Mathematical Sciences at Harvard
2008 — 2009
Information for undergraduates about courses, concentrations, and resources in mathematics, applied mathematics, computer science, and statistics.
Contents
A description of the four undergraduate concentrations in the mathematical sciences: Mathematics, Applied Mathematics, Computer Science, and Statistics
A guide to the introductory and to some intermediate courses in the mathematical sciences offered by the Department of Mathematics, the School of Engineering and Applied Sciences, the Department of Statistics, and others
A. Introductory Mathematics
B. IntermediateLevel Mathematics
C. Computer Science
D. Statistics
E. Alphabetical List of Courses Described in Chapter II
Information about various activities and resources
A. Lectures, Colloquia, and Clubs
B. Computing Services and Facilities
C. Libraries and Databases
Sample programs satisfying concentration and honors requirements
A. Mathematics
B. Applied Mathematics
C. Computer Science
D. Statistics
Names, addresses, telephone numbers, and email addresses
Although some students who concentrate in fields such as Economics, Engineering Sciences, or Physics use the mathematical sciences extensively, most students interested in pursuing the mathematical sciences in depth concentrate in one of the four fields described in this pamphlet  Mathematics, Applied Mathematics, Computer Science, and Statistics.
The concentration in Mathematics, offered by the Department of Mathematics, serves a variety of interests and goals. A major concern of mathematics is understanding the beautiful and profound concepts that lie behind our understanding of numbers, space, and the quantitative relationships between them. These ideas and their generalizations form the basis for some of the major fields of mathematics  algebra, geometry, topology, and analysis. The logical structure of these fields, and the unity and generality of the conclusions that can be drawn from them, are central to pure mathematics; students who wish to explore these fundamental ideas in depth should major in Mathematics. The broad outlook and reasoning skills provided by a grounding in pure mathematics may also be valuable to students whose principal interests lie elsewhere: for example, in mathematical physics, computer science, or probability and statistics. Harvard has long been a world center for mathematics and the Department offers students excellent opportunities for studying the subject and glimpsing current research frontiers.
Applied Mathematics, an interdepartmental concentration offered by the School of Engineering and Applied Sciences, also serves many roles. Students concentrate in Applied Mathematics because they wish to understand the mathematical concepts and techniques used to analyze, explain, or predict information (whether it relates to scientific and engineering measurements or the less precise data that describe certain aspects of our social, economic or ecological environment); to study relations between models and observations; or to examine the mathematical foundations and limitations of models and techniques and develop extensions. Undergraduates who are interested in applicationoriented mathematical fields such as numerical analysis or decision and systems analysis major in Applied Mathematics, along with many interested in the more analytical subfields of the natural and social sciences, computer science, statistics, and engineering.
The goal of the concentration in Computer Science, offered by the School of Engineering and Applied Sciences, is to give students a broad understanding of the concepts underlying computational systems and processes and of their applications. Mathematical concepts provide a core underpinning to the rapidly developing science of computation, and the program stresses how these concepts are utilized within computer science and their relevance and applications of computers to other disciplines in the natural sciences and engineering. The concentration is intended not only for those who plan graduate careers in computer science and engineering but also for students headed toward many of the professions (and other academic disciplines) in which a thorough knowledge of computing is central for developing, organizing, and transmitting information.
The concentration in Statistics, offered by the Department of Statistics, aims to give students an understanding of the quantitative methods for the analysis of data, the making of rational decisions under uncertainty, the design of experiments, and the modeling of randomness in the social and natural sciences. Most students concentrating in the mathematical sciences will want to take at least one course in probability or statistics. A concentration in Statistics prepares a student for careers in industry and government, for graduate study in a very broad array of social and natural sciences, and for professional study in law, medicine, business, or public administration.
It should be emphasized that an early choice of concentration need not be a final one. Although advanced courses differ in subject matter and approach, concentrators in the four programs described above, as well as in engineering and several of the natural sciences, start with the same mathematics courses. Students can therefore quite easily design course programs that will permit them to transfer from one program to another at the end of the sophomore year or even later. Changes to fields that require a sophomore tutorial, such as Philosophy or Economics, are more difficult but not impossible.
Because the ability to reason carefully and logically and the quantitative skills developed in the mathematical sciences are universally valued, these four programs draw many concentrators who do not plan careers in the mathematical sciences. Some concentrators enter graduate school in fields that range from biology to engineering, and from economics to sociology; many others go on to professional careers in law, medicine, or business.
To accommodate those students who want to teach mathematics in the public school system after graduation, the Mathematics Department offers a subconcentration "Mathematics and Teaching." Students choosing this option must be simultaneously enrolled in the Undergraduate Teacher Education Program (UTEP). More information about this option can be obtained from Professor Peter Kronheimer, Director of Undergraduate Studies in Mathematics (4955745), Svetlana Alpert in the Department of Mathematics (4959116), or the Undergraduate Teacher Education Program, Harvard Graduate School of Education, Longfellow Hall, 13 Appian Way, Cambridge, MA 02138, 4952783, utep@fas.harvard.edu, www.fas. harvard.edu/~utep/.
The requirements for all four concentrations can be found in Fields of Concentration (contained in the Handbook for Students). In Chapter IV of this pamphlet some of the types of programs concentrators complete are illustrated. However, students contemplating their choice of concentration should not rely on these sources, but should seek out more detailed information and advice available from each concentration. Joint concentrations involving one or more of the mathematical sciences programs are also possible.
Computer Science, Mathematics, and Statistics all offer both basic and honors programs while Applied Mathematics offers only an honors program. To qualify for Honors in Mathematics or in Statistics a Senior Thesis is required. Honors in Applied Mathematics and Computer Science may be awarded on the basis of distinguished course records alone, but all honors candidates are encouraged to participate in research by undertaking an individual project or writing a Senior Thesis. For ground rules regarding High or Highest Honors consult concentration literature and representatives.
