If you have completed 18.03, any subject listed on the first row of the chart is suitable as a next mathematics subject. Of these, the subjects 18.04 (Complex Variables) and 18.06 (Linear Algebra) are taken most often closely followed by 18.05 (Probability and Statistics). (In fact, 18.05 and 18.06 require only 18.02 as a prerequisite.) Each of these subjects is useful for students majoring in a wide variety of disciplines.

Each of these subjects is also suitable for students who plan to major in mathematics. For such students, or for those who simply wish to select a somewhat more demanding subject, there are further possibilities:

• For students interested in applied mathematics, a good possibility is the two-term sequence 18.310-18.311 (Principles of Applied Mathematics). Another possibility is 18.112, which is a more demanding version of 18.04.
  
  
• For students interested in probability and statistics, an alternative to 18.05 is the more demanding sequence 18.440 (or 18.313) followed by 18.441.
  
  
• For students interested in theoretical mathematics, there are several possibilities. One is 18.700, which is a more theoretical version of 18.06. Another is the basic subject in analysis, 18.100, which in a sense is a high-level, rigorous version of 18.01. This subject comes in two versions; of these, 18.100A is less abstract than 18.100B.

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When you get beyond the beginning subjects in any of the listed areas of mathematics, the chart can give you some guidance as to an appropriate next subject. You will need to be sure you have the necessary prerequisites. You should also be aware of the level of difficulty and/or abstraction of the subject. Now, let us turn to a description of these subjects themselves:

ANALYSIS

Real Analysis (18.100, 18.101, 18.103, 18.152) deals with real-valued functions of real variables, as in calculus, but with complete rigor and considerable abstraction. 18.100 is concerned with functions of a single variable and their derivatives and integrals, in the context of metric spaces. It is a prerequisite for many other subjects. (Some bright sophomores do well in this subject, on the other hand, it is probably more frequently dropped than any other math subject at MIT. It helps if the student has already had some experience with proofs.) There is a version of 18.100 (called "Option A") that is somewhat slower paced and less abstract. 18.101 treats functions of several variables; topics include derivatives and integrals in n dimensions, and the n-dimensional version of Stokes' theorem (involving differential forms rather than vector fields). 18.103 treats the Lebesgue integral (which is different from the familiar Riemann integral of calculus) and Fourier analysis. (18.125 also treats the Lebesgue integral, but much more abstractly; it is a graduate subject.) 18.152 is a mathematical treatment of partial differential equations; it is a logical sequel to 18.03 or 18.034.

Complex Analysis (18.04, 18.112, 18.115, 18.075) deals with complex-valued functions of a complex variable. Although the concepts studied may seem familiar—derivatives, integrals, and the like—the results are remarkably different from those in real analysis. The applications are many and varied; the so-called "imaginary" numbers have important real-life applications to physics and engineering. 18.04 is an elementary introduction to the subject, taken by many EE majors. 18.112 is a deeper and more extensive treatment, mathematical rigor is not stressed however. 18.115 is a theoretical, theorem-proving treatment, it is a graduate subject, which is taken by many advanced undergraduates.

Overlapping 18.100 and 18.112 and 18.103 is the "Advanced Calculus for Engineers" sequence 18.075-18.076. Although designed with engineering students in mind, it is taken by a few other students as well. It covers topics in both real and complex analysis that are useful in applications. Both 18.04 and 18.075 may not be taken for credit.

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ALGEBRA

Linear Algebra (18.06, 18.700) deals with solving systems of linear equations, both concretely using matrix algebra and more abstractly using vector spaces and linear transformations. Its subject matter is basic in mathematics, and everyone should have some knowledge of it, 18.06 emphasizes the concrete approach using matrices, while 18.700 is more abstract and proof oriented. (Linear algebra is also treated briefly in 18.701.)

Modern Algebra (18.703, 18.701-18.702) is the study of certain abstractly defined systems (called groups, rings, and fields) whose properties generalize those of integers, polynomials, and matrices. Using these concepts one can prove, for instance, that the general angle cannot be trisected by a Euclidean construction, and that there is no formula like the quadratic formula for solving polynomial equations of the 5th degree. There are modem applications to such subjects as coding theory and cryptography. 18.703 is a standard introduction to the subject. 18.701-18.702 is a more intensive high-level sequence, the student should have some experience with proofs (as in 18.100 or 18.700) before taking this subject.
Number Theory (18.781) uses the tools of modem algebra to study properties of the set of positive integers.

