phd level math problems

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List of problems for graduate topics?

When I study a new topic, I never feel satisfied until I have spent some time solving a long list of problems.

I am looking for either a problem book or a list of problems on graduate math topics. While there is an abundance of problem books on undergraduate math topics (such as various websites on quals or books like Berkeley Problems in Mathematics), there seems to be fewer books at the graduate level with a lot of problems. There are books like Evan's PDE book or do Carmo's Riemannian Geometry book which has a good number of problems, but again, I feel like they are in the minority.

The closest thing to what I am looking for is the Cambridge Tripos III .

To clarify, the following is what I am looking for:

• A book with ≥ 10 problems for a particular topic.
• By "graduate topic," I mean anything that requires standard undergraduate curriculum (single/multivariate calculus, basic/Fourier analysis, ODEs, linear/basic algebra, point set topology, basic manifold theory, curves and surfaces, say) as a prerequisite. I am particularly interested in problem books for "advanced topics" whose prerequisites are standard graduate topics (algebraic/differential topology/geometry, measure theory, real/complex analysis, commutative algebra, representation theory, say).
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• 12 $\begingroup$ What's the advantage of specific "problem books" as opposed to the more traditional exercises within a textbook (which are also common even at the graduate level)? $\endgroup$ –  Sam Hopkins Commented May 19, 2021 at 3:07
• 5 $\begingroup$ What about solving old phd qualifyling exams from various universities? There is a lot of content online, and not all universities evaluate the same topics. $\endgroup$ –  efs Commented May 19, 2021 at 4:54
• $\begingroup$ @EFinat-S Sure, but qual exams usually don't go far beyond the "standard graduate topics" I described above. I was hoping there are lists of problems for more advanced topics as well. $\endgroup$ –  user676464327 Commented May 19, 2021 at 9:40
• 1 $\begingroup$ @SamHopkins Well for one, it is nice to have everything in one place; I can also work with one consistent notation/convention. I also think longer "problem books" have more interesting problems, especially towards the end of the list. (Just compare any two undergraduate textbooks on, say abstract algebra.) $\endgroup$ –  user676464327 Commented May 19, 2021 at 9:42
• 3 $\begingroup$ The Cambridge Part III Guide to Courses 2020-21 has suggested literature for each course, though the courses rarely follow a particular book. Past Tripos questions are also available, though the courses change year to year. $\endgroup$ –  Henry Commented May 21, 2021 at 22:19

9 Answers 9

Pólya-Szegő seems unsurpassed as a graduate level problem book on classical function theory. Other classic examples are:

P. Halmos, A Hilbert space problem book,

A. Kirillov and A. Gvishiani, Theorems and problems in functional analysis, available in English and French, bseides the Russian original, and

I. Glazman and Yu. Lyubich, Linear analysis in finite-dimensional spaces, translated from the Russian.

You could try M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory and/or M. Ram Murty Problems in Analytic Number Theory .

• 3 $\begingroup$ There's also "Problems in the Theory of Modular Forms" by M. Ram Murty, Michael Dewar and Hester Graves which was written in the same spirit. $\endgroup$ –  Anurag Sahay Commented May 26, 2021 at 3:32

Just yesterday I was looking at Clark Barwick's 121 Exercises on Locally Compact Abelian Groups: An Invitation to Harmonic Analysis . The opening sentence is "This is a collection of challenging exercises designed to motivate interested students of general topology to contemplate Pontryagin duality and the structure of locally compact abelian groups."

László Lovász, Combinatorial Problems and Exercises: Second Edition . Over 600 pages, divided into Problems, Hints, and Solutions.

An extensive list is at MSE , including pointers to dedicated web sites, such as this one. A particularly comprehensive collection is in Problems and Solutions in Mathematics , with a list of advanced topics (including Galois theory, homotopy/homology, differential geometry of manifolds, measurability and measure, PDE's).

I might add

The Stanford Mathematics Problem Book by George Polya and Jeremy Kilpatrick.

These 20 sets of intriguing problems test originality and insight rather than routine competence. They involve theorizing and verifying mathematical facts; examining the results of general statements; discovering that highly plausible conjectures can be incorrect; solving sequences of subproblems to reveal theory construction; and recognizing "red herrings," in which obvious relationships among the data prove irrelevant to solutions.

• 2 $\begingroup$ What's the level of mathematics you need to solve these problems? Do they go beyond the standard undergraduate math? $\endgroup$ –  user676464327 Commented May 19, 2021 at 9:57

For differential geometry, there is (now in its second edition):

Gadea, Pedro M.; Muñoz Masqué, Jaime; Mykytyuk, Ihor V. , Analysis and algebra on differentiable manifolds: a workbook for students and teachers , Problem Books in Mathematics. London: Springer (ISBN 978-94-007-5951-0/hbk; 978-94-007-5952-7/ebook). xxv, 617 p. (2013). ZBL1259.53002 .