A. Introductory Mathematics
Identifying the mathematics course best suited to your background and interests is not always easy. Starting with the proper course — one that is neither too elementary nor too advanced for you  is a very important matter. Take it seriously and solicit advice. You may obtain advice from several sources: the informational meetings and individual counseling for freshmen at the Science Center the week before classes begin; individual faculty from the Mathematics Department and the School of Engineering and Applied Sciences who are available for consultation before the fall term begins. Your scores on the Harvard Mathematics Placement Test, which is designed to measure preparation rather than ability, will provide some indication of where you should start. Representatives of the Mathematics Department are available to assist in the interpretation of these scores, whose significance has been validated by long experience.
If you have just come to Harvard and are considering enrolling in a Mathematics course beyond the level of 21a55a, you should definitely confer with the Director of Undergraduate Studies in Mathematics. No more than a handful of students entering each year will have had sufficient training in the calculus to start work in some other area of mathematics. The vast majority will choose one of the introductory courses listed
Introductory halfcourses 
Mathematics Xa (F) 
First and second level halfcourses in calculus 
Mathematics Xb (S) 
Third and higher level half courses in calculus 
Mathematics 19a (F&S) 
in this section. (Throughout this chapter, the semester in which a course is offered is indicated as follows: (F) denotes a fallterm course, (S) a springterm course, and ([]) a course not offered in 2008–2009.) Any entering student considering enrollment in an Applied Mathematics course numbered above 100 should consult the Director of Undergraduate Studies in Applied Mathematics.
Mathematics Xa (F) and Xb (S) form a fullyear sequence. They cover a semester of precalculus material and a semester of calculus in an integrated format. Students who enroll in Mathematics Xa in the fall should enroll in Mathematics Xb the next semester. After Mathematics Xb, students are ready for Mathematics 1b.
If you have had the usual secondary school courses in algebra, geometry, and trigonometry  and you can now handle the concepts and manipulations these courses cover  you are ready for Mathematics 1a (the standard first course in calculus). If you are in doubt, your score on the first part of the Mathematics Placement Test should give you a good idea of whether your preparation is adequate.
If you have already started to study calculus, you should exercise care in finding the right course since secondary school calculus curricula vary considerably. The second part of the Mathematics Placement Test is intended to help you find the calculus course that is most suitable for you. Many students should start with Mathematics 1b, although some will be ready for Mathematics 21a or Applied Mathematics 21a. Note that Mathematics 1b covers more than what is tested on the BC Calculus Advanced Placement Examination. Some students with high BC Calculus scores who skip Mathematics 1b may find themselves unprepared for calculus applications in other courses, especially in Applied Mathematics and Physics courses.
Both Mathematics 21a,b (F&S) and Applied Mathematics 21a,b (F, S) are aimed towards students in the natural and applied sciences, including applied mathematics, engineering, earth and environmental sciences, chemistry, biology, and physics. Topics in Mathematics 21a and Applied Mathematics 21a are closely parallel; they are coordinated with Physics 11a and 15a so that they may be taken concurrently. Both Mathematics 21b and Applied Mathematics 21b study linear algebra and differential equations, from different perspectives. Students who begin the calculus sequence with Mathematics 1b in the fall term of their first year should normally continue with Mathematics 21a in the spring term. Mathematics 21a has special sections (in Fall and Spring) geared towards students who are thinking of concentrating in Physics, in Biochemistry and in the Social Sciences. In the spring term, Mathematics 21b will have a special section for students with Biology, Biostatistics and Statistics interests. Applied Mathematics 21 has a different format and a somewhat more applicationsmotivated focus than does Mathematics 21; except in some pure mathematics courses, it can serve as a prerequisite wherever Mathematics 21 is called for.
Mathematics 19a,b are courses that are designed for students concentrating in the life sciences, chemistry and the environmental sciences. (These courses are recommended over Math 21a,b by the various life sciences, environmental science, and chemistry concentrations.) Math 19a focuses on differential equations, related techniques and modeling with applications to the life sciences. Math 19b teaches linear algebra, probability and statistics with a focus on life science examples and applications. Math 19a can be taken either before or after Math 21a,b. Math 19b requires some multivariable calculus background, and should not be taken with Math 21b. If you passed Mathematics 1b (or have the permission of the instructor), you can take Mathematics 19a,b.
Mathematics 20 is a onesemester course, which covers selected topics from Mathematics 21a and 21b for students particularly interested in economic and social science applications. Mathematics 20 is excellent preparation for an economics concentration; but students wishing to preserve the option to concentrate in Applied Mathematics specializing in mathematical economics should take Mathematics 21a,b or Applied Mathematics 21a,b to obtain sufficient background for later mathematics courses. If you received a grade of 5 on AB Calculus or 3 on the BC Calculus Advanced Placement Examination, you should be adequately prepared for Mathematics 20.
Accelerated courses in calculus and linear algebra are Mathematics 23, 25 and 55. Only students with a thorough grasp of firstyear calculus and a strong interest in and enjoyment of mathematics should elect these courses. Mathematics 23 treats the topics in Mathematics 21 from a more abstract and rigorous point of view. It develops proofbased mathematical tools that go well beyond Mathematics 21. It is not correlated closely with Physics 11 or 15. Mathematics 23 requires more sophistication than Mathematics 21 but is less homework intensive than Mathematics 25 or Mathematics 55.
Mathematics 25 is a rigorous course in and beyond calculus; it introduces students to the techniques of abstraction and proof on which pure mathematics builds. Some preparation beyond firstyear calculus is helpful, but not necessary. Mathematics 55 is very advanced, and is designed for students who already have significant experience in collegelevel mathematics. Permission of the instructor is mandatory for Mathematics 55. Both courses meet for three lectures and one section each week and require extensive additional work.
B. IntermediateLevel Mathematics
1. Applicable Mathematics
There are two categories of intermediatelevel mathematics courses. The first consists of courses offered by the Department of Mathematics and the School of Engineering and Applied Sciences that cover areas of widely applicable mathematics. These courses are specifically intended to serve students who are not concentrating in the mathematical sciences as well as those who are. They are open to anyone who has passed Mathematics 19a,b Mathematics 21a,b, Applied Mathematics 21a,b, Mathematics 23a,b, Mathematics 25a,b, or Mathematics 55a,b. (Those who have completed Mathematics 25b or 55b would find that they have covered much of Mathematics 101, 112, 121 and 152, and should not enroll in these courses.)
