Analysis is the study of various concepts that involve the idea of taking limits, such as differentiation, integration, and notions of convergence. Princeton’s emphasis on analysis is reflected by the fact that two of the three introductory courses for math majors (MAT215 and MAT218) deal with the subject. Analysis has applications ranging from physics to number theory, and underlies many branches of applied math. After taking the introductory courses, most students interested in analysis proceed to the four core analysis courses called the “Stein sequence”, described below. The department also offers courses on the applications of analysis to other fields, including MAT 407 (Mathematical Methods of Physics) and MAT 415 (Analytic Number Theory).
Introductory Analysis Courses [Show]Introductory Analysis Courses [Hide]
MAT 215: Single Variable Analysis
This introductory class covers the first eight chapters of Walter Rudin’s “Principles of Mathematical Analysis”. Many of the concepts such as differentiation and integration might already be familiar to students. However, the emphasis of the course is on the rigor of proofs. The second chapter on point-set topology is especially crucial in the sense that it lays foundation for the rest of the course as well as for any future analysis classes. It might be beneficial to go through every definition a few times to fully internalize the ideas. A general difficulty in studying this class is about writing proofs. In the first half of the class, most of the proofs more or less “write themselves” once you have unraveled the definitions, so the most important part is again trying to understand what the statement is really saying. For instance, as soon as you write out definitions of continuity and closedness, it should be quite clear that “the zero set of a continuous real function is closed”. Chapter 7 of Rudin’s book touches on a subtle but important issue in analysis: when can you interchange two limiting processes? For example, if a sequence of function f_n converges pointwise to a function f, is the limit of the integral (by definition a limit process) of f_n the same as the integral of the limit of f_n? This turned out to be a recurring theme in many later courses, and the study of which leads to many fascinating ideas such as uniform convergence and convergence theorems of Lebesgue integration.
MAT 218: Analysis in Several Variables
This course is the last in Princeton’s introductory sequence for math majors. It builds on the single variable analysis theory developed in MAT 215, but also uses linear algebra tools from MAT 217, such as vector spaces, linear transformations, and determinants. In recent years, MAT 218 has been the least popular of the three introductory courses, and in fact the math department is considering switching to a two-semester introductory sequence. More information about the content and structure of MAT 218 can be found here on the math department’s website.
Analysis Sequence [Show]Analysis Sequence [Hide]
At the core of Princeton’s program in analysis is the analysis sequence, known as the “Stein sequence.” The Stein sequence is based on a series of four books written by Elias Stein, a famous analyst and professor emeritus at Princeton. The courses in the Stein sequence are Analysis I: Fourier Analysis, Analysis II: Complex Analysis, Analysis III: Real Analysis, and Analysis IV: Special Topics in Analysis. It is convenient, but not necessary, to take these classes in order. For example, both the Complex Analysis and Real Analysis courses deal with aspects of Fourier analysis. Having said this, the courses are advertised as being self-contained, so taking them in other orders is also fairly common.
MAT 325: Fourier Analysis and Partial Differential Equations
This course covers the fundamental concepts in Fourier series and Fourier analysis, as well as their applications to differential equations, number theory, physics, and other topics. It begins with a study of the behavior of a vibrating string, which motivates the idea of decomposing a periodic function into an infinite sum of sines and cosines. It then discusses the basics of Fourier series, such as the Dirichlet and Fejér kernels, Parseval’s theorem, and the Riemann-Lebesgue lemma, with a particular focus on issues of exactly when and how the Fourier series of a function converges to that function. The course then moves on to Fourier analysis, covering the Fourier transform, the inverse transform, Plancherel’s theorem, and the Poisson summation formula. These results are first shown on the real line, and are then easily generalized to n-dimensional space. Finally, the course studies Fourier analysis on an arbitrary abelian group, using the particular case of the unit group of Zn to provide an elegant proof of Dirichlet’s theorem on arithmetic progressions. Along the way, these results are applied to various interesting topics, such as the solutions of the wave and heat equations, the isoperimetric inequality, the existence of a function which is continuous everywhere and differentiable nowhere, and the Heisenberg uncertainty principle.
The course closely follows Elias Stein and Rami Shakarchi’s textbook Princeton Lectures in Analysis I: Fourier Analysis. The only true prerequisite for the course is a solid understanding of high school calculus, but an introductory analysis course such as MAT 215 is certainly helpful. Although Fourier analysis is more naturally done in the context of Lebesgue integrable functions, Stein and Shakarchi choose not to introduce measure theory until Book III of their series, so this course is taught within the context of Riemann integrable functions. The course grade is based on weekly problem sets, which tend to consist of exercises from the textbook, as well as a midterm and final exam. Professor Stein usually teaches the course every other year, and he is notorious for including final exam questions that require you to write out proofs of major theorems from the textbook, so it is important to set aside some time during reading period to memorize the most important proofs.