From the preface of the 2009 (revised first) edition: “This book is intended to cover the exercises of standard courses on analysis and algebra on differentiable manifolds for advanced undergraduate and graduate years, with specific focus on Lie groups, fibre bundles and Riemannian geometry.”

• 2 $\begingroup$ Note that this “Problem Books in Mathematics” series has quite a few more volumes, most with reviews linked from the ZBL page above. $\endgroup$ –  Francois Ziegler Commented May 19, 2021 at 13:10

There is John Dixon's book Problems in Group Theory . But I do not know whether you will classify it as graduate level or not. (It has the advantage that it is quite cheap.)

Instead of Cambridge Part III, you can look at Oxford's Part C and post-graduate courses .

Some of the books that I can remember from my PhD reading list under Peter Petersen at UCLA were: Characteristic Classes , by Stasheff, Morse Theory , Milnor, Dimension Theory , Hurewicz and Wallman, The Topology of Fibre Bundles , Steenrod.

We both agreed that Spivak's $5$ volume Comprehensive Introduction to Differential Geometry was quite useful. There was Galot, Hulin and La Fontaine's Riemannian Geometry .

I'll update this if I remember anything else.

It seems to me that Milnor's Topology from the Differentiable Viewpoint might have been also, and Spivak's Calculus on Manifolds probably wasn't, but I enjoyed it.

I also had a copy of his Riemannian Geometry in manuscript form, which is now available in the GTM series.

A good place to get these was from "Book Scientific", where Spivak himself used to pick up the phone. They had an $800$ number.

Perhaps finally, I don't think it's necessarily that important whether there are a lot of problems, because you should be primarily grappling with ideas and concepts, as opposed to just working problems. He did also give me a list of unsolved problems, which I have long since lost (maybe that's more what you were looking for!)

He also rated the journals, and recommended reading those. Annals of Mathematics and Journal of Differential Geometry, there were journals from Duke and Indiana university, there was the Pacific Journal of Mathematics, among others. Inventiones Mathematicae is also one of the best.

I think Lang's Algebra was probably on the list too. If not, it probably should have been.

• 3 $\begingroup$ When you say "grappling with ideas and concepts," what specifically are you doing? Are you working through examples and calculations that you come up with while reading the book? $\endgroup$ –  user676464327 Commented May 20, 2021 at 2:20

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Sample Qualifying Exams

Below are samples of recent qualifying exams offered in the department.  Descriptions of the topics covered on these exams can be found in the graduate handbook .  Current students who want access to more exams should consult Graduate Program Manager Kaitlyn O'Konis ( @email ).

Basic Exams

Adv calc/linear algebra, applied statistics, probability, advanced exams, advanced statistics version i, advanced statistics version ii, applied math, stochastics.

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PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Autumn 2024.

Tuesday, September 17: Applied Math , Room 384I and Algebra , Room 384H

Wednesday, September 18: Real Analysis , Room 384H

Thursday, September 19: Geometry and Topology , Room 384H

Department of Mathematics

Qualifying exams.

The Ph.D. qualifying examination in Mathematics is a written examination in two parts. The purpose of the Ph.D. qualifying examination is to demonstrate that the student has achieved a degree of mathematical depth and maturity in the core areas of real analysis and abstract linear algebra, has additionally cultivated advanced problem solving skills in graduate level mathematics, and is poised to undertake independent mathematical research. The content and timing of the qualifying exam allows this determination to be made within the first two years of graduate study.

The two parts of the examination are as follows.

• Part 1 covers the material presented in the core course MTH 511, Real Analysis
• Part 2 covers the material in MTH 543, Abstract Linear Algebra.

The qualifying exam is given twice each year, near the beginning of Fall and Spring terms. The two parts of the exam are usually given one or more days apart. A student may take each part of the Ph.D. qualifying examination a maximum of three times, with one additional free attempt before a student's first term in the program. To advance in the Ph.D. program, a student must pass both parts, but they do not need to be passed at the same time. A student must pass both parts of the exam by the end of spring term of the student’s second year of study.

Questions about the qualifying exam can be directed to the Chair of the Qualifying Examination Committee ( [email protected] ).

Fall 2024 Qualifying exams

Real analysis.

Tuesday, September 17, 2024, from 4:00-8:00 pm.

Linear Algebra

Thursday, September 19, 2024, from 4:00-8:00 pm.

Related Stories

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Making math inclusive: 2023 Math For All satellite conference

GEM Fellowship allows parent Ph.D. student space to thrive

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

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PhD Preliminary Exams

Last updated May 2, 2022.

The preliminary exams, or "prelims," are the examinations required by the department for admission to official candidacy for the PhD degree. Old prelims are available on-line .