They treat both the mathematical structure of a field and associated calculational techniques. Proofs are presented but they are not the focus of these courses. Mathematics 101, 106, 113, 115, 121 and 154, and Applied Mathematics 105a, 105b, 106, 107, 111 (described in Sec. C.3), 115, 120, and 147 fall in this category. Students whose last mathematics course was Mathematics 21b may find many of the 100level Mathematics courses difficult if they had no additional prior exposure to proofs. A good grounding in proofbased reasoning can be obtained in Mathematics 23, 25, 55, 101, 112, or 121. Experience with proofs can also be obtained in Mathematics 152 although this is not its purpose.
Applied Mathematics 50 provides an introduction to the problems and issues of applied mathematics. This will be accomplished both through the reading of papers that use mathematical arguments to have substantial impact on some field of human activity; as well as guest lecturers from around Harvard to discuss how mathematics is used in their field.
Mathematics 101 (S) is designed to introduce students who have little prior experience with careful mathematical proofs to the three main branches of abstract mathematics (geometry, analysis, and algebra) and to teach them to read and write proofs. Mathematics concentrators who have taken Mathematics 21 are strongly encouraged to take Mathematics 101 before enrolling in other 100level Mathematics courses.
Mathematics 106 ([]) studies analytical, numerical and qualitative approaches to understanding linear and nonlinear ordinary differential equations. Applications to biology, physics and the social sciences are stressed.
Mathematics 113 (S) is a course in analytic functions of one complex variable. The theory of these functions is perhaps the greatest achievement of 19th century mathematics and a source of techniques and applications that pervade applied mathematics, engineering, and physics. The theory of complex functions combines great beauty and elegance and is used often in the sciences.
Applied Mathematics 105a (F) also treats complex variables (including mapping, integration, and branch cuts). In addition it covers Fourier series, Fourier and Laplace transforms, the Fast Fourier Transform, the solution of linear differential equations by convolution, series and transforms, and elementary probability theory.
Applied Mathematics 105b (S)  which may be taken before or after Applied Mathematics 105a or Mathematics 113  discusses ordinary differential equations, power series solutions, special functions and eigenfunction expansions, the algebra and calculus of dyadics and tensors, and elementary partial differential equations, including separation of variables and series solutions.
Applied Mathematics 147 (S) studies nonlinear ordinary differential equations, and their applications to electrical, mechanical, chemical and biological systems.
Students who wish to study the application of mathematics to human endeavors 
economics, engineering systems, and rational decision making  are also urged to explore Applied Mathematics 121 (S) and the graduate courses in Decision and Control, Engineering Sciences 201 (S), 202 (F), and 210 (S).
Mathematics 116 (F) is a course on convexity and real analysis that introduces these topics via convex programming, optimal control theory and the calculus of variations.
Mathematics 121 (F) and Applied Mathematics 120 (F) are courses in linear algebra that pick up where Mathematics 21b leaves off. Linear algebra is used in all mathematical subjects, including economics and other social sciences. Mathematics 121 covers the theory of vector spaces and linear transformations and some of their applications to economics and physics; it stresses the coordinatefree approach to vector spaces. Applied Mathematics 120, offered in alternate years, takes an algorithmic approach to the study of matrix theory and of its applications. Both courses discuss linear equations, linear inequalities, optimization, eigenvalues and singular values. Although the two courses treat many similar topics, they approach them quite differently. Students who have completed Mathematics 23b, 25b or 55b should not enroll in Mathematics 121, but may enroll in Applied Mathematics 120, if appropriate.
Mathematics 152 (F) introduces finite groups, finite fields, finite geometry, discrete probability and graph theory in order to study symmetries of threedimensional objects.
Mathematics 153 (F) is a course on Mathematical Biology and Evolutionary Dynamics.
Applied Mathematics 106 ([]) is an introductory course on algebra. It treats the basic algebraic structures  groups, rings, lattices, and fields  that appear throughout the mathematical sciences, using illustrative examples from automata, encryption, coding, and other applied areas. Mathematics 122 (described below) covers similar mathematical material but places more emphasis on proofs and on concepts that appear throughout pure mathematics. (Mathematics 122 and Applied Mathematics 106 have substantial overlap and both may not ordinarily be taken for credit.)
Applied Mathematics 107 (S) surveys topics in discrete and combinatorial applied mathematics important in computer science. Topics include graph theory, discrete probability theory, and techniques for enumerating combinatorial objects.
Applied Mathematics 106 and 107 provide a firm grounding in important areas of applicable mathematics not treated in the courses discussed in Sec. A.
Applied Mathematics 115 (F&S) is concerned with the creation, analysis and critical evaluation of mathematical models. Problems from a variety of scientific contexts are approached from different relevant mathematical perspectives, without presuming any specific scientific or mathematical preparation. Ideally taken in the junior year by students with backgrounds typical of Applied Mathematics concentrators, subjects studied may prove to be a fruitful source of potential senior thesis topics. The course may also serve as a valuable capstone experience for seniors who do not undertake an individual project or thesis.
Applied Mathematics 91r is a lettergraded individual or small group supervised reading and research course culminating in a substantial expository or research paper on topics of mutual interest to faculty and students. Applied Mathematics 99r facilitates preparation of a senior thesis. Both presume that relevant lecture courses will have been taken previously.
Mathematics 115 (F), is an intensive course in the techniques of mathematical analysis. This course, which is designed for Physics concentrators, is also recommended to Mathematics concentrators with an interest in analysis. Mathematics 115 covers some complex function theory, classical methods for solving linear partial differential equations with Fourier series and other eigenfunction expansions, and the calculus of variations.
2. Abstract Mathematics
The Mathematics Department also offers a second category of intermediatelevel courses for its concentrators and other undergraduates. These courses deal with more abstract topics and study mathematical structures primarily for their intrinsic beauty and elegance. Abstract mathematics has its own technical terminology and concepts; few are even mentioned in standard secondary school courses and not many figure significantly in elementary calculus courses. Consequently, for most students modern pure mathematics is an entirely new mode of thought.