MAT 335: Complex Analysis
The second course in the Stein sequence deals with complex analysis. This branch of analysis deals with analytical properties of holomorphic functions of complex variables. A function complex-valued function of a complex variable is holomorphic at a point if it has a complex derivative there. This turns out to be a strong condition, one which makes complex analysis a much richer theory than, say, analysis of a single variable. The many beautiful properties of holomorphic functions give complex analysis a unique flavor.
MAT 335 follows Stein’s well-written Complex Analysis. The course usually covers most of the book’s ten chapters, in order. The first three chapters establish the many elegant basic properties enjoyed by a holomorphic function. Chapters 4, 5, 8 are three stand-alone chapters that cover fascinating individual topics in complex analysis. Highlights include Hadamard’s factorization theorem (entire functions are more or less determined by their zeros) and Riemann mapping theorem (given any two proper simply connected domains in the complex plane, there is some holomorphic function mapping one bijectively to the other). Chapter 6, 7, 9, 10 are strongly flavored in number theory. It is a wonder that complex analysis has become such a successful tool in tackling many problems in number theory, a field concerned with the properties of integers. Chapters 6 and 7 introduce the zeta function (the link between number theory and complex analysis) and prove the Prime Number Theorem, one of the milestones in analytic number theory. Chapters 9 and 10 briefly touch on the vast subject of elliptic functions and theta functions.
MAT 425: Real Analysis
The third course in the analysis sequence is Real Analysis. This course adds another layer of sophistication to the theories of integration and differentiation covered in MAT 215, extending them to a more general context. The course covers the first five chapters of Professor Stein’s book Real Analysis: measure theory, integration, and Hilbert spaces.
The first chapter of the book establishes the Lebesgue theory of measure, which underlies the theory of Lebesgue integration. The measure of a set is in some sense its “volume.” The Lebesgue measure drastically generalizes this intuition. For example, the measure of the set of rational numbers is zero. The material in this chapter is similar to that of MAT 215, with its technical, but often elegant, epsilon-delta arguments.
The second chapter covers the theory of Lebesgue integration. The starting point of this theory is the characteristic function of a set, which is 1 on the set and 0 elsewhere. The integral of such a function is defined to be the measure of the corresponding set. The Lebesgue integral is then built up from this foundation. The generality of the Lebesgue measure makes the Lebesgue integral a significant improvement on the Riemann integral. For example, the characteristic function of the rationals on [0, 1] is not Riemann integrable, but is Lebesgue integrable. The rest of chapter 2 discusses the properties of the Lebesgue integral.
Chapter 3 explores analogues of the fundamental theorem of calculus in the setting of the Lebesgue integral. An important result is the Lebesgue differentiation theorem, which states (loosely) that the derivative of the integral of a function is the function itself. It turns out that it is more difficult to understand the integral of the derivative of a function. The class of functions of bounded variation is introduced; it is these functions whose derivatives are integrable. A stronger assumption called absolute continuity is needed to guarantee that the integral of the derivative is the function itself.
Chapters 4 and 5 deal with Hilbert spaces, which are vectors spaces endowed with inner products, separability, and completeness. Chapter 4 covers Hilbert spaces and operators on these spaces, two very fundamental topics. Chapter 5 covers applications of Hilbert spaces, including L^2, the space of square-integrable functions, which is the link between Hilbert spaces and integration theory.
MAT 425 is an interesting course, but also a very difficult one for many. The course proceeds at a rapid pace, and thus reading before lecture is strongly advised. The problem sets are the hardest part of the course, and doing well on them requires a significant time commitment. The problem sets are very important in understanding the course material, but some problems are quite tricky. Working in groups on the problem sets is advisable, both to split the difficulty and to discuss the material with other people.
MAT 520: Special Topics in Analysis
This topics course varies in content from year to year. In Fall 2012, this course will cover functional analysis, following Prof. Stein’s recently published book on the subject.
Other Analysis Courses [Show]Other Analysis Courses [Hide]
MAT 314: Introduction to Real Analysis
The course goes over the basics of measure theory and Lebesgue integration. It begins with a quick introduction of algebras, sigma algebras, measures and their properties. Then the course develops the basic theory of Lebesgue integrals over the real line including Fatou’s Lemma, Dominated Convergence Theorems, Monotone Convergence Theorems, the Egorov Theorem, and others. In the middle of the semester the course discusses abstract measure spaces and integrals with a focus on the Carathéodory extension theorem, Hahn decomposition theorem and Radon-Nikodym theorem. In the second half of the semester the course discusses Lp spaces, inner product spaces with an emphasis on L2. Near the end of the semester the course goes over the basic properties of Hilbert spaces and Fourier series.
