phd qualifying exam math

The first milestone in the Mathematics PhD program are the qualifying exams. Exams are offered in Fall (before the academic year begins) and in Spring. PhD students must pass at least one exam before the start of their 4th quarter. All exams must be completed before the start of the student's 7th quarter. Failure to meet these deadlines is cause for dismissal from the program. Carefully read the Guidelines for Graduate Qualifying Exams document.

Exam requirements are different depending on which program a student is in. Please refer to the UCSD catalog for specific requirements:  https://www.ucsd.edu/catalog/curric/MATH-gr.html .

During any examination period the student may take as many exams as he or she chooses. The qualifying exams are written and graded by the faculty who teach the courses. The scores are brought before the Qualifying Exam Appeals Committee (QEAC) and the grades are discussed. The final decision as to whether the student has failed or passed (and at what level) is made by QEAC. This decision is based upon exam performance, and performance in exam cognate coursework, though the QEAC is free to consider additional circumstances in rendering its decision. After the QEAC meeting, the PhD staff advisor will inform students when/how they can find out their results.

Students can request to see their exams after grading in order to find out what they did well/poorly on. Students who wish to see their exam for purpose of contesting the grading should be advised that there will be a very strong burden of proof needed to sustain a grade appeal on a graduate exam because of the nature of the exam writing and grading process. Such an appeal is most likely not going change the exam result.

Qualifying Exam Requirements, Old and New

The Department of Mathematics has undertaken a reform of our Qualifying Exams. This brief note explains the old/current system, the new system, and how the changes are being phased in. These requirements apply to PhD students in Mathematics ; Statistics and CSME PhD students have separate requirements administered by the faculty.

Qualifying Exam Courses and Areas

There are 7 qualifying exams administered each Spring and Fall. Each corresponds to a three-quarter graduate course. They are organized into three Areas.

Old/Current Requirements

For PhD students who entered our program in Fall 2023 or earlier, the following are the current requirements to complete the qualifying exams.

• Each exam is assigned one of four grades: PhD Pass, Provisional PhD Pass (also known as PhD- Pass), Masters Pass, and Fail. The grade cutoffs are determined by the instructors who create/grade the exams; those cutoffs are not released to students.
• At least one exam must have a PhD Pass.
• At least one additional exam must have a Provisional PhD Pass or better.
• At least one additional exam must have a Masters Pass or better.
• Students must pass at least one exam from Area 1 , and at least one exam from Area 2 .
• Students must have two exams, each with a Provisional PhD Pass or better, from two different Areas .
• Students must pass at least one exam with a Provisional PhD Pass or better before the start of their 4th quarter .
• Students must complete all the qualifying exams before the start of their 7th quarter

New Requirements

For students who enter our program in Fall 2024 or later, the following are the requirements to complete the qualifying exams.

• PhD Area Pass indicates readiness to begin research in that area. This grade is equivalent to PhD Pass in the current system.
• PhD General Pass indicates sufficient familiarity with the subject to begin research in a different area. This standard is lower than Provisional PhD Pass, and higher than Masters Pass .
• Masters Pass is only relevant for Masters students. A Masters Pass no longer counts towards completion of qualifying exams for PhD students.
• At least one exam must have a PhD Area Pass.
• At least two additional exams must have a PhD General Pass or better.
• Students must complete qualifying exams from at least two different Areas .
• Students must pass at least one exam before the start of their 4th quarter .
• Students must complete all the qualifying exams before the start of their 7th quarter .

Principal Differences

The new system has more flexibility: students no longer have to take quals from both Areas 1 and 2, simply from 2 distinct Areas among 1, 2, and 3. The standards for completion are simplified. Although Masters Pass is no longer a sufficient standard for PhD students, the PhD General Pass standard is lower than the old Provisional PhD Pass standard, and more consistent with the intent of the exams: to prepare students for focused research in one main area.

Phasing In Period

Any current PhD students (who entered in Fall 2023 or earlier) still progressing towards completing the qualifying exams may satisfy either the current or the new requirements . To be precise:

• Each Spring and Fall (in fact starting this past Fall 2023), qual instructors will select cutoffs corresponding to all five possible grades:

PhD Pass = PhD Area Pass > Provisional PhD Pass > PhD General Pass > Masters Pass > Fail

• At each qual session, each PhD student’s file will be evaluated using both the current and the new requirements. It will be judged complete if it satisfies the current requirements or if it satisfies the new requirements.

