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