The intermediatelevel mathematics courses are the principal entrance to this domain. (Mathematics 23, 25 and 55, discussed above, also are presented abstractly and fall in this category.) Some of these courses require a knowledge of calculus and some do not. Most students who have had little exposure to abstract mathematics will have difficulty orienting themselves in any of these courses even if they have studied calculus.
Topics in analysis are covered in the 110series of courses:
Mathematics 112 (S) covers the foundations of the real number system and the theory of metric spaces and uses these ideas to treat rigorously infinite series, improper and multiple integrals and the existence and uniqueness theorem for differential equations. Students who have completed Mathematics 25b or 55b should not enroll in this course.
Mathematics 114 (F) covers measure theory, which underpins the mathematical theory of integration, as well as topics in functional analysis and related aspects of topology.
Mathematics 118r (S) studies dynamical systems, beginning with existence and uniqueness theorems for flows, and moving on to examine equilibria, attractors, the behavior of iterated maps, and bifurcation theory.
The other courses in this series, Mathematics 113, 115, and 116 are described above.
Topics in algebra are covered in the 120series of courses. Aside from Mathematics 121 mentioned earlier, these are:
Mathematics 122 (F), mentioned above, deals with the fundamental concepts of algebra from a rigorous viewpoint. (Mathematics 122 and Applied Mathematics 106, described above, ordinarily cannot both be taken for credit.)
Mathematics 123 (S), a continuation of Mathematics 122, emphasizes Galois theory. Among the applications of Galois theory studied is the famous theorem that 5th degree equations are not solvable by radicals.
Mathematics 124 (F) is a course in number theory, the arithmetic of ordinary integers. It studies prime numbers, primality tests and their use in codes, and other topics in number theory, leading up to the quadratic reciprocity theorem, a truly remarkable theorem of mathematics.
Mathematics 129 (S) is an advanced course in number theory. This course requires Mathematics 124 or equivalent background in number theory as a pre requisite.
Topics in geometry are presented in the 130series of courses:
Mathematics 130 (S) is a course on Euclidean and nonEuclidean geometry, including spherical and hyperbolic geometry.
Mathematics 131 (F) is a course in topology, the wholly abstract study of shape. This is the theory of geometry without distances or angles; only the glue that keeps nearby points near each other is left.
Mathematics 132 (S) presents "differential" geometry in which smooth variation can be defined and calculus can be applied. Mathematics 132 stresses the interaction of topology and calculus.
Mathematics 136 (F) studies the geometry of curves and surfaces in space.
Mathematics 137 (S) presents an algebraic approach to geometry, in which the principal objects of study are curves defined analytically by polynomials. Lines and conics are familiar curves defined by polynomials of degree one and two. Curves defined by polynomials of higher degree exhibit new phenomena that are best understood with the algebra of polynomials.
Mathematics 130 (S) studies the classical Euclidean and nonEuclidean 2 and 3dimensional geometries.
Topics in logic are covered in the 140series of courses:
Mathematics 141 (F) is a course on mathematical logic and set theory. It starts with a critical analysis of what is meant by a mathematical proof and goes on to develop some remarkable theorems on the limitations of mathematics.
Mathematics 143 ([]) examines the foundations of set theory, including the work of Godel and the continuum hypothesis. Mathematics 144 (S) and Philosophy 144 (S) provide other perspectives on topics in logic.
Students of logic may consider crossregistering at MIT where many courses in mathematical logic are offered every year.
Mathematics 154 (S), as mentioned above, delves into the mathematics of probability theory.
Mathematics 155r (F) is a theoretical course that studies elements of combinatorics and group theory.
Mathematics 60r (F&S) is a reading course taken by seniors writing theses. Mathematics 99r (F&S) consists of small group tutorials on topics not covered by other courses. For a list of current offerings, consult the Undergraduate Coordinator for Mathematics, Svetlana Alpert, Science Center 503.
C. Computer Science
The School of Engineering and Applied Sciences offers most courses related to computing under the Computer Science, Applied Mathematics, and Engineering Sciences labels. The Mathematics and Philosophy Departments offer a few courses relevant to the theory of computing. The courses may be separated into three groups:
1. Computer systems  the organization of computers and the extension of their capabilities by systems programming.
2. Advanced and applicationoriented modeling and programming techniques  the modeling and simulation of phenomena such as intelligence, perception, and physical objects. This area covers the topics of artificial intelligence, robotics, and graphics.
3. The theory of computation and scientific computing  what can and cannot be computed, in principle, and how efficiently computations can be made. The theory of computation studies models that are useful for understanding computation in all its variety; the field of scientific computing examines the issues that arise in complex numerical computations. Both use mathematics to study algorithms.
The sections below describe the undergraduate courses in these three areas. Many students go on to take graduate courses (200level) in these areas as well. In addition, opportunities for independent or supervised reading and research are available through the Computer Science 91r course. Many students writing research theses in Computer Science take one semester, or occasionally two semesters, of Computer Science 91r.
1. Computer Systems
Computer Science 50 (F) is the usual first course for students who intend to work in disciplines where a knowledge of computing, and of computer programming, is desirable. It has no prerequisites. The programming language used is C, and in addition to programming exercises there are exercises involving network usage and other nonprogramming aspects of computer science. The course is sectioned by experience level to accommodate the wide variety of backgrounds that students coming into it have. Students who are unsure if they should take Computer Science 50 should contact Professor Steven Gortler, sjg@seas.harvard.edu.
Computer Science 51 (S) is the gateway to the more advanced systems courses in computer science. Students should have taken Computer Science 50, or equivalent, before enrolling in this course. It covers a range of basic scientific and engineering subjects related to computing. In particular, Computer Science 51 aims to develop more advanced and specialized programming skills (in C++, Java and Lisp), and to teach students about the internal structure of computer systems software. Much time in Computer Science 51 is spent on the logical architecture of digital computers, on fundamental issues of data representation, and on the organization and control of computer programs, especially large programs.
