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.

Keith Merrill (spring)

**
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.

Staff

**
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.

Staff

**
MATH
40a
Introduction to Applied Mathematics
**

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

Introduces the problems and issues of applied mathematics, with emphasis on how mathematical ideas can have a major impact on diverse fields of human inquiry. Usually offered every year.

Thomas Fai (spring)

**
MATH
47a
Introduction to Mathematical Research
**

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*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 year.

Staff

**
MATH
98a
Independent Research
**

Usually offered every year.

Staff

**
MATH
98b
Independent Research
**

Usually offered every year.

Staff

**
MATH
99a
Senior Research
**

Usually offered every year.

Staff

**
MATH
99b
Senior Research
**

Usually offered every year.

Staff

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

Courses numbered 131 and above are ordinarily taken by graduate students; interested undergraduates 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 15a or 22a, or permission of the instructor. Students may take MATH 28a or 100a for credit, but not both. *

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

An Huang (fall)

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

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*Prerequisite: MATH 100a or permission of the instructor. Students may take MATH 28b or 100b for credit, but not both.*

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

Konstantin Matveev (spring)

**
MATH
102a
Introduction to Differential Geometry
**

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]

*Prerequisites: MATH 23b and either MATH 20a or 22b or permission of the instructor.*

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.

Mark Adler (spring)

**
MATH
104a
Introduction to Topology
**

[
sn
]

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

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

Staff

**
MATH
108b
Introduction to Number Theory
**

[
sn
]

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

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

Omer Offen (fall)

**
MATH
110a
Introduction to Real Analysis, Part I
**

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sn
]

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

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

Alan Mayer (fall)

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

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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. *

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

Alan Mayer (spring)

**
MATH
121a
Mathematics for Natural Sciences
**

[
sn
]

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

Introduces a set of mathematical tools of great applicability to the natural sciences. It will prepare students to use these tools in concrete applications. Topics include complex numbers, power series, calculus of variations, and Laplace transform. Usually offered every second year.

Jonathan Touboul (spring)

**
MATH
122a
Scientific Computing
**

[
sn
]

*Prerequisites: MATH 15a or MATH 22a and MATH 20a or MATH 22b, and MATH 40a.*

Introduces some of the widely used techniques in scientific computing through examples chosen from physics, chemistry, and biology. The purpose of this course is to introduce methods that are useful in applications and research. Usually offered every second year.

Thomas Fai

**
MATH
123a
Principles of Mathematical Modeling and Applications to Biology
**

[
sn
]

*Prerequisites: MATH 15a or MATH 22a, MATH 20a or MATH 22b, and MATH 37a.*

Provides the basic concepts and approaches for modelling in physics and biology. The course will be developed around examples of central research interest in biology and related fields. Usually offered every second year.

Jonathan Touboul (fall)

**
MATH
124a
Convex Optimization
**

[
sn
]

*Prerequisites: MATH 15a or MATH 22a, MATH 20a or MATH 22b, and COSI 21a or MATH 23b.*

Introduces fundamental algorithms for linear programming and convex optimization, and studies their global convergence and time complexity. Topics include duality, the simplex algorithm, gradient descent, and interior-point methods. Usually offered every year.

John Wilmes

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

[
sn
]

*Prerequisites: MATH 15a, 20a, and 36a. *

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
]

*Prerequisites: MATH 100a and 100b or permission of the instructor. *

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.

Olivier Bernardi (fall)

**
MATH
131b
Algebra II
**

[
sn
]

*Prerequisite: MATH 131a or permission of the instructor. *

Continuation of MATH 131a. Usually offered every year.

Kiyoshi Igusa (spring)

**
MATH
140a
Geometric Analysis
**

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sn
]

*Prerequisites: MATH 110a and 110b or permission of the instructor. *

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

Alan Mayer (fall)

**
MATH
140b
Differential Geometry
**

[
sn
]

*Prerequisite: MATH 140a or permission of the instructor. *

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

Daniel Ruberman (spring)

**
MATH
141a
Real Analysis
**

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sn
]

*Prerequisites: MATH 110a and 110b or permission of the instructor. *

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

Dmitry Kleinbock (fall)

**
MATH
141b
Complex Analysis
**

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sn
]

*Prerequisite: MATH 141a or permission of the instructor. *

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.