Caveat : students who entered in Fall 2022 or earlier already have qualifying exams graded only using the old cutoffs. Qualifying exams from Spring 2023 or earlier will not be regraded to compute PhD General Pass cutoffs.

Other Aspects of Qualifying Exam Reforms

In addition to the logistical changes described above:

• Faculty will be undertaking the creation of standardized syllabi for all seven qualifying exams, to be available to PhD students upon entry. This is a process that will take the faculty significant time and energy to complete, and is planned to be available starting in Fall 2024 .
• In the meantime, qualifying exam course instructors will give detailed syllabi in each course (as always, per Academic Senate regulations), and content cutoffs for the exams will be communicated to students by the Graduate Advisor in advance of the qualifying exams. The same content cutoffs will apply to both Spring and Fall qualifying exams, as has been standard.
• There will be closer coordination of mentoring efforts by course advisors and the Vice Chair for Graduate Affairs. All advisors for first-year PhD students will formulate plans for course enrollment for the full year, as well as plans for which qualifying exams to take in Spring 2024 . Advisors should meet again with their advisees before the beginnings of Winter and Spring quarters, and possibly make adjustments at those times.
• Preliminary full year course and qualifying exam plans should be submitted by the advisors to the Graduate Vice Chair by the end of Week 1 of the Fall quarter.

Spring 2024 exam schedule

Topology - Monday, May 13  1:00 - 4:00 AP&M 6402

Real Analysis - Tuesday, May 14 1:00 - 4:00 AP&M 6402

Statistics - Wednesday, May 15 1:00 - 4:00 AP&M 6402

Complex Analysis - Monday, May 20 9:00 - 12:00 AP&M 6402

Numerical Analysis - Tuesday, May 21 9:00 - 12:00 AP&M 6218

Algebra - Wednesday, May 22 9:00 - 12:00 AP&M 6402

Applied Algebra - Thursday, May 23 9:00 - 12:00 AP&M 6218

Sample Qualifying Exams

Algebra (Math 200A/B/C): SP04 ,  SP05 ,  SP06 ,  FA06 ,  SP07 ,  FA07 ,  SP08 ,  FA08 ,  SP09 ,  FA09 ,  FA10 ,  SP11 , FA11 ,  SP12 ,  SP13 ,  FA13 ,​​​​ SP14 ,  FA14 ,  SP15 ,  SP16 ,  SP17 ,  FA17 ,  SP18 , FA18 ,  SP19 ,  FA19 ,  SP20,    FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23 , FA23

Applied Algebra (Math 202A/B/C): SP04 ,  FA04 ,  SP05 ,  SP06 ,  SP08 ,  FA06 ,  SP07 ,  FA07 ,  FA11 ,  SP11 ,  SP13 ,  SP15 ,  SP17  ,  FA17 ,  SP18 ,  FA18 ,  SP19 ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23A , SP23B , FA23A , FA23B , FA23C

Complex Analysis (Math 220A/B/C): SP04 ,  SP05 ,  FA05 ,  SP06 ,  FA06 ,  SP07 ,  FA07 ,  SP08 ,  FA08 ,  SP09 ,  FA09 ,  FA10 ,  FA11 ,  FA15 ,  SP11 ,  SP12 ,  SP13 ,  FA13 ,  SP15 ,  FA16 ,  SP17 ,  FA17 ,  SP18 ,  SP19 ,  FA19 ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23 , FA23

Numerical Analysis (Math 270A/B/C): SP99 ,  SP00 ,  FA00 ,  SP01 ,  FA01 ,  SP02 ,  FA02 ,  SP03 ,  FA03 ,  SP04 ,  FA04 ,  SP05 ,  FA06 ,  SP06 ,  FA07 ,  SP07 ,  SP08 ,  FA08 ,  SP09 ,  FA09 ,  FA10 ,  SP11 ,  SP13 ,  FA15 ,  SP17 ,  FA17 ,  SP18 ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23 , FA23