Starting in 2018, the prelims will be administered twice a year: once in September (1st try) and once over the spring break (2nd try).   See “Exam Process” section below for changes to the eligibility criteria.

In 2022, the prelims will tentatively be offered September 12-16

• Algebra: Monday, September 12
• Analysis:  Wednesday, September 14
• Manifolds:  Friday, September 16

The second try for all of the prelims for 2023 will be announced at a later date.

Exam Process

These exams are offered every year in the mathematical subjects treated by the designated core courses: algebra,  analysis, and geometry/topology. They are intended to serve as an objective measure of a student's mastery of basic graduate level mathematics, as a milestone on the road to becoming an independent mathematician.

The written preliminary exams are given twice a year: once during the week in September that precedes the start of Autumn Quarter, and once during the break that follows Winter Quarter. Only the students who have completed the corresponding year-long core sequence with a grade of at least 3.0 each quarter are eligible for the second try.

Each exam is written and graded by a group of faculty that is kept confidential.  Examiners arrive at a recommendation that is given to the Graduate Program Committee, which makes a final decision on who has passed the exam. These decisions are communicated to students in brief memos roughly a week after the exam. 

Students who wish to have additional feedback on an exam are encouraged to do so by requesting a review with a faculty member in the area of the exam, and submitting a request that the Graduate Program Coordinator release the student's exam to that faculty member. If, as a result of this process, it is believed that an exam was graded incorrectly, then a student may submit a written request that the Graduate Program Committee reconsider its decision. The final decision on prelim results rests with the Graduate Program Committee.

A student may substitute completion of a full three-quarter sequence of a designated core course with a high enough grade for the passing of the corresponding preliminary exam.

Normally, students take two exams in September of the beginning of their second year, but students are welcome to attempt exams in September of year one.   A student is expected to pass two of the three exams by the end of the student's second year in the Ph.D. program. If you are concerned about not meeting this milestone, please speak with your preliminary advisor and/or the Graduate Program Coordinator.

Students' experiences in the program and on the tests are important to us. If you have a temporary health condition or permanent disability that requires accommodations for exams (conditions include but not limited to mental health, attention-related, learning, vision, hearing, physical or health impacts), you are welcome to contact DRS at 206-543-8924 or [email protected] or disability.uw.edu. DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions.  Reasonable accommodations are established through an interactive process between you, DRS, and the GPC.   If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to the GPC at your earliest convenience.

Exam Topics

There are three exams:

• Algebra: Topics at the level of 402-3-4 and 504-5-6.
• Analysis: Topics at the level of 424-5-6, 524-5, and 534.
• Manifolds: Topics at the level of 544-5-6.

Each syllabus below lists certain topics that have appeared on the exams. This list is advisory only – it is intended to suggest the level of the exams, not to prescribe exactly the material that will appear. Past exams can be a useful source of practice questions, but a student need not master all material that has been covered on these exams. A student who knows the material in the syllabus and who has spent some time solving problems should do well on the exams. 

• Linear algebra: vector spaces and linear operators, characteristic and minimal polynomials, eigenvalues and eigenvectors, Cayley-Hamilton theorem, Jordan canonical form, rational canonical form.
• Commutative rings : PIDs, UFDs, modules over PIDs, prime and maximal ideals, Noetherian and Artinian rings and modules, Hilbert basis theorem, local rings and Nakayama lemma, localization, Integral extensions, Noether normalization lemma, Hilbert Nullstellensatz, prime ideal spectrum.
• Rings and modules : simple modules, composition series, Jordan-Holder theorem for modules, semi-simple rings, Artin-Wedderburn theorem, tensor product.
• Group theory : nilpotence, solvability and simplicity, composition series, Sylow theorems, group actions, free groups, simple groups, permutation groups, and linear groups, direct and semi-direct product of groups, presentations in terms of generators and relations.
• Representation theory : group algebras, irreducible representations, Schur's lemma, Maschke's theorem, character theory.
• Field theory : roots of polynomials, finite and algebraic extensions, algebraic closure, splitting fields and normal extensions, finite fields, Galois groups and Galois correspondence, solvability of equations.
• Category theory and homological algebra : categories and functors, natural transformations, universal properties, products and coproducts, exact and  split exact sequences, 5-lemma and snake lemma, projective and injective modules, resolutions, (left and right) exact functors, adjoint functors, adjointenss of Hom and Tensor, Tor and Ext.

References: Dummit and Foote, Abstract Algebra , second edition; Lang, Algebra ; Herstein, Topics in Algebra ; Hungerford, Algebra ; J.-P. Serre, Linear Representations of Finite Groups ; M. Atiyah, I. Macdonald, Introduction to Commutative Algebra ; M. Reid, Undergraduate Commutative Algebra .