The course is intended for sophomores and juniors who require a basic introduction to real analysis. The material is presented carefully and rigorously; students are consistently motivated by examples that test their understanding. For the most part the course follows the book by Royden – “Real Analysis”, although the topics are not followed in class in the same order as they appear in the book. The material on Fourier series is usually taken from a different text and Professor Warren often provides additional material either in lecture or in homework sets.
There are weekly problem sets that account for about 30% the grade in the course. Most of them aim to improve student understanding of the theory and can be quite challenging. Lectures are usually spent proving results, emphasizing their role in the theory as well as the techniques employed in the proofs. In addition, there is a midterm and a final (both take-home) that account for 70% of the grade. The exams are usually much more difficult than the homework sets, but the material tested is always within the range of what has been taught.
The course is exceptionally challenging for people who have not taken MAT 215 (or equivalent) before as basic properties of sequences are assumed to be known. Although MAT 202 and MAT 201 (or equivalent) are listed as prerequisites, there is hardly any use of linear algebra or multivariable calculus, although it is still advisable to check this with Professor Warren.
MAT 407: Mathematical Methods of Physics
The content of this class varies from year to year depending on the instructor. In particular, it depends on whether a physicist or a mathematician teaches the class. The following are descriptions of MAT 407 for two recent years.
Professor: Elliot Lieb
Prof. Lieb covered basic Hilbert Space theory, distributions, Fourier transforms, and briefly introduced the class to unbounded operators, which are important for quantum theory. Then, he spent a significant amount of time studying Trace class operators and their utility in quantum statistical mechanics. Prof. Lieb explored mathematically the concept of entropy in thermodynamics, along with some important inequalities such as the Peierls-Bogoliubov. Towards the end of the course, he also discussed some representation theory and showed where it was relevant for quantum mechanics. The course was fast paced, but provided a great overview of some important working fields in mathematical physics. For additional inspiration, a few of the problems Professor Lieb assigned were related to some of his papers. These papers can be found in the book Inequalities: Selecta of Elliot Lieb.
Professor: Chris Herzog
Prof. Chris Herzog, a string theorist who is no longer at Princeton, taught MAT 407 in Spring 2011. The class was structured as a broad survey of mathematical topics applied to problems in physics. Topics included linear algebra and basic operator theory (quantum mechanics), linear ordinary differential equations and Green’s functions (related to Prof. Herzog’s research in AdS/CFT correspondence), special polynomials (e.g. of the Hermite, Laguerre, Legendre varieties, each with applications to several areas of physics), group theory, basic representation theory – in particular character theory (determining the vibrational modes of molecules from their symmetries), and Lie algebras (elementary particle physics). Coverage emphasized breadth over depth; in the interest of time, proofs of only the most important results were presented. The prescribed textbook was Sadri Hassani’s Mathematical Physics, but Stone and Goldbart’s Mathematics for Physics is also a very good reference. As for prerequisites, it is sufficient to have knowledge of linear algebra, elementary quantum theory, and complex analysis.
MAT 451: Advanced Topics in Analysis
Prof Lieb teaches this class using his textbook “Analysis”. The class is quite self-contained. It starts by reviewing concepts of measure theory and Lebesgue integration. Then it goes on to touch a wide variety of topics, including L^p spaces, integration inequalities, Fourier transformation etc. If the first half of the course focuses more on “tools” that analysts should master, the second half shifts more towards using of these tools to tackle problems in partial differential equation and mathematical physics. The highlight of the course is the study of distribution and Sobolev spaces and their application to solving some partial differential equations such as the heat equation and the Schrodinger’s equation.
MAT 390/MAT 391: Probability Theory and Random Processes
These two probability theory courses are taught over two semesters by Prof Sinai. These two classes count towards the math department’s real analysis requirement because they have a substantial measure theory component. Hence, like MAT 425, they also offer an introduction to the theory of measure and Lebesgue integration. Please see the Probability and Statistics section for detailed summaries and discussions of these two courses.
MAT 415: Analytic Number Theory
This course is devoted to analytic techniques in number theory. Please see the Number Theory section for a discussion of this course.
MAT 427: Ordinary Differential Equations
This differential equations course recently underwent a switch from a 300-level course to a 400-level course, which suggests that it will become more oriented towards theory. However, not much more is known at this time. Please check the math department page and the registrar’s page for more information.
Contacts [Show]Contacts [Hide]
Gene Katsevich ’14 (ekatsevi[at]princeton[dot]edu) for general questions
Tengyao Wang ’12 (tengyaow[at]princeton[dot]edu) for general questions
Rohan Ghanta ’13 (rghanta[at]princeton[dot]edu) for mathematical physics
Kai Sheng Tai ’13 (ktai[at]princeton[dot]edu) for mathematical physics