Real Analysis (Math 240A/B/C): SP04 ,  FA04 ,  FA05 ,  SP06 ,  FA06 ,  SP07 ,  FA07 ,  SP08 ,  SP09 ,  FA09 ,  FA10 ,  FA11 ,  SP11 ,  SP13 ,  SP15 ,  FA16 ,  SP16 ,  SP17 ,  FA17 ,  SP18 ,  FA18 ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23 , FA23

Statistics (Math 281A/B): SP99 ,  FA99 ,  SP00 ,  FA00 ,  SP01 ,  SP02 ,  FA02 ,  SP03 ,  FA03 ,  SP04 ,  SP05 ,  SP06 ,  SP07 ,  SP08 ,  SP09 ,  FA10 ,  SP11 ,  SP13 ,  FA15 ,  SP17 ,  FA17 ,  SP18 ,  SP18 Formulas ,  SP19 Part A ,  SP19 Part BC ,  FA19 (Part A) ,  FA19 (Part BC) ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23AB , SP23C , FA23AB , FA23C

Topology (Math 290A/B/C): SP00 ,  SP01 ,  SP02 ,  FA02 ,  FA03 ,  SP04 ,  FA04 ,  SP05 ,  SP06 ,  SP07 ,  FA06 ,  FA07 ,  SP08 ,  FA08 ,  FA09 ,  SP10 ,  FA10 ,  SP11 ,  SP13 ,  FA15 ,  SP17 ,  FA17 ,  SP18 ,  FA18 ,  FA19 ,  SP20 ,  FA20 ,  SP21 , FA21 , SP22 , FA22 , SP23 , FA23

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Department of Mathematics

• Qualifying Exams

• Prospective Graduate Student FAQ
• Graduate Board Oral Exam
• Graduate Courses
• Recent PhD Theses
• Graduate Awards

Core Qualifying Exams

The core qualifying exams consist of two written exams, one in algebra and one in combined real analysis and complex analysis. These are two- to three-hour exams covering the core material in each subject. The exams are administered twice a year, in September and May. Exams may be taken one at a time. Entering students are invited to attempt the September exams in their first year. Both exams must be passed by September of the second year.

Qualifying exam syllabi:

• Algebra Exam Syllabus
• Analysis Exam Syllabus

Old qualifying exams:

• Old Algebra Exams (combined PDF file)
• Old Analysis Exams (combined PDF file)

Fall 2024 Exams

• Analysis: Tuesday, August 27, 12:00 noon – 3:00 pm.
• Algebra: Thursday, August 29, 12:00 noon – 3:00 pm.

Spring 2024 Exams

• Analysis: Tuesday, May 7, 12:00 noon – 3:00 pm.
• Algebra: Thursday, May 9, 12:00 noon – 3:00 pm.

Major Oral Exam

In addition to the core qualifying exams, there is an oral major exam in the intended area of research. Examining committees consist of two faculty members (including the intended adviser). The intended adviser will determine in advance the exam material and the second committee member. Students have two attempts to pass this exam. The major exam must be passed by April 8 of the third year.

Qualifying Exams

The Qualifying Examination is an oral exam given by a committee of three faculty members. Each student chooses three qualifying exam topics and discusses the content with suitable examiners. The topics must be in distinct, relatively broad areas of mathematics. The major topic is usually chosen in consultation with the prospective thesis advisor. After passing the qualifying exam, students must designate a thesis advisor, most often the examiner in the major topic.

Each minor topic should correspond to a single-semester graduate subject or equivalent content. The major topic should go into significantly greater depth. If either minor topic is related to the major topic, it should be covered deeply enough to contain sufficient disjoint material and the other minor topic should be largely unrelated.

Students have two chances to pass. The exam can be taken as soon as the student feels ready; however the first attempt should be scheduled before the start of the fourth semester. The exam must be completed by the last day of the fourth semester.

In order for a student to pass the qualifying exam, at least one faculty member must be willing to serve as thesis advisor. Therefore students are strongly encouraged to talk to a potential advisor when assembling their qualifying exam committee to confirm in advance that there is a faculty member willing to serve as advisor if the student passes.

Instructions

At least one month prior to the examination, students must obtain approval of the examination topics and the composition of the examining committee as follows.