Students who are interested in learning about computer programming but are not planning to concentrate in Computer Science might consider taking Quantitative Reasoning 20 (S) or Computer Science 1 (S).
Computer Science 61 (F) provides a solid background in systems programming and a deep understanding of lowlevel machine organization and design.
Computer Science 152 []) presents higherlevel ways of thinking about programming and about programming languages. To develop expertise, students expend significant effort writing short programs in a variety of languages, using the three major programming paradigms: functional, objectoriented, and logic. The course emphasizes the design elements that go into the languages (“ingredients”), not the full languages themselves (“recipes”); students learn to recognize and exploit these elements wherever they are found. To add depth, the course introduces the underpinnings of modern programming languages from both theoretical and practical points of view, with topics ranging from lcalculus and type systems to garbage collection and its performance.
Computer Science 153 [] presents the theory of compilers (programs for translating into executable form programs written in higherlevel languages) and gives extensive practice in the actual writing of a compiler.
Computer Science 161 (S) introduces the subject of operating systems. Understanding the principles that govern the design and analysis of these complex programs is essential for the practicing computer scientist. Much of the course is devoted to writing large parts of a model operating system.
All of the courses described above are concerned with how the primitive operations carried out by computer hardware are combined and organized to perform far more powerful operations. By contrast, Computer Science 141 (F) is a course in computing hardware (i.e., the internal design and operation of computers): how the physical parts of the machine communicate with each other, how “wiredin” arithmetic operations work, and why designers and manufacturers choose to organize the basic components of a computer in different ways. Although Computer Science 141 has a laboratory for the study of digital circuits, it is not a course in electronics. Students interested in electronic circuits should look into Engineering Sciences 50 (S), 154 (F), and Physics 123 (F&S).
Computer Science 143 (S) presents the principles and engineering practice of computer and telecommunications networking, especially the emerging field of very highspeed networking. It is accessible to students who have taken Computer Science 51. Students having a background in physics who are interested in communication may wish to consider Engineering Sciences 158 ([]).
Computer Science 144r (F) is a course in computer network design that is taught in seminar format with considerable student participation. Students explore realworld network design concerns through the relevant literature, and by developing a project.
Computer Science 148 (S) treats the design of VLSI chips. Students learn the basic principles governing the layout and structure of these microelectronic devices and gain experience with computer programs that assist the design and simulation of chips for various functions.
2. Advanced Techniques
All of the software courses described above deal with the structure of systems software: that is, with programs that give the user the illusion that the machine can perform more powerful primitive operations than it actually can. The courses described below treat the modeling and simulation of higherlevel phenomena (such as intelligent behavior, perception, and locomotion) for the purpose of better understanding of the phenomena and better engineering of artifacts that simulate or make use of them.
Computer Science 175 (F) discusses the specialized hardware, software, and mathematics of computer graphics — the production of realistic pictures by computers.
Computer Science 171 (S) studies how complex data is transformed into meaningful and perceptually intuitive representations for computer display.
Computer Science 182 ([]) is an introduction to the field of Artificial Intelligence focused on “higher level” and general issues of reasoning and planning. By contrast Computer Science 181 ([]) focuses on problems of perception, machine learning and reasoning under uncertainty.
Computer Science 187 (F) studies human language using the tools and techniques of computer science, with applications to a variety of naturallanguageprocessing problems.
3. Theoretical Computer Science
Computer Science 121 (F) is a first course in the theory of computation. The major topic is computation by automata — formal mathematical models for computing machines. Other mathematical formalisms for computation are discussed and shown to be equivalent to the most general type of automaton, the Turing machine. The main goal of the course is to show that there is a single mathematical notion of algorithm that can be defined formally in several equivalent ways.
Computer Science 124 (S) covers the theory and practice of data structures — how data are organized in computer memories to permit rapid retrieval, modification, and analysis. The subject blends the “systems” and “theoretical” parts of computer science by examining sophisticated mathematical methods widely used to develop better programs and to predict their behavior.
The basic courses in algebra, Applied Mathematics 106 ([]) and Mathematics 122 (F), in combinatorics, Applied Mathematics 107 (S) and Mathematics 192r (F), and in logic, Mathematics 141 (F), 143 (S), and 144 ([]), are described in Sec. B. These subjects are important for computer science students.
The first course in scientific computing is Applied Mathematics 111 (S). It covers both mathematical and computational aspects of the solution of numerical problems and presupposes some knowledge of continuous applied mathematics and some prior, but no specific, computer programming experience. In addition to writing programs, students learn to use existing numerical software. Applied Mathematics 205 (F) and 211 (F) are introductory courses with particular perspectives which require more mathematical and computational background. Students who wish to emphasize scientific computing should, at an early stage in their careers, master the material on linear algebra covered in Applied Mathematics 120 or Mathematics 121 (see Sec. B).
D. Statistics
The courses offered by the Department of Statistics range from largely theoretical to predominantly applied. When applications are stressed, attempts are made to select examples from widely differing areas, both to serve the needs and interests of a broad spectrum of students and to demonstrate that principles of statistical design, data analysis, probability modeling, and inference are widely relevant. Departmental courses are organized to accommodate students with differing degrees of mathematical preparation.
The firstlevel statistics courses have no mathematical prerequisites. The main offering, Statistics 100 (F&S), draws its examples from many fields. It introduces principles for designing surveys and experiments, techniques for analyzing statistical data, and elements of probability theory as applied to statistical modeling and inference. Experience with interactive computing for data analysis and simulation of random systems is provided.
In addition to Statistics 100, the Statistics Department offers three different courses that are designed to serve as introductions to quantitative methods: Statistics 101 (F&S), 102 (F), and 104 (F&S). Statistics 101 covers the same ground as Statistics 100, but emphasizes analysis of variance, which is widely used in experimentallyoriented subjects such as psychology and biology. Statistics 104 combines the contents of Statistics 100 and 101 and moves somewhat faster, assuming a stronger quantitative orientation. Statistics 102 emphasizes applications in the health sciences. Only one of the above four courses may be counted toward the Statistics concentration; none may be counted for credit in the other concentrations discussed in this pamphlet.
