This page is the central point for finding the statements and solutions to students' qualifying exam questions. The questions are split up by major subjects; hopefully these will make relevant questions easy to find. We've also put up the qual questions from our old site, though not all of them have solutions provided (feel free to add one, if you'd like).
When you post a question(s), please include your name (or anonymous); the three main members of your committee; the date; and the topic this question came from on your qual syllabus. For example:
Student X; Professor A (Chair), Professor B, Professor C; May 8, 2004; Number theory (minor).
Question: Are there infinitely many primes, and if so, exhibit at least 3 proofs of this fact. If there are finitely many, can you list them all?
Answer: There are infinitely many primes. The first proof is usually attributed to Euclid. Blah blah blah.
Atmosphere of the Qual
Contrary to how this page may make it seem, the qual is not a firing squad where the bullets are complicated mathematical questions. Much of the qual has a more conversational flavor, where the committee asks you a question and you may have some ideas to what they are asking for and make some comment. They then usually throw in their own comments or hints, until you arrive at the solution. Many quals do not go for the full three hours, and can end in as little as two hours or less.
Most people do not fail their quals. This is sort of a self-selecting phenomenon; by the time you take your qual, you will have studied intensively for two or more months and will know the material far better than you think. Even if you do fail, you will usually be given a second chance to take it, and almost no one has gotten kicked out from not passing their qual. If you do well on one part of the exam but poorly on another part, the committee may only require you to retake the part you failed. Many people feel during their qual like their performance is a lot more disappointing than it is — don't lose hope, quals are meant to be hard!
Advice on Studying
The Graduate: Studying For The Qualifying Exam
- Algebra (General)
- Algebraic Geometry
- Algebraic Topology
- Banach Spaces and Spectral Theory
- C^* and von Neumann algebras
- Combinatorial Game Theory
- Combinatorics and Combinatorial Algorithms
- Commutative Algebra
- Complex Analysis
- Computation Complexity
- Computational Biology
- Differential Topology
- Dynamical Systems (Symbolic and Otherwise)
- Fields and Galois Theory
- Fluid Mechanics
- Foundations and Recursion Theory questions, from the Logic Quals
- Geometry, Riemannian and Symplectic
- Group Theory
- Harmonic Analysis
- 3-manifolds, 4-manifolds, knots
- Measure Theory and Real Analysis
- Model Theory
- Non-commutative Ring Theory
- Number Theory
- Numerical PDE's and ODE's
- Numerical Linear Algebra
- Ordinary Differential Equations
- Partial Differential Equations
- Quantum Groups
- Quantum Mechanics
- Representation Theory, Lie Groups, Lie Algebras
- Semiclassical Analysis
- Set Theory
- Topology (General)
- Universal Algebra and Category Theory
See here for questions from Princeton's equivalent of the qualifying exam.