Students must submit by e-mail to Davesh Maulik (for Pure Mathematics) or to Jon Kelner (for Applied Mathematics), with a copy to the examiners:

• Approximate Date of Exam
• Major Topic; Examiner; Description by course number(s) and/or subject matter from text or readings (only required for the major topic if the Examiner is outside the department)
• Minor Topic 1; Examiner; Description by course number(s) and/or subject matter from text or readings
• Minor Topic 2; Examiner; Description by course number(s) and/or subject matter from text or readings

Scheduling the Qualifying Exam

Having obtained approval by return e-mail, students should file the following forms, complete with all signatures, to Math Academic Services .

• Application for Qualifying Examination
• Proposal for Qualifying Examination Coverage

The examination usually takes place in the office of an examiner. Students are responsible for making all the scheduling arrangements.

Reporting the Results of the Qualifying Exam

Students should bring the Report on Qualifying Examination Form to the quals. This form is to be signed by the three examiners at the completion of the quals and then returned to the Math Academic Services office.

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

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] ).

Spring 2024 Qualifying exams

Real analysis.

Wednesday, April 3, 2024, from 4:00-8:00 pm.

Linear Algebra

Friday, April 5, 2024, from 4:00-8:00 pm.

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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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Sample Qualifying Exams

Some old departmental qualifying exams are available here (all links are PDF’s)

Some old qualifications questions from 1990-2002 [PDF] Some PDF files of questions arranged by topics.

• PDE’s
• C* algebras
• probability
• algebraic geometry
• differential topology
• representation theory
• algebraic topology

Collected by Danny Calegari and Tom Coates.

• banach spaces
• fluid mechanics
• combinatorics
• foundations
• measure theory
• commutative algebra
• galois theory
• noncommutative rings
• universal algebra
• complex analysis
• game theory
• numerical analysis

Ph. D. Qualifying Exams

This page contains a list of topics for the written qualifying exams in algebra, analysis and geometry/topology and copies of past exams. Hard copies of past exams from 2006-present are available in the graduate office.

Geometry/Topology

Ph.D. Program Overview

Description.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

Core Courses

A normal course load for a beginning graduate student is three courses per term. 

There are no qualifying exams, but the program requires that all students pass four courses to be selected from the six core courses. First-year students are allowed to place out of some (possibly, all) of the core courses. In order to place out of a course, students should contact the faculty member who is teaching the course during the current academic year, and that faculty member will make a decision. The minimum passing grade for the core courses is B-; no grade is assigned for placing out of a core course.

At least two core courses should be taken (or placed out) by the end of the first year. At least four core courses should be taken (or placed out) by the end of the second year (cumulative). These time requirements can be waived for students with health problems or other significant non-academic problems. They can be also waived for students who take time-consuming courses in another area (for example, CS) and who have strong support from a faculty; requests from such students should be made before the beginning of the spring semester. 

The core courses  are distributed among three main areas: analysis, algebra and topology/geometry. A student must pass at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. 

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

Students who are not ready to take some of the core courses may take MATH 4130-4140, Introduction to Analysis, and/or MATH 4330-4340, Introduction to Algebra, which are the honors versions of our core undergraduate courses.

"What is...?" Seminar

The "What Is...?" Seminar is a series of talks given by faculty in the graduate field of Mathematics. Speakers are selected by an organizing committee of graduate students. The goal of the seminar is to aid students in finding advisors.

Schedule for the "What Is...?" seminar

Special Committee

The Cornell Graduate School requires that every student selects a special committee (in particular, a thesis adviser, who is the chair or the committee) by the end of the third semester.

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. 

Entering students are not assigned special committees. Such students may contact any of the members on the Advising Committee if they have questions or need advice.

Current Advising Committee

Analysis / Probability / Dynamical Systems / Logic: Lionel Levine Geometry / Topology / Combinatorics: Kathryn Mann Probability / Statistics:  Philippe Sosoe Applied Mathematics Liaison: Richard Rand

Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her special committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the core graduate courses. 

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or B Exam. All doctoral students take a Final Examination (the B Exam, which is the oral defense of the dissertation) upon completion of all requirements for the degree, no earlier than one month before completion of the minimum registration requirement.

Masters Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for this degree, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are explained on the referenced page.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

Graduate Student Funding

Funding commitments made at the time of admission to the Ph.D. program are typically for a period of five years. Support in the sixth year is available by application, as needed. Support in the seventh year is only available by request from an advisor, and dependent on the availability of teaching lines. Following a policy from the Cornell Graduate School, students who require more than seven years to complete their degree shall not be funded as teaching assistants after the 14th semester.

