Analysis

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 MAT216) 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, students interested in analysis often 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 493/PHY 403 (Mathematical Methods of Physics).

In order to graduate with a mathematics degree, it is required to complete at least one real analysis course and one complex analysis course. On this page, every non-introductory course except those numbered MAT 33x should count towards the real analysis departmental.

Introductory Courses

MAT 215: Honors Analysis (Single 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.
Offered every semester.

MAT 216/218: Accelerated Honors Analysis
This two-semester sequence, taught out of Robert C. Gunning’s An Introduction to Analysis, has also been taught for the last many years by Professor Gunning himself. Beginning with basic set theory and continuing to elementary abstract algebra, topology, linear algebra, sequences and convergence, differentiation, differential geometry, and integration, and culminating in the proof of the generalized Stokes’ theorem, MAT216/218 touches on a great deal of mathematics. A primary benefit of these courses is that it entrenches a familiarity with so many topics, which is useful in upper-level courses, which can frequently draw from ideas in a number of seemingly-unrelated fields. Familiarity and comfort with proofs and rigor are strongly recommended in order to take these courses; once you’ve gone through the gauntlet, you’ll be an even more capable student of mathematics.
MAT 216 (part I) is a Fall course and MAT 218 (part II) is a Spring course.

The “Stein Sequence”

At the core of Princeton’s program in analysis is the analysis sequence, sometimes known as the “Stein sequence” because it is taught from a series of four books written by Elias Stein, who was a famous analyst and professor at Princeton. The texts 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, aside from Functional Analysis (MAT520), which requires prior completion of MAT425 (and preferably 335 and a working understanding of Fourier Analysis as well). 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: Analysis I: Fourier Series 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.
Offered in the Spring.

MAT 335: Analysis II: 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 in order, though depending on the semester, the course may skip certain earlier sections or not make it all the way through the ten chapters. Highlights of the course include properties of holomorphic functions which Stein calls the “three miracles;” Hadamard’s factorization theorem; and the Riemann mapping theorem. Multiple chapters are strongly flavored in number theory, and it is itself 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. Highlights of this portion of the course include the properties zeta function (which gives the link between number theory and complex analysis) and the Prime Number Theorem, one of the milestones in analytic number theory. Chapters 9 and 10 also briefly touch on the vast subject of elliptic functions and theta functions.
Offered in the Fall.

MAT 425: Analysis III: Integration Theory
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. In recent years, however, the course has stopped at chapter 4.
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.
Offered in the Fall.

MAT 520: Functional Analysis
This course typically covers the first four chapters in the fourth volume in Stein’s series, as well as supplemental resources for topics covered outside the Stein text. Please note that the course is offered intermittently, and unlike the previous courses in the sequence the content may change slightly with each offering. The course starts with a discussion of analysis in L^p spaces, and reviews basic measure theory. It also introduces Banach spaces (complete normed vector spaces) and dual spaces of an L^p space, as well as linear functionals, and discusses relevant theorems, specifically Hahn-Banach [Chapter 1].  Next, first and second category sets (and the Baire Category theorem) are discussed, along with the uniform boundedness principle for functionals and several Banach theorems (Open Mapping Theorem, Closed Graph Theorem, etc) [Chapter 4]. Afterwards, compact operator theory was discussed with reference Chapters 6 and 7 of Eidelman’s Functional Analysis text. In particular, there was a focus on self-adjoint operators, as well as the spectrum of an operator, and Fredholm theory was discussed. After a brief aside on Gauss’s Theorem and Green’s Theorem (no text reference), we discussed distributions (test functions) and Sobolev spaces [Chapter 3], and concluded with a small discussion on harmonic analysis [first half of Chapter 2], including Poisson kernels. The Fourier Transform was a key tool in the discussion of results in Chapters 2 and 3 of the course. The problem sets were exceptionally challenging, moreso than the exercises in MAT335 and 425 and require key mastery of the concepts in the prior courses–truly an undertaking.

Other Courses

MAT 320: Introduction to Real Analysis
Offered in the Fall.

MAT 330: Complex Analysis with Applications
Offered in the Spring.

MAT 385: Probability Theory
Offered in the Spring, but not offered in Spring 2022.

ORF 526: Probability Theory
This course is a measure-theoretic based probability course. Prerequisites are ORF309 (or sufficient background in probability) along with some prior analysis coursework (usually MAT320, though standard track analysis usually suffices, and prerequisites are not strict for course entry). It begins with discussions of measurable sets, sigma-algebras and Borel-sets, and key theorems in Measure Theory (Fatou’s Lemma, Borel-Cantelli, the Monotone Convergence Theorem, Dominated Convergence Theorem). The course then discusses standard probability results through measure theory, such as the law of large numbers, central limit theorem, Markov chains, ergodic theory, conditional expectations, and martingales. This course has weekly problem sets, though they are slightly shorter than most standard weekly mathematics problem sets and take around 7-8 hours on average. This course is a popular way to satisfy the real analysis requirement, and is helpful for study of probability theory and associated and a prerequisite for higher level ORF classes so this is nice to take in Junior year if possible. Offered in the Fall.

MAT 427: Ordinary Differential Equations

MAT 429: Topics in Analysis
This course’s content varies from year to year. When Professor Klainerman teaches it, often in the Spring, it is usually entitled “Distribution Theory, PDE and Basic Inequalities of Analysis.”

MAT 522: Introduction to PDE
This course roughly follows the first eight chapters in Evans’ PDE text, with most of the time spent on 2 and 5-7. Problem sets were extremely challenging, though somewhat infrequent. However, this course, as an irregular offering, may change somewhat from year to year, so information may be outdated. Major topics discussed included the Fourier Transform, properties of harmonic functions, the maximum principle, sobolev spaces, elliptic PDE, and fundamental solution forms of PDE. The heat equation was discussed in depth, though other major PDE such as the transport equation, wave equation, and Laplace’s equation were also discussed. There was a main focus on elliptic PDEs and Sobolev spaces throughout the course though. A key supplemental reference for the course is the textbook by Gilbarg-Trudinger.

MAT 52x (x>0)
Each semester there is an assortment of graduate-level courses in analysis offered.

Contacts

Sumanth Maddirala ’22 (for general questions)

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