Alan Mayer (spring)

**
MATH
151a
Topology I
**

[
sn
]

*Prerequisite: MATH 104a or permission of the instructor. *

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

Ruth Charney (fall)

**
MATH
151b
Topology II
**

[
sn
]

*Prerequisite: MATH 151a or permission of the instructor. *

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

Corey Bregman (spring)

**
MATH
180a
Combinatorics
**

[
sn
]

Emphasis on enumerative combinatorics. Generating functions and their applications to counting graphs, paths, permutations, and partitions. Bijective counting, combinatorial identities, Lagrange inversion and Möbius inversion. Usually offered every second year.

Staff

**
MATH
180b
Topics in Combinatorics
**

[
sn
]

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

Konstantin Matveev (fall)

#### (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.

Dmitry 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.

Staff

**
MATH
202a
Algebraic Geometry I
**

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

Joël Bellaïche (fall)

**
MATH
202b
Algebraic Geometry II
**

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

An Huang (spring)

**
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.

Omer Offen (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. May not be counted toward one of the lecture courses that is required in the second and third years. Usually offered every year.

Rebecca Torrey (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.

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.

Mark Adler (spring)

**
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.

Daniel Ruberman (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.

Staff

**
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.

An Huang (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.

Staff

**
MATH
250b
Complex Algebraic Geometry II
**

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

Staff

**
MATH
299a
Readings in Mathematics
**

Staff

**
MATH
399a
Advanced 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

**
COSI
130a
Introduction to the Theory of Computation
**

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sn
]

*Prerequisite: COSI 29a. May not be taken for credit by students who took COSI 30a in prior years.*

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.

James Storer

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

[
sn
]

*Prerequisite: COSI 121b 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.

Staff

**
ECON
184b
Econometrics
**

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qr
ss
]

*Prerequisites: ECON 83a. Corequisite: ECON 80a or permission of the instructor. Students must earn a 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.

Elizabeth Brainerd, Linda Bui, and Davide Pettenuzzo

**
NPHY
115a
Dynamical Systems
**

[
sn
]

*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.

Irving Epstein

**
PHIL
106b
Mathematical Logic
**

[
hum
sn
]

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.

Berislav Marušić

**
PHYS
100a
Classical Mechanics
**

[
sn
]

*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.

Richard Fell

**
PHYS
110a
Mathematical Physics
**

[
sn
]

*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

#### Cross-Listed in Applied Mathematics

**
BCHM
102a
Quantitative Approaches to Biochemical Systems
**

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sn
]

*Prerequisite: BCHM 100a or equivalent and Math 10a and b or equivalent.*

Introduces quantitative approaches to analyzing macromolecular structure and function. Emphasizes the use of basic thermodynamics and single-molecule and ensemble kinetics to elucidate biochemical reaction mechanisms. Also discusses the physical bases of spectroscopic and diffraction methods commonly used in the study of proteins and nucleic acids. Usually offered every year.

Maria-Eirini Pandelia

**
BCHM
104a
Classical and Statistical Thermodynamics
**

[
sn
]

*Prerequisites: MATH 10a,b or equivalent, PHYS 11 or 15.*

Covers basics of physical chemistry underpinning applications in BCHM 104b. Focus is placed on quantitative treatments of the probabilistic nature of molecular reality: molecular kinetic theory, basic statistical mechanics, and chemical thermodynamics in aqueous solution. Not offered in 2013-2014.

Douglas Theobald

**
BCHM
145a
How to Decide: Bayesian Inference and Computational Statistics
**

[
sn
]

*Prerequisites: Math 10a and b.*

A calculus-based courses that teaches the theory and practice of modern statistical methods used by experimental scientists. Topics include Bayesian inference, maximum likelihood estimation, and computational resampling methods. The course consists of a mixture of small lectures and in-class computational exercises. Usually offered ever third year.

Jeff Gelles and Douglas Theobald

**
COSI
123a
Statistical Machine Learning
**

[
qr
sn
]

*Prerequisite: COSI 29a and MATH 10a.*

Focuses on learning from data using statistical analysis tools and deals with the issues of designing algorithms and systems that automatically improve with experience. This course is designed to give students a thorough grounding in the methodologies, technologies, mathematics, and algorithms currently needed by research in learning with data. Usually offered every year.