Topics in Real Analysis: Metric spaces. General measure and general integration theory, Lebesgue integral, convergence theorems. Banach spaces, Hilbert spaces, L p -spaces. Differentiation and its relation to integration in R n , signed measures, the Radon-Nikodym Theorem, representation of bounded linear functionals on C_0(X) for locally compact Hausdorff spaces X.

References : Folland, Real Analysis ; Royden, Real Analysis ; Rudin, Real and Complex Analysis ; Stein and Shakarchi, Real Analysis .

Topics in Complex Analysis: Basic theory of analytic functions from complex numbers to power series to contour integration, Cauchy's theorem and applications such as the maximum principle, Schwarz Lemma, argument principle, Liouville theorem etc.  

References : Ahlfors, Complex Analysis ; Conway, Functions of One Complex Variable , vol. 1 ; Marshall, Complex Analysis ; Rudin, Real and Complex Analysis (the chapters devoted to complex analysis).

Topics: Elementary manifold theory; the fundamental group and covering spaces; submanifolds, the inverse and implicit function theorems, immersions and submersions; the tangent bundle, vector fields and flows, Lie brackets and Lie derivatives, the Frobenius theorem, tensors, Riemannian metrics, differential forms, Stokes's theorem, the Poincare lemma, de Rham cohomology; elementary properties of Lie groups and Lie algebras, group actions on manifolds, the exponential map.

References: Lee, Introduction to Topological Manifolds , 2nd ed. (Chapters 1-12) and Introduction to Smooth Manifolds , 2nd ed. (all but Chapters 18 and 22);  

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Guide for Topics for the Qualifying Exams

The following describes the format and scope of Qualifying Exams in each of the six areas of graduate study. It is department policy that qualifiers be based on curriculum from the first year graduate sequences and any undergraduate prerequisites. Students, who have mastered those courses, should be able to pass the exams. Faculty members, who write the exams, are expected to implement this policy, and to adhere conscientiously to the guidelines that follow. Students, in turn, are expected to interpret each exam problem in a reasonable fashion, so as not to trivialize any solution. Copies of past exams and a record of previous passing scores are available from the department by request.

Qualifying Exams (affectionately known as Quals) are given twice a year and typically take place the week or two before classes begin each semester. A precise schedule is posted months in advance. Students are allowed six hours to take the exam. Food can be brought in to help fuel the brain. Faculty, who grade the exams, are expected to release the results before the last date for students to drop or withdraw from courses without receiving a DR or W on their transcripts, and within two weeks in any case.

The books listed for each area below should be more than sufficient to cover topics that will appear on the exam. It should be emphasized, however, that the exams are intended to test general knowledge and competence rather than any particular set of books or courses.

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Galois Theory

• Field extensions including: algebraic and transcendental elements, finite/algebraic/Galois/simple/separable/purely inseparable field extensions, separable and inseparable polynomials.
• Splitting fields and algebraic closures.
• The fundamental theorem of Galois theory.
• Examples including: finite fields, polynomials of degree at most 4, composite extensions.
• Primitive elements.

Reference text Dummitt and Foote’s Abstract Algebra book Chapter 13 (exlcuding 13.3) and Chapter 14 (excluding 14.7, 14.8, and 14.9). (110 pages).

General Algebra

You should know the meaning of and be able to give examples and non-examples of:

• Left/right/two-sided ideals, left and right modules, bimodules
• Annihilator of a module
• Matrix ring, quaternion ring, group ring
• Division ring, simple ring, zero-divisor
• Modules: Exact sequences of modules, tensor products, Hom, localization of modules, flat/projective/free modules, support of a module

References text For a commutative ring specifically, see the references below. For a not necessarily commutative ring, see Dummitt and Foote, Chapters 7 and 10-12. (220 pages).

Commutative Algebra

• Rings and ideals: prime/maximal/radical ideals, quotient rings, integral domains, localization of rings, local rings, polynomial rings, zero-divisors, nilpotent elements, nilradical, fraction fields, Nakayama’s Lemma.
• Modules: see the list in “general algebra.”
• Noetherian rings, including chain conditions and the Hilbert Basis Theorem.

Reference text In recent years 742, which concentrates on commutative algebra, has been taught from Altman-Kleiman’s A term of commutative algebra , and the relevant chapters would be 1-5 and 8-13. (55 pages). An alternate source would be Atiyah-MacDonald’s Introduction to Commutative Algebra chapters 1-3 and 6-7. (88 pages).