Special Requests

Students who have special requests should first discuss them with their Ph.D. advisor (or with a field member with whom they work, if they don't have an advisor yet). If the advisor (or field faculty) supports the request, then it should be sent to the Director of Graduate Studies.  

PhD Qualifying and Preliminary Exams (Semesters)

Qualifying exams .

All PhD students must pass one of the two following qualifying exams.

• Mathematics   based on the two-semester sequence Advanced Calculus MATH6001-6002 and the one semester course Advanced Linear Algebra MATH6003. 
• Statistics   based on the two two-semester sequences Mathematical Statistics STAT6021-6022 and Applied Statistics STAT6031-6032.

All incoming PhD students are required to take at least one qualifying exam before the beginning of their first semester. Students who do not pass this exam at the PhD level are placed in the appropriate 6000 - level courses. In order to remain in the PhD program, the student must pass the qualifying exam (at the PhD level) by the beginning of the Fall Semester following their admission into the program.

Preliminary Exams

All PhD students must also pass four of the following ten preliminary exams. Exams are primarily offered in May and August of every academic year and can be attempted at most twice. Exams taken by incoming students in August do not count against this two-attempt rule. All four preliminary exams must be passed within one and a half years of completing the qualifying exam.

• Real Analysis
• Complex Analysis
• Ordinary Differential Equations
• Partial Differential Equations
• Multivariate Analysis
• Theory of Linear Models
• Statistical Methods
• Probability

Note: From 2013-2018 there were five Prelim exams covering two subjects each:

• Algebra and Topology
• Differential Equations
• Linear Models
• Probability and Statistics

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Qualifying Exams

New style qualifying examination materials (2014-present):.

Analysis Qualifying Exam Syllabus   (updated 2023)

Algebra Qualifying Exam Syllabus  (updated 2020)

Below are packets containing samples of the Old-Style Qualifying exams (Pre-2014):

Analysis Qualifying Exams Packet  

Algebra Qualifying Exams Packet  

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email  [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at:  Adobe Acrobat Reader .

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Phd qualifying / masters comprehensive exams.

Information on exam requirements for the masters and PhD degrees in mathematics can be found in the General Information on Graduate Programs (GIOGP) under Degree Requirements . This page contains the current syllabi and past exams for the PhD qualifying / Masters Comprehensive exams.

• Qualifier Syllabus
• January 2023
• January 2022
• January 2021
• January 2020
• January 2019
• January 2017
• January 2016
• January 2015
• January 2014
• January 2012
• January 2008
• January 2007
• June 2007 901-902
• January 2006
• January 2005
• January 2004
• January 2004 871-953
• January 2003
• January 2002
• January 2001
• January 2000
• January 1999
• January 1998
• January 2018
• January 2011
• January 2010
• January 2009
• January 1997

Differential Equations

• January 2017 830_842
• January 2008 850-871
• June 2007 852-871
• January 2006 852-871
• January 2006 850-871
• June 2004 852-871
• January 1995 b
• January 1995

Mathematics

Phd qualifying and ms comprehensive exam archives.

PhD qualifying exams/MS comprehensive exams are given every year in January and August.

For more information about the exam requirements, consult the handbook for students enrolled after fall 2020.

Exam archives

• Algebra–Fall 2005
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Numerical analysis

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The Written Comprehensive Exams

General information.

All graduate mathematics students are required to take the written comprehensive examination in the three following subjects: Advanced Calculus, Complex Variable, and Linear Algebra.

Past examinations are a recommended guide to the level and typical content of the examinations. A complete set of examinations is available electornically by contacting [email protected] or by hard copy at the departmental office, room 623, Warren Weaver Hall. 

The examinations are given twice a year, in late August or early September and in early January, just before the beginning of the fall and spring terms, respectively. They are scheduled on two consecutive days, with three hours allotted to each subject examination. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

Students are required to apply for the August examination by the end of the previous spring term, and for the January examination by the end of the previous fall term; specific application deadlines, pertaining to each academic year, are found in the departmental Academic Year Calendar, distributed to students prior to each fall term (and also available at both fall and spring registration periods). Application forms are available at the Mathematics Department Office; they will be mailed to students upon request.