Pengyu Hong

**
COSI
132b
Networked Information Systems
**

[
ss
]

*Prerequisite: COSI 131a.*

Explores the fundamental concepts in design and implementation of networked information systems, with an emphasis on data management. In addition to distributed information systems, we will also study modern applications involving the web, cloud computing, peer-to-peer systems, etc. Usually offered every second year.

Olga Papaemmanouil

**
COSI
177a
Scientific Data Processing in Matlab
**

[
sn
]

*Prerequisite: COSI 11a. MATH 15a is recommended.*

Introduces scientific computing using Matlab. Programming concepts such as data types, vectors, conditional execution, loops, procedural abstraction, modules, APIs are presented. The course will present scientific techniques relevant to computational science, with an emphasis on image processing. Usually offered every second year.

Antonella DiLillo

**
COSI
178a
Computational Molecular Biology
**

[
sn
]

*Open to advanced undergraduate students and graduate students.*

Information and computing technologies are becoming indispensable to modern biological research due to significant advances of high-throughput experimental technologies in recent years. This course presents an overview of the systemic development and application of computing systems and computational algorithms/techniques to the analysis of biological data, such as sequences, gene expression, protein expression, and biological networks. Hands-on training will be provided. Usually offered every other year.

Pengyu Hong

**
COSI
180a
Algorithms
**

[
sn
]

*Prerequisites: Undergraduates and combined BA/MA students must complete COSI 21a and COSI 29a, and received a B+ or better. Other graduate students should have a background that includes at least what is covered in these two courses. *

Basic concepts in the design and analysis of algorithms. Usually offered every second year.

James Storer

**
ECON
181b
Game Theory and Economic Applications
**

[
ss
]

*Prerequisites: ECON 80a, ECON 83a, and MATH 10a or equivalent.*

Analysis of decision making in multiperson settings. Studies models of equilibrium and various kinds of games under perfect and imperfect information. The applications include business strategy and competition, auctions, and risk sharing. Usually offered every second year.

Nelson Sa

**
ECON
184b
Econometrics
**

[
qr
ss
]

*Prerequisites: ECON 83a. Corequisite: ECON 80a or permission of the instructor. Students must earn a 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.

Elizabeth Brainerd, Linda Bui, and Davide Pettenuzzo

**
NBIO
136b
Computational Neuroscience
**

[
sn
]

*Prerequisite: MATH 10a and either NBIO 140b or PHYS 10a or approved equivalents.*

An introduction to concepts and methods in computer modeling and analysis of neural systems. Topics include single and multicompartmental models of neurons, information representation and processing by populations of neurons, synaptic plasticity and models of learning, working memory, decision making and neural oscillations. The course will be based on in-class computer tutorials, assuming no prior coding experience, with reading assignments and preparation as homework. Usually offered every second year.

Paul Miller

**
NPHY
115a
Dynamical Systems
**

[
sn
]

*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.

Irving Epstein

**
PHYS
100a
Classical Mechanics
**

[
sn
]

*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.

Richard Fell

**
PHYS
110a
Mathematical Physics
**

[
sn
]

*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

**
PHYS
163b
Principles of Soft Materials Theory
**

[
sn
]

*Prerequisite: PHYS 163a or the equivalent.*

Introduces non equilibrium statistical mechanics and applications of equilibrium and non equilibrium statistical mechanics to understanding emergent phenomena in soft materials such as colloids, polymers and liquid crystals. Usually offered every second year.

Staff

#### Courses of Related Interest

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

**
BIOL
51a
Biostatistics
**

[
sn
]

An introductory level biostatistics class providing an overview to statistical methods used in biological and medical research. Topics include descriptive statistics, elementary probability theory, basic concepts of statistical inference, hypothesis testing, regression and correlation methods, as well as analysis of variance. Emphasis will be on applications to medical problems. Usually offered every year.

Staff

**
BIOL
107a
Data Analysis and Statistics Workshop
**

[
qr
sn
]

The interpretation of data is key to making new discoveries, making optimal decisions, and designing experiments. Students will learn skills of data analysis through hands-on, computer-based tutorials and exercises that include experimental data from the biological sciences. Knowledge of very basic statistics (mean, median) will be assumed. Usually offered every second year.

Stephen Van Hooser

**
PHIL
138b
Philosophy of Mathematics
**

[
hum
]

*Prerequisite: A course in logic or permission of the instructor.*

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.

Palle Yourgrau