Group Theory

You should know the meaning of, and be able to give an example and a non-example of the following:

• order (of a group)
• order (of a group element)
• normal subgroup
• quotient group
• abelian group, nilpotent group, lower central series, solvable group, simple group, perfect group
• commutator subgroup, centralizer, normalizer, conjugacy class
• group homomorphism
• group action, orbit, stabilizer, transitive action, faithful action
• free group, finitely presented group
• p-group, symmetric group, permutation group, alternating group, dihedral group, general linear group

You should be able to:

• State and apply the orbit-stabilizer theorem;
• Compute the conjugacy classes of a finite group;
• Work fluently with free groups, matrix groups, and symmetric groups

Reference text Dummit and Foote Chapters 1, 2.1-2.4, 3.1-3.3, 4.1-4.3, 5.1-5.2, 5.4, 6.1, and 6.3.

Linear Algebra

• Eigenvalue, eigenvector, generalized eigenspace
• Jordan normal form
• dual vector space, transpose, bilinear form, Hermitian form
• orthogonal matrix, symplectic matrix
• tensor product of vector spaces

References text Dummit and Foote, chapters 11-12.

The Analysis Qualifying Exam involves the tools from a) advanced calculus, b) Math 721, and c) one of the two courses: Math 722 (Complex Analysis) and Math 725 (Real Analysis). Choose one at the time of exam registration.

The exam usually consists of nine questions and six are to be attempted. There will be at least two from each of a), b) and c), though some problems may involve tools from more than one area. The content of 721, 722, and 725 certainly varies somewhat from instructor to instructor. Questions for 2018-2019 will come from the topics and tools below.

Recommended texts Function theory of one complex variable by Greene and Krantz ; Functions of one complex variable by J.B. Conway . Good sources for additional reading and problems: old qualifying exams , Gamelin’s Complex Analysis,  Rudin’s Real and Complex Analysis, Stein-Shakarchi : Princeton Lectures in Analysis II: Complex Analysis.

• Analytic functions and Cauchy-Riemann equations. Elementary functions, branches and principal branches.
• Line integrals. Cauchy’s theorem and Cauchy’s formula.
• Cauchy’s estimates, Liouville’s theorem, Morera’s theorem, Goursat’s theorem.
• Power series, Laurent series, and isolated singularity. Residue calculus.
• Argument principle, Rouche’s theorem, Hurwitz’s theorem, open mapping theorem.
• Simply connected domains. Normal families and Montel’s theorem.
• Conformal mappings of the unit disc and upper half-plane, fractional linear transformations. Schwarz’s lemma. Elementary conformal mappings. The Riemann Mapping Theorem.
• Harmonic functions, the mean value property and maximum principle, Harnak’s lemma and principle, subharmonic functions.
• Dirichlet problem on the unit disc. Schwarz reflection principle. Perron’s theorem.
• Mittag-Leffler’s theorem, Runge’s theorem.

Recommended texts The principal reference is Folland’s Real Analysis:  Modern Techniques and Their Applications, Chapters 1-5. Good sources for additional problems:  old qualifying exams,  Rudin’s Real and Complex Analysis , Chapters 1-8.  Chapter 2 of Rudin’s Functional Analysis (for problems on the Baire Category Theorem). Stein-Shakarchi: Princeton Lectures in Analysis III: Real Analysis.

Reference Chapters 1 and 2 of Folland.

Reference Chapter 3 of Folland, excluding functions of bounded variation.

• Basic point set topology, commensurate with Chapter 4 of Folland, particularly non-metric topologies, locally compact and locally convex spaces.

Reference Chapter 5 of Folland.

Recommended texts Details are given in the list of topics. Folland, Chapters 6-9. Rudin’s Functional Analysis , Chapters 6-8. (Distribution theory is typically taught at the level of Rudin’s  Functional Analysis , rather than Folland. Stein and Shakarchi’s  Princeton Lectures in Analysis IV: Functional Analysis , Chapter 4. (Good reference and problems for further consequences of the Baire Category Theorem.)

Reference text Chapter 6 of Folland.

Reference text Chapter 5 of Folland.

Chapter 8 of Folland, Chapter 7 in Rudin.

Chapters  1,  6, 7  Rudin.

9.3 of Folland.

Chapter 4 of Stein and Shakarchi (omitted in 2021)

Applied Math

The Applied Mathematics Qualifying Exam consists of six problems, all of which are to be attempted. The exam is based on material usually covered in undergraduate ordinary differential equations, partial differential equations, complex variables, and the first-year graduate sequence in Applied Mathematics (Math 703-704).

References Churchill, Fourier Series and Boundary Value Problems Gelfand and Fomin B, Calculus of Variation Kevorkian, Partial Differential Equations Levinson and Redheffer, Complex Variables Pinsky B, Partial Differential Equations and Boundary Value Problems Stakgold, Green’s Functions and Boundary Value Problems Strang, Introduction to Applied Mathematics Zanderer B, Partial Differential Equations

Computational Math

The Computational Mathematics Qualifying Exam is offered at the beginning of every fall and spring semester. Students have 6 hours to complete the exam of about 5-6 problems. The exam is typically 120 points in total. The material is based on Math/CS 714 and Math/CS 715. The students taking Math 714 / 715 are assumed to have a basic understanding of the undergraduate level of numerical analysis (covered in Math 513 / 514).