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GEOMETRY AND TOPOLOGY

Topology (18.901 ) deals with quite general objects called "topological spaces" which are more general than the metric spaces of analysis and the surfaces and manifolds of geometry. It studies such familiar notions as compactness and connectedness, and then goes on to such topics as metrization theorems and product spaces. Writing proofs is emphasized.

Differential Geometry (18.950) is an introduction to modem geometry. The basic objects of study are smooth curves and surfaces in Euclidean space. Topics include curvature and geodesics, and their relation to the nature of the curve or surface.

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LOGIC

Logic forms the very foundation of mathematics. One studies how mathematical statements are formulated, what it means for them to be true, what a proof is, and whether all true statements in a mathematical system can be proven within the confines of that system. Either 18.510 or 18.511 constitutes a basic introduction. The first halves of these two courses are the same; but the second half of 18.510 is formal set theory and the second half of 18.511 is recursion theory. 18.423J is a subject that explores the connections between logic and computer science, it is jointly offered with Course 6.

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CONTINUOUS APPLIED MATHEMATICS

Continuous applied mathematics deals most commonly with the mathematics of classical physics and continuum mechanics. 18.311 is an elementary introduction-, it builds on the basics of calculus and differential equations to provide insight into more difficult problems and to give the flavor of the methods used in applied mathematics. 18.303 is the basic high-level subject in this area. It concentrates on the partial differential equations of physics and applied mathematics, along with their applications and methods of solutions.

Nonlinear Dynamics (18.353-18.354) deals with understanding the differential equations of science and engineering. 18.353 studies maps and ordinary differential equations, explaining the qualitative properties of solutions, and conditions under which they can become chaotic. 18.354 concerns the mathematical principles underlying continuum systems. Fundamental ideas include averaging, singular perturbations, instability, and turbulence.

Numerical Analysis (18.330) deals with finding efficient methods of determining numerical solutions to problems generated by engineering and science. Sample problems include approximating an integral or finding roots of an algebraic equation. The calculations are usually done with the aid of a computer or calculator.

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DISCRETE APPLIED MATHEMATICS

18.310 is an elementary introduction to this area- calculus is the only prerequisite, but linear algebra might be helpful. Topics vary, but have included linear programming. coding theory, scheduling, sorting, and game theory. It is skewed toward applications to real world problems. 18.062J is somewhat more elementary, it treats those aspects of mathematics that are basic for computer science.

Combinatorics (18.312, 18.314) 18.314 deals With such topics as graph theory, enumeration problems, and sorting; there are applications to computer science. This is the basic introduction to combinatorics at MIT, though it has few prerequisites, it is distinctly not elementary. 18.312 deals with the applications of modem algebra to combinatorics; it is a bit more advanced.

Combinatorial Optimization(18.433) gives a thorough treatment of linear programming, which is one of the basic tools of operations research and is heavily used in business and industry.

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THEORETICAL COMPUTER SCIENCE

This subject deals with the mathematical aspects of the field of computer science.

Theory of Computation (18.400J, 18.404J) treats the theoretical foundations underlying modem computers, topics include Turing machines, automata, and complexity. 18.400J is an introduction to the subject, oriented toward applications; 18.404J provides a more extensive and theoretical treatment of the same material.

Algorithms (18.410J, 18.421) treats the design and analysis of efficient problem solving algorithms. 18.410J is an applications-oriented introduction; 18.421 deals more specifically with algorithms for solving problems from classical algebra and number theory.

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PROBABILITY AND STATISTICS

Probability and Statistics constitute one of the most widely used fields of mathematics, being of great importance in the experimental sciences. 18.05 is an elementary introduction to both areas; it is taken primarily by non-math majors.

Probability (18.313, 18.440) deals with measuring the likelihood of the occurrence of a given event, or series of events. Both 18.313 and 18.440 cover basic probability; they are equivalent for prerequisite purposes. 18.440 is the standard introduction. 18.313 is an alternative version; it is taught in a lecture-recitation format and is somewhat more demanding mathematically.

Statistics (18.441, 18.443) deals with analyzing of data and drawing inferences from the results. 18.441 is the standard introduction. 18.443 is an alternative version that is oriented more towards the applications of statistics and less toward theory. Both require a background in probability

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SEMINARS

The upper-level seminars in mathematics are designed for students who plan to do graduate work in mathematics. In such a seminar the student studies a particular area of mathematics under the guidance of a faculty member, learning to read mathematics independently and to organize and present the results. Topics covered in the seminars in analysis 18.104), logic (18.504), and algebra (18.704), vary from year to year. The seminar in topology (18.904) is an introduction to algebraic topology, in which one uses tools of algebra to study problems of topology, by studying the fundamental group of a topological space.

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