Passing the written comprehensive examinations is part of one of the alternatives for fulfilling the requirements for the master of science degree in mathematics. Passing with the grade of A is a prerequisite to the oral preliminary examination for the Ph.D. degree in mathematics. Students may take the examination twice without special permission; a third try will require the permission of the Director of Graduate Studies. In the fall term, the Department offers a series of workshops, taught by an advanced graduate assistant, to help students prepare for the written examinations.

The Examinations

The following lists suggest the scope of the examinations but are not necessarily complete:

Advanced Calculus

Real numbers. Functions of one variable: continuity, mean-value theorems, convergence, differentiability, maxima and minima, integrals, fundamental theorem of calculus, inequalities, estimation of sums and integrals, elementary functions and their power series. Functions of several variables: partial derivatives, chain rule, MacLaurin expansion, critical points, Lagrange multipliers, inverse and implicit function theorems, Jacobian, divergence and curl, area and volume integrals, Green and Stokes theorems.

Complex Variables

Complex numbers, analytical functions, Cauchy-Riemann equations, Cauchy's integral and applications, power series, maximum principle, Liouville's theorem, elementary functions and their conformal maps, bilinear transformation, classification of singularities, residue theorem and contour integration, Laurent series, Rouche's theorem, number of zeros and poles.

Linear Algebra

Vector spaces, linear dependence, basis, dimension, inner product, linear transformation, systems of linear equations, matrices, determinants, ranks, eigenvalues, diagonalization of matrices, quadratics forms, symmetric and orthogonal transformations.

Cooperative preparation is encouraged, as it is for all examinations. You may also find the following books helpful:

• Buck,  Advanced Calculus
• Courant and John,  Introduction to Calculus and Analysis
• Strang, Linear Algebra
• Churchill,  Complex Variables and Applications

PhD Area Exam

The Area Examination is a departmental requirement for students in the PhD program. The Area Examination will assess the student's breadth of knowledge in their particular area of research. It is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

This requirement must be completed by the end of the Winter quarter of the student’s third year in the program.

The Area Examination committee is comprised of three members: the dissertation advisor and two additional examiners. All members are normally on the Stanford Academic Council. Emeritus faculty are also eligible to serve as examiners. The appointment of an examining committee member who is not on the Academic Council may be approved by the chair of the department if that person contributes an area of expertise that is not readily available from the faculty.

Once the Area Examination date is confirmed, the student must email the Student Services Manager with the date, time, location, and names of the members of their committee. The student must reserve a room through the Room Reservation Form . 

Recording Completion

Once the Area Examination is successfully completed, the student’s advisor must sign the Area Exam Form which must be submitted to the Student Services Manager. The completion will be recorded in the student’s file and Axess record.

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Computational Biology

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Computational  Biology as a science is in the midst of a major transition, as modern experimental methods are generating data at an unprecedented rate. The availability of this data is leading to the development of quantitatively detailed models of complex biological systems and associated computational approaches to the study of biology.

Masters and Doctoral Program Requirements in the Computational Biology Track The standard professional degree in computational biology and biomedical sciences is a Ph.D. The computational biology track also offers a M.S. degree, consisting of 30 credits of graduate coursework with no thesis. The core courses required for the M.S. degree in computational biology are also taken by all PhD students in this track in preparation for the PhD qualifying examinations. For more details about requirements for the Ph.D., please see  Ph.D. Requirements .

Course Requirements

The required courses provide computational biology students with the fundamentals of both biology and applied mathematics, as well as in specific methods and applications in computational biology. It is expected that the student will choose a set of electives to provide in depth specialization. Students with less formal training in either math or biology may wish to audit an undergraduate course concurrently with, or prior to, taking the graduate level courses. We refer to the  Graduate Bulletin  for further information about university and department-wide requirements.