Covered Materials for Math Students

All materials that may appear in the exam are listed below. The components that are marked “advanced” are unlikely to appear, but we do not rule out the possibility. (Please contact the most recent instructor of 714/715 for details.)

• Basic ODE Theory: well–posedness
• Explicit and implicit methods, stability Runge-Kutta and multistep methods, stiff problems
• Numerical differentiations, uniform and nonuniform meshes
• Consistency, stability and convergence
• Multidimensional problems: ADI and fractional step methods
• Linear hyperbolic equations and their numerical discretizations
• Basic theory for nonlinear hyperbolic equations: shock formation, weak solution and entropy condition, Riemann problem
• (advanced) Shock capturing methods: Godnov and Roe methods, slope limiters, flux-splitting
• (advanced) Hamilton-Jacobi equations and the level set method for front propagation
• Fast Fourier transform
• Fourier spectral method, pseudospectral methods, Chebyshev method
• Direct and iterative methods for linear systems, eigenvalue problems, sparse matrices
• Conjugate gradient methods, nonlinear algebraic equations
• Variational formulation, Galerkin methods, energy estimate and error analysis, implementation
• (advanced) Discontinuous Galerkin, multigrid methods, boundary element method

References Many textbooks cover similar topics. If textbook (A) is listed under topic (b), that means we believe (A) organizes materials (b) better than other textbooks. However, every student is different. Ultimately please choose textbooks according to your own preferences. We only list recommendations below. Basics: Basic Numerical Analysis Suli and Mayer, An Introduction to Numerical Analysis Finite Difference Methods LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007. Spectral Methods Trefethen, Spectral Methods in MATLAB, SIAM, 2000. Gottlieb and Orzag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977. Finite Element Methods: Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009. Larson and Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013. Advanced: Finite Volume Methods LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. Monte Carlo Methods: Kalos and Whitlock, Monte Carlo Methods, J. Wiley & Sons, New York, 1986.

Geometry/Topology

Logistics of the exam:

When registering for the exam, students must choose either the algebraic topology option or the differential topology option. The algebraic topology option is based on the courses Math 751/752, and the differential topology option is based on Math 751/761.

The exam consists of two parts, Part I and Part II. Each part has three questions. Part I is the same on both exams, and covers material from 751.  Part II of the Algebraic Topology option covers material from 752, and Part II of the Differential Topology option covers material from 761.

Students are asked to answer two questions from Part I and two questions from Part II.

The exam is based on (a) background material usually covered in advanced calculus, undergraduate topology (e.g. 551) and undergraduate algebra courses (e.g. 541), and (b) topics from the first year graduate topology sequence (751, 752, 761), as identified below. Note that familiarity with basic concepts of point set topology (e.g. metric spaces, completeness, connectedness, and compactness) will be assumed, although these may not be treated in 751, 752, 761.

Reference texts: The reference text for 751 and 752 is Allen Hatcher’s Algebraic Topology. The reference text for 761 is John Lee’s Introduction to Smooth Manifolds. Ch 1-6 8 up to Lie brackets 9 up to Lie derivatives 12,14 15 up to Orientations of Manifold 16 up to Stoke’s Theorem 17 up to Homotopy Invariance Additional reference texts for 761 are Frank Warner’s Foundations of Differentiable Manifolds and Lie Groups  and Spivak’s A Comprehensive Introduction to Differential Geometry, Volume I

Description of advanced material covered by the exam.

Part I. The student should be prepared to:

• Work with the standard constructions in algebraic topology, such as homotopies, chain complexes, quotients, products, suspensions, retracts, and deformation retracts.
• Effectively use the fundamental tools of homology, reduced homology, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence for homology.
• Compute the fundamental group of an explicitly given cell complex.
• Compute the fundamental group of a space using the Seifert-Van Kampen Theorem.
• Compute the homology of an explicitly given cell complex using the definition of cellular homology.
• Make use of the standard cell structures of spheres and real and complex projective spaces in all dimensions.
• Know the fundamental and homology groups of spheres and real and complex projective spaces in all dimensions.
• Compute the homology of a space using the Mayer-Vietoris sequence.
• Make use of the long exact sequence in homology to make computations.
• Compute the Euler characteristic of a space.
• Construct finite covering spaces of an explicitly given cell complex.
• Construct covering spaces with prescribed group of deck transformations by constructing a corresponding quotient of the fundamental group.
• Use contractibility of the universal cover to deduce that certain maps are null-homotopic.
• Use local homology to distinguish two spaces.
• Use the Lefschetz fixed point theorem to find a fixed point of a continuous map.
• Combine the above machinery and techniques to solve problems.