Many other departments (e.g., Anthropology, Economics, Government, and Sociology) offer disciplinespecific introductory statistics courses. None of these courses may be counted for credit in the four concentrations discussed in this pamphlet.
A second group of introductory courses begins with Statistics 110 (F) on probability and Statistics 111 (S) on statistical inference. These courses, which require multivariable calculus and linear algebra, erect a basic mathematical foundation for probability theory and statistical techniques such as regression, analysis of variance, and the design of surveys and experiments.
Applied Mathematics 101 (F) requires Applied Mathematics 21b or Mathematics 21b but no prior course in probability and statistics. It is intended as a survey of material covered more thoroughly in Statistics 110, 111 and 139. The course presents applications of probability and statistics, emphasizing the design of experiments and the parameterization of models drawn from problems in the natural sciences.
Mathematics 154 (S), discussed in section B, and Engineering Sciences 150 (S) are alternative introductions to probability theory and its applications.
The secondlevel courses typically require Statistics 110 or 111 as prerequisites. Statistics 115 (S) will cover basic problems, algorithms, and data analysis approaches in computational biology. Statistics 131 (F) discusses modeling, inference and forecasting with time series data. Statistics 135 (F) introduces the major statistical software packages: SPlus, R, and SAS. Statistics 139 (F) provides the basic theory associated with leastsquares regression, one of the most widely used of all statistical techniques. Statistics 140 (S) introduces the principles of randomized experiments, the concept of confounding, and some structural inference procedures. Statistics 149 (S) describes models used to study categorical and other nonGaussian data. Statistics 155 ([]) will introduce spatial statistics with applications to social science and public health research. Statistics 160 ([]) covers concepts and tools used in the design and analysis of sample surveys. Statistics 170 (S) will provide an introduction to derivative pricing. Statistics 171 (S) investigates the major classes of stochastic processes used as models in many of the natural and social sciences.
In 2008–2009 the Statistics Department introduced two new secondlevel courses. Statistics 105 (S) aims to instill an appreciation of statistical principles and reasoning by examining reallife situations such as financial investing, online dating, clinical trials, etc. Statistics 120 ([]) is an intermediate course in Biostatistics.
E. AlphabeticallyOrdered List of Courses Described in Chapter II
Appl. Math. 21a (F) 
Comp. Sci. 124 (S) 
Engrg. Sci. 210 (S) 
Math. 114 (F) 
Phil. 144 (S) 
(F) Fall
Course
(S) Spring
Course
([]) Not
offered in 20082009
In addition to the undergraduate courses listed above and described here, adequately prepared undergraduates often enroll, especially in their junior and senior years, in many of the 200level courses designated primarily for graduate students. Examples are included in the sample programs listed in Chapter IV. Short descriptions of these courses may be found in the appropriate sections of Courses of Instruction.
A. Lectures, Colloquia, and Clubs
Harvard faculty and visitors from all parts of the world frequently give colloquia, series of lectures, and even whole courses, on topics of current research. Regular colloquia series include: a mathematics colloquium organized jointly with Brandeis and MIT, which meets on Thursday afternoons at 4:30, moving from one institution to another; an applied mathematics colloquium, which meets on Fridays at 2:00 in Maxwell Dworkin; an applied mechanics colloquium, which meets on Wednesdays at 4:00 in Pierce 209; a statistics colloquium, which meets on Mondays at 4:00 in the Science Center; and Computer Science colloquia, which usually meet at 4:00 on Thursdays in Maxwell Dworkin. Several large seminars meet regularly each year as do many smaller informal seminars organized by faculty members and graduate students.
A number of these colloquia and seminars are listed each week in the Harvard University Gazette and are advertised on a central bulletin board in the Science Center. A mathematics calendar, issued each week, lists many of the mathematics colloquia scheduled for the Boston area. A copy is usually posted on the Mathematics Department seminar bulletin board and can normally be accessed on the department's Web page (http://math.harvard.edu). Students can receive announcements of the Computer Science colloquia by contacting the Director of Undergraduate Studies' office. Students can receive electronic announcements from the Statistics Department by contacting the department office (rinkel@stat.harvard.edu). Students who are interested in the teaching of mathematics may find some planned seminars on the theory and practice of teaching useful. These will be run at 1:00 p.m. on most Thursdays. For more information contact Robin Gottlieb at gottlieb@math.harvard.edu.
Monday through Thursday, undergraduate concentrators, graduate students, fellows, and faculty from the Mathematics Department congregate in the common room at 4:00 for tea. A mathematics club for undergraduates and graduate students meets every week at Mather House to dine and to listen to a talk by a student or faculty member. Two Rogers Prizes are awarded each year for the best undergraduate talks. Meeting announcements are posted in the Science Center and on the undergraduate bulletin board opposite Room 320. General messages for concentrators may be posted near Room 503. Concentrators and potential concentrators are encouraged to give their names to Svetlana Alpert (svetlana@math.harvard.edu) to receive information of interest, including announcements about summer jobs, research opportunities, scholarships, graduate schools, and postgraduation employment.
Many Harvard undergraduates take part in the William Lowell Putnam Mathematical Competition, a test with two threehour sessions, offered nationally in early December. Teams and individuals can win substantial prizes. Harvard students have done very well in the recent past. A team from Harvard also shone the past two years in a nationwide mathematical modeling contest sponsored jointly by MAA and SIAM. Further information about these contests may be obtained from Svetlana Alpert.
The Director of Undergraduate Studies in Computer Science provides semiregular announcements concerning activities, academic programs, and jobs in the computer science field. Computer Science concentrators receive these announcements automatically; others who wish to be on the mailing list should contact Tricia Ryan at (617) 4952833 (ryan@seas.harvard.edu).
Each year the Harvard Computing Contest Club (HC3) sends two teams of students to a Computer Programming Contest sponsored by the Association for Computing Machinery. Harvard's team competes with schools in the Northeast Region for the coveted right to go to the international competition, and has both gone to, and won, this worldwide competition. Students interested in joining the team should contact Dr. Robert Walton (walton@seas.harvard.edu).