Core Courses Fundamentals of Applied Math AMS 507 Introduction to Probability AMS 510 Analytical Methods for Applied Mathematics and Statistics AMS 533 Numerical Methods and Algorithms in Computational Biology

Fundamentals of Biology CHE 541 /MCB 520 Graduate Biochemistry (an alternate graduate level cell or developmental biology course may be substituted with permission)

Methods of Computational Biology  AMS 531 Laboratory Rotations in Computational Biology (2 semesters)  AMS 532  Journal Club in Computational Biology (3 semesters) AMS 535 Introduction to Computational Structural Biology and Drug Design AMS 537  Biological Networks and Dynamics CSE 549 Computational Biology

Electives AMS 530  Principles in Parallel Computing AMS 534  Introduction to Systems Biology AMS 536  Molecular Modeling of Biological Molecules AMS 538 Methods in Neuronal Modeling  CHE 538 Statistical Mechanics CHE 533 Chemical Thermodynamics PHY 558 Physical and Quantitative Biology  Electives may also be chosen from any area relevant to computational biology, based on the specific interests of the student. The student is encouraged to consult with a faculty adviser in advance of choosing electives. Likely areas of specialization may include: - Computational applied math - Optimization and simulation of complex systems - Structural biology/biochemistry - Developmental/cell biology - Biostatistics 

Qualifying Examinations

As with the other tracks in the department, the qualifying exam for the doctoral program consists of two parts: the Common Exam taken early in the spring semester of the first year, and the Area Exam taken a year later. Both cover the material from the core course sequence.

Common Exam

The common exam is a written exam, consisting of two parts. Part A of the exam covers the fundamentals of Linear Algebra and Advanced Calculus, as covered in AMS 510; all doctoral students in AMS take this portion of the exam. Part B of the exam is specific to Computational Biology, and consists of questions covering the basics of Computational Structural Biology (AMS 535) and Bioinformatics (CSE 549).

The area exam in computational biology is an oral exam based on the student's specific course sequence. The student is examined by a panel of at least three faculty and must answer questions from those courses the student has covered in each of these three key areas: fundamentals of applied mathematics; fundamentals of biology and/or biochemistry; methods and applications in computational biology as well as in the particular elective courses the student has taken. The oral format is chosen to allow greater flexibility in dealing with a range of students having different focus areas and in assessing the student's understanding of biological systems. Students should consult with the examination committee well prior to the exam in order to determine which areas/courses will be emphasized.

Suggested Course Sequence for the Computational Biology Track 

For a student entering in the fall semester, the following outline provides a suggested sequence through the first four semseters of the track.

SEMESTER 1 - Fall

• AMS-510 (3 credits), Analytical Methods for Applied Mathematics and Statistics
• CSE-549 (3 credits), Computational Biology
• AMS-535 (3 credits), Intro to Computational Structural Biology & Drug Design (R. Rizzo)
• Elective (3 credits)
• AMS-531 (0/3 credits), Lab Rotations
• AMS-532 (0/1 credits), Journal Club
• AMS-539 (0/1 credits) Intro to Physical & Quantitative Biology

*** Please note that many students enroll in AMS 599 (Research) to fill their elective credits, and international students may be required to enroll in an ESL course. Both these options fulfill the need to maintain a full-time course load, but may not be counted towards the 30 graduate credits required for the MS degree.

COMMON EXAM (Ph.D. Students; first week of Spring semester)

• Part A (Linear Algebra) 3 of 4 written questions based on AMS-510
• Part A (Calculus) 3 of 4 written questions based on AMS-510
• Part B (Comp. Bio.) 3 of 4 written questions based on AMS-535 and CSE-549
• Make-up exam is held at end of the semester

SEMESTER 2 - Spring

• AMS-533 (3 credits), Numerical Methods and Algorithms in Computational Biology (D. Green)
• AMS-537 (3 credits), Biological Networks and Dynamics (T. MacCarthy)
• AMS-532 (0/1 credits), Journal Club (includes Responsible Conduct of Research)
• PHY-561 (1/3 credits) Introduction to Biology for Physical and Quantitative Scientists

SEMESTER 3 - Fall

• AMS-507 (3 credits), Introduction to Probability
• CHE-541/MCB-520 (3 credits), Graduate Biochemistry
• AMS 532 (0/1 credits), Journal Club

*** Suggested elective PHY 558 Physical Biology

QUALIFYING AREA EXAM (Ph.D. Students; first week of Spring semester)

• An oral exam consisting of questions based on the graduate courses the student has taken that are not covered by the common exam.

SEMESTER 4 - Spring

• AMS 699 Research
• JRN 565 (3 credits) Communicating Your Science

*** For PhD students, the courses here should consist of 3 to 9 credits of thesis research augmented by elective courses. For MS students, 9 credits of elective courses should be taken.