Part II.   Algebraic option. The student should be prepared to:

• Compute the cohomology of an explicitly given cell complex using the definition of cellular cohomology.
• Effectively use the fundamental tools of cohomology, reduced cohomology, cup product, cap product, cross product, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence.
• Apply the Universal Coefficient Theorem in computations.
• Compute cup products of cohomology classes.
• Distinguish the homotopy types of two spaces using the Cohomology Ring.
• Make effective use of Poincaré duality.
• Make elementary computations of homotopy groups using the Hurewicz Theorem.
• Know the homotopy groups of the n-sphere through dimension n.
• Know the homotopy groups of the 2-sphere through dimension 3.
• Build continuous maps between cell complexes inductively using high-connectivity of the target: e.g. “Using the fact that Y is k-connected, construct a map from the given X to Y.”
• Make effective use of Whitehead’s Theorem.
• Recognize and construct fiber bundles.
• Use the long exact sequence of homotopy groups of a fibration.
• Know the standard examples of fiber bundles of spheres over spheres arising from the unit spheres in the real division algebras.

Part II. Differential Option. The student should be prepared to:

• Work with the standard concepts in differential topology, including smooth manifolds, local coordinates, transversality, regular values, the Inverse Function Theorem, tubular neighborhoods, vector fields, flows, differential forms, orientation, integration of forms, distributions, basic de Rham cohomology, and Stokes Theorem.
• Perform computations with differential forms, including integration of explicit forms over given submanifolds.
• Perform computations with the Lie derivative.
• Make use of Sard’s theorem.
• Distinguish de Rham cohomology classes given explicit forms on an explicit manifold.
• Show that a given manifold admits a smooth structure.  For example, the student should be able to show that spheres, projective spaces, Grassmannians, the special linear group, and the orthogonal group admit smooth structures.
• Construct trivializations of explicit vector bundles, such as the tangent bundle of the 3-sphere.

The Logic Qualifying Exam will consist of (usually 6) questions based on the content of the two introductory graduate courses: 770 and 773.

Students should be prepared to answer questions on the following topics. Since these topics may be presented in different ways from year to year, the student should read broadly from the references to supplement the course work.

First-order logic syntax and semantics, Completeness and Compactness Theorems, Löwenheim–Skolem Theorem, Incompleteness Theorem, decidable and undecidable theories, basic properties of ordinals and cardinals.

References Ebbinghaus, Flum and Thomas: Mathematical Logic (Chs. 1–6 and 10) Kunen: The Foundations of Mathematics Kunen: Set Theory (1980 Elsevier edition, Chs. 1 and 3)

Computability Theory

Computable sets and (partial) computable functions, Recursion Theorem, computably enumerable sets, halting problem, Turing reducibility, Turing degrees and jump, arithmetical hierarchy, index sets, low and high degrees, Martin’s high domination theorem, Friedberg and Shoenfield jump inversion, minimal degrees, exact pairs, 1-generic, hyperimmune, and hyperimmune-free degrees, diagonally non-computable functions, Π01-classes, PA degrees, low and hyperimmune-free basis theorems, finite injury, Friedberg-Muchnik theorem, Sacks Splitting theorem, priority trees, infinite injury, Sacks jump inversion, computable ordinals, Kleene’s O, hyperarithmetical hierarchy.

References Soare: Recursively Enumerable Sets and Degrees (Chs. 1–8) Ash/Knight: Computable Structures and the Hyperarithmetical Hierarchy (Chs. 4.5-5.3)

Model Theory

Elementary chains and extensions, preservation theorems, ultraproducts, quantifier elimination, model completeness, types, saturated and special models, small theories, countable categoricity, strong minimality, Baldwin-Lachlan characterization of uncountably categorical theories.

References Hodges: A Shorter Model Theory Marker: Model Theory, An Introduction (up to Ch. 6.1) Tent, Ziegler: A Course in Model Theory (Chs. 1-5)

Qualifying Exams

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Qualifying Examination

The Qualifying Examination (QE or orals) in Mathematics is an oral examination that covers three principal topics, two of which are designated as major topics, and one as a minor topic; the minor topic is examined in less depth than the major topics. The intent of the QE is to ascertain the breadth of the student's comprehension in the selected subject areas, and to determine whether the student has the ability to think incisively and critically about the theoretical and practical aspects of the topics. The exam is administered by a faculty committee of four members and lasts approximately 3 hours. The result of the exam is decided via a committee vote. Possible results include, pass, partial fail, and total fail. Students who receive a partial or total fail may be granted, by the committee, a second opportunity to take the exam on some or all of the topics covered, or may be recommended for dismissal from the Ph.D. program.  