The Harvard Computer Society (hcs@fas.harvard.edu) disseminates information of broad interest, including news about the University's computer systems, personal computers, and popular software packages. It also sponsors popular lectures, and organizes users' groups for owners of Macintosh and PCcompatible computers.
The Harvard College Engineering Society is an undergraduate group aimed at promoting engineering and crossdisciplinary collaboration on campus through participating in competitions and working on various engineering projects. For more information on this group, please contact/visit:
http://www.hcs.harvard.edu/~hces/.
The Harvard Society for Mind, Brain, and Behavior is an undergraduate organization that fosters exchange across disciplines: among students who are formally involved in Mind/Brain/Behavior program tracks (neurobiology, psychology, philosophy, computer science, history of science, biological anthropology, and linguistics) or less informally interested in MBBrelated topics or courses. Programs include dinner discussions with faculty, movie nights, field trips, symposia, and other special events. For more information on this group, please contact/visit:
hsmbb@hcs.harvard.edu
http://www.hcs.harvard.edu/~hsmbb
TECH's mission is to advance the understanding
and practice of translating science and technology into societal benefit. TECH is
both a real and virtual space for students, faculty, alumni and industry leaders
to learn together, collaborate, and innovate. TECH encourages the extracurricular
exercise of students' innovative and entrepreneurial spirit while they continue the pursuit of their formal education. It is a place where technologies may find application ideas, ideas may find implementation technologies, and both may find the people they need to succeed. For more information in this group, please contact/visit:
tech@seas.harvard.edu
http://www.seas.harvard.edu/tech
WICS is devoted to fostering a sense of community among women engaged in computer science and related field at Harvard College. For more information please contact one of the advisors.
Radhika Nagpal (rad@eecs.harvard.edu);
B. Computing Services and Facilities
FAS Computer Services provides a variety of computing services and facilities to students, faculty, and staff of the Faculty of Arts and Sciences (FAS) and its affiliates. Most services are distributed via the FAS Network, which connects student residences, faculty and administrative offices, libraries, laboratories, and public areas. FAS Computer Services has specialists dedicated to providing for the needs of instruction, student communication, office automation, faculty interaction, and research. Except for a small fee for network laser printing, computer services are provided to students at no cost.
FAS Computer Services offers laboratory facilities and computing support to undergraduate and graduate students within FAS and to students enrolled in computerbased courses in the Extension and Summer Schools. Student services include Internet access, Unix accounts for email, and a support system of User Assistants based on a model of "students helping students". FAS Computer Services also maintains the Harvard Technology Showcase, an advanced multimedia computing facility. Students have access to the FAS Network through the Computer Labs in the basement of the Science Center, residential labs in the Houses and some dormitories, and numerous computer kiosks around campus. They may also connect personal computers (Macintosh or PC compatible) to the FAS Network directly from their rooms via both wired and wireless connections.
Students may ask computing questions or request an appointment with a User Assistant for personal computer assistance by contacting the Help Desk (Science Center B13, 59000, help@fas.harvard.edu). Students living in Harvard houses or dormitories are also encouraged to contact the User Assistants assigned directly to their residence.
For additional information about FAS Computer Services, please visit the Computer Services website at http://www.fasit.fas.harvard.edu/.
Computer Science facilities in Maxwell Dworkin include many UNIX workstations, peripherals, and a telecommunications conference room. The Mathematics Department maintains a network of workstations and advanced mathematical software. Additional workstations, located in Maxwell Dworkin and Pierce Hall, are used for undergraduate instruction and research in computer graphics, VLSI chip design, and robotics.
C. Libraries and Databases
1. Principal Collections
Four libraries house the main collections in the mathematical, statistical and computer sciences. The table at the end of this section indicates their locations, hours, and special services. Cabot Science Library has an extensive collection of books and journals, periodical indices and reference materials, among them a useful selection of handbooks and tables. The Birkhoff Mathematical Library and the Statistics Library are noncirculating reading rooms with more specialized journal selections. The Gordon McKay Library (Pierce Hall) covers applied mathematics and computer science. Widener Library maintains extensive collections of journals of learned societies.
2. Catalogs
The key access point to these collections is the HOLLIS catalog which lists all materials acquired by Harvard libraries since 1978, the majority of earlier materials, and all current serial subscriptions. The HOLLIS catalog is available on the Web through the Harvard Libraries website (http://lib.harvard.edu). Instruction in the use of the HOLLIS catalog, and guides to the catalog, are available at the Cabot and Gordon McKay libraries.
3. Electronic Resources
Electronic journals, abstracts and indices in mathematics and computer science are available under EResearch on the Harvard Libraries website (http: //lib.harvard.edu). The databases include MathSciNet (Mathematical Reviews on the Web), INSPEC (Computer and Control Abstracts, Electrical and Electronics Abstracts, Physics Abstracts, and Information Technology), CIS (Current Index to Statistics) and Science Citation Index/Web of Science (from ISI). The Gordon McKay staff and the Cabot reference staff offer training in the use of these and other science information resources.
The Mathematics department Web page (http://math.harvard.edu) has a great many links to various mathematical sciences databases.
4. Interlibrary Loan Services
The Cabot and Gordon McKay libraries provide document delivery service for materials not in Harvard's collections.
5. Math Help
The Mathematics department staffs the Math Question Center in Loker Commons 810 pm Sunday through Thursday to help students enrolled in Mathematics Xa,b, 1a,b and 21a,b.