For Current Students

Students in the Mathematics Ph.D. program are required to attempt the QE before the start of their third year in the program (within 24 months of matriculation). The Graduate Office has accommodated requests for short extensions of this time frame so that students may take their orals in the fall term of their third year. Such requests must be made in advance to and approved by the Vice Chair for Graduate Studies. Any further request for an extension must be approved by Committee Omega. Most QE exams take place over the fall or spring terms as faculty are often away from campus during the summer months. ***For international students, it is quite crucial that students pass their orals and advance to candidacy before the start of the fall term of their third year. Advancing to candidacy initiates a 3-year (calendar year) Nonresident Supplemental Tuition (NRST) reduction. In most circumstances, the program only agrees to pay NRST for international students for 4 semesters, so to avoid paying NRST on one's own in the third year, students must advance to candidacy and trigger the NRST reduction.***

Steps to Scheduling the Qualifying Examination

• Identify a Faculty Advisor (Dissertation Chair): To arrange the QE, you must first settle on an area of concentration, and identify and secure a prospective Dissertation Chair (Faculty Advisor) — someone who agrees to supervise your dissertation research if the examination is passed.
• Committee Formation:   Two months before the date of the exam, you must constitute the QE committee. Students should seek advice from their prospective dissertation chair (faculty advisor) when putting together their exam committee. All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee.
• Create an Exam Syllabus: In consultation with the Chair of the QE committee, you will write up an exam syllabus. Sample syllabi and a list of faculty who have served as outside members (formally called Academic Senate Representatives (ASR)) in the past are all available in the QE Resource Folder linked below. The syllabus must indicate the departmental sections for which you will be examined and must be approved by the QE Committee. To properly format your syllabus, please use the template provided in the QE Resource Folder linked below.
• Distribution of Exam Syllabus to Faculty: 6 weeks before the exam, you should send a PDF copy of the QE syllabus to the Graduate Advisor who will distribute it to all faculty in the two (or three) sections in which you will be examined. You should allow two weeks for a response from the faculty. If feedback on the syllabus content is received, the student must then consult the Chair of the QE Committee and their prospective advisor before proceeding with changes to the syllabus. Starting in Fall 2024 you can submit the  Application for the Qualifying Exam  (hyperlinked) to notify the Math Grad Office.
• Exam Logistics: 4 weeks before the date of the exam, the student should reserve a room using the room reservation calendar or schedule a remote meeting via Zoom. Any department space with chalk or whiteboard (aside from 1015 Evans Hall) should be suitable for the exam. If a room in the department is not available you can ask Main Office staff to reserve a room in Evans from the central Scheduling Office. If you are taking the QE on Zoom, please review these Best Practices for Zoom Qualifying Exams .

Four weeks before the exam you should submit the Graduate Division Application for the QE which is an eForm housed in CalCentral . The eForm is called the “Higher Degrees Committees Form.” Once your QE is formally approved at the Graduate Division level you will see a notice of approval on the My Academics tab of your CalCentral account. Please note that the QE will not be valid until approved at the Graduate Division level. Failure to submit the application four weeks prior to your QE may require you to reschedule the exam!

Two or three days before the exam date, the student should send a reminder to all committee members with all exam details (date, location, time) and a copy of the syllabus.

Other Details

At the conclusion of the exam, the QE Committee submits the exam results to the Graduate Office, which is then submitted to the Graduate Division.

Once you have passed the QE you are required to advance to candidacy by the end of the following semester. Students who fail to advance to candidacy in the stated time will have an enrollment hold placed on their account.

QE Resources

The QE Post-Exam Survey: Students are sent the survey via email by the advising staff shortly after completing any exam attempt. Responses include data on a student's advisor, committee members, topics examined, questions asked, and general exam experience. The response sheet is viewable by all current students here. Students may sort by creating a filtered view in the "Data" drop-down menu. You must be logged into your  @berkeley.edu account to access this resource.

CLICK HERE  TO VIEW RESPONSES  TO THE QE POST-EXAM SURVEY

QE Resource Folder:   The QE Resource Folder has been replaced by the post-exam questionnaire, but will remain active and accessible. The folder includes copies of past student syllabi, a QE syllabus template, a list of faculty from outside of the Math Department who have served as Academic Senate Representatives (Outside Member), and other useful documents.  You must be logged into your @berkeley.edu email account to access this folder. 

CLICK HERE TO ACCESS RESOURCE FOLDER

CalCentral eForm Support Resources

Having trouble with the CalCentral Higher Degrees Committee eForm? Use the guides below for assistance.

• Click here for a YouTube tutorial on how to apply for the QE
• Click here for a step-by-step guide for completing the Higher Degrees Committee eForm

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