Library 
Hours 
Special Services 
Cabot Science Library 
Monday  Thursday: 8:30 am  midnight 
Reference, interlibrary loan, course reserves, group study rooms, video viewing rooms, copier, database searching 
George Birkhoff Library 
Monday  Friday: 9 am  5 pm 

Statistics Library 
By appointment (61402) 

Gordon McKay Library 
Monday  Thursday: 9 am  10 pm 
Reference, interlibrary loan, course reserves, copiers, group study room, database searching 
Click here to download the Cabot Science Library Mathematics Textbook Reference Shelf
A. Mathematics
A Mathematics concentration requires twelve onesemester courses, eight of which must be in Mathematics. The other four may be either in Mathematics or in a related field. A list of approved related courses may be found in Fields of Concentration. The Director of Undergraduate Studies will often approve other courses that make substantial use of mathematics. Concentrators must take at least one course beyond Mathematics 21b in each of the areas of algebra, analysis, and geometry. To qualify for honors, students must also write senior theses, described more fully in a pamphlet available from Svetlana Alpert, Undergraduate Coordinator in Mathematics. Students wishing to learn more about the concentration should contact Svetlana Alpert who can provide various pamphlets describing the concentration. Some sample programs for students with different interests and degrees of preparation are listed below.

1st Year 
2nd Year 
3rd Year 
4th Year 
Program for 
Math 1a 
Math 21a 
Math 112 
Math 130 
Program for 
Math 21a 
Math 101 
Math 122 
Math 60r 
Abstract 
Math 23a/25a/55a 
Math 113 
Math 114 
Math 221 
Mathematical 
Math 21a /23a/25a/55a 
Math 115 
Math 136 
Math 132 
Honors concentrations in Mathematics may be combined with secondary concentrations in other fields, e.g., Computer Science and Philosophy. In such cases, honors theses should draw upon both fields. Sample programs for combined concentrations might involve:

1st Year 
2nd Year 
3rd Year 
4th Year 
Mathematics 
Math 21a 
Math 101 
Math 122 
AM 107 
Mathematics 
Math 21a 
Math 112 
Math 122 
Math 113 
The Mathematics department offers 12 small group tutorials (Math 99r) each semester. Students typically gain a tremendous amount from these tutorials and are encouraged (but not required) to take one.
B. Applied Mathematics
Since concentrators may specialize in the mathematical, physical, biological, social, engineering, and management sciences, programs are diverse and flexible. Faculty advisers in these different fields help students design suitable individuallytailored honors programs consistent with the broad requirements outlined in Fields of Concentration. The examples listed below illustrate how students may fulfill requirements in several fields. In each case, students should seriously consider taking Applied Mathematics 115 in the third or fourth year. It is not unusual for concentrators to take programs that are more intensive and advanced. Further details are available in the Applied Mathematics Concentration Guidelines document, and related material, available from the Academic Office, Pierce Hall 110.
Specialty 
1st Year 
2nd Year 
3rd Year 
4th Year 
Economics 
Math 1b 
Math 21b 
Math 121/AM 120 
Econ 1723 
Computer 
Math 21a/AM 21a 
Stat 110 
Stat 171 
AM 111/205 
Biology 
Math 1b 
Math 21b 
AM 101 
AM 111/205 
Chemistry 
AM 21a/Math 21a 
AM105a 
AM 120/Math 121 
AM 111/205 
Decision and 
Math 21a/AM21a 
AM 101/Stat 110 
AM 120/Math 121 
ES 201 
Environmental 
Math 1b 
Math 21b 
AM 120/Math 121 
AM 105a 
Biomedical Sciences 
AM 21a/Math 21a 
AM 105a 
AM 120/Math 121 
AM 101 
Geophysical 
AM 21a/Math 21a 
AM 105a 
AM 111/205 
AM 120/Math 121 
C. Computer Science
Concentrators in Computer Science may pursue either a basic (12 halfcourses) or an honors (14 halfcourses) program. (Students with Advanced Placement credit in Mathematics can reduce the requirements somewhat; it is quite possible to complete an honors program starting in the sophomore year.) Both programs require a few specific courses in mathematics and computer science, a selection of additional courses in theoretical computer science and in computer systems, and a coherent selection of technical electives in computer science. Please consult the CS Undergraduate Program Guide and the handbook for more information

1st Year 
2nd Year 
3 rd Year 
4 th Year 
Program for Students who have Not had Calculus 
Math 1a Math 1b CS 50 CS 51 
Math 21a Math 21b CS 121 CS 124 
AM107 CS 152 CS 153 
CS 143 CS 144r CS 164 CS 181 
Program for 
Math 21a 
CS 121 
AM 107 
CS 143 
Students can combine Computer Science with other fields and design programs that include a thesis involving both fields. In recent years, students have combined Computer Science with Philosophy, Linguistics, Psychology, Economics, and Physics. A program combining Mathematics (as the major field) with Computer Science is shown in Section IV, A.
It is possible for an Advanced Standing student to apply for admission to the SM program and to complete the Master’s degree in Computer Science during the fourth year in residence. Students interested in pursuing this option should consult with the Director of Undergraduate Studies early on about preparing themselves appropriately.
More information about concentration programs and requirements may be found on the Computer Science program website (http://www.seas.harvard.edu) and at the Academic Office of the School of Engineering and Applied Sciences, Pierce Hall 110.
D. Statistics
The flexible concentration requirements in Statistics reflect the diverse applications of the subject. Twelve halfcourses (seven in Statistics and up to five in "related" fields) are required. For honors, two additional halfcourses (which may be "related") and a thesis are required. Statistics now offers "tracks" or areas of specialization, in Bioinformatics and Computational Biology and in Quantitative Finance. Statistics also now offers a Secondary Field in Statistics. A sample honors program follows:

1st Year 
2nd Year 
3rd Year 
4th Year 
Statistics 
Stat 104 
Math 21b ** 
Math 121/AM 120 
Stat 149/155 
** Students are encouraged to complete the four courses marked with two asterisks by the end of their sophomore year so that they can optimize their choices in subsequent years. Fields of Concentration lists the many qualifying alternative courses in Statistics, Applied Mathematics, Computer Science, Engineering Sciences, Psychology, and Economics.
Contact 
Telephone 
Email address 
Svetlana Alpert 
59116 

Peter Kronheimer 
55745 

Ellen Holloway 
51524 

Steven Gortler 
53751 

Michael P. Brenner 
53336 

Betsey Cogswell 
55497 

Marie Dahleh 
51485 

Joseph K. Blitzstein 
62985 

David P. Harrington 
58710 