Applied mathematics exam

The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra.

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Material

Linear algebra

•  Undergraduate linear algebra
• Spaces: Normed spaces, inner product spaces, dual spaces, fundamental subspaces, rank-nullity theorem, direct sums
• Matrices and operators: linear, bounded, self-adjoint, unitary, normal, positive-definite, and compact operators; projections, adjoints, matrix exponential
• Spectra: Eigenvalues, eigenvectors, Rayleigh quotient, eigendecomposition, singular value decomposition, Schur decomposition, generalized eigenvectors
• Orthogonality: orthogonal projections and reflections, orthogonal complement, orthonormality, Gram-Schmidt orthogonalization, orthogonality of range and null space of operators and their adjoints
• Hilbert spaces: completeness, Riesz representation theorem, Parseval’s identity

Ordinary differential equations

• ODE theory: Local existence and uniqueness of solutions to initial-value problems, continuous dependence on initial conditions, continuous dependence on parameters, Sturm-Liouville theory for one-dimensional boundary-value problems
• Dynamical systems: continuous and discrete systems, equilibrium states and their stability, flows, orbits, limit sets, invariant sets; linear systems, stable, unstable, center spaces; nonlinear systems, linearization, topological equivalence, center manifold theory, nullclines, Lyapunov function, limit cycles, Poincaré–Bendixson theory, stability of periodic solutions, Poincaré map.
• Bifurcation theory: structural stability, local bifurcations of equilibrium states, genericity, codimension, center manifold theory, global bifurcations (homoclinic, heteroclinic), bifurcations in 1-dimensional maps.

Partial differential equations

• PDE theory: classification of PDEs, uniqueness of solutions, energy estimates
• Initial-value problems: First-order linear and quasilinear PDEs, method of characteristics, heat equation, wave equation, separation of variables, series solutions, Fourier transform
• Boundary-value problems: Laplace equation, Poisson equation, Sturm–Liouville theory, Green’s functions, fundamental solutions, Dirichlet and Neumann boundary conditions, variational formulations of boundary-value problems.

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Textbooks. The following textbooks are recommended:

•  Friedberg, Insel, Linear algebra
• Roman, Advanced Linear algebra
• Cheney, Analysis for applied mathematics
• Perko, Differential equations and dynamical systems
• Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering
• Hirsch, Smale, and Devaney, Differential equations, dynamical systems, and an introduction to chaos
• Teschle, Ordinary differential equations and dynamical systems
• Bleecker, Csordas, Basic partial differential equations
• Evans, Partial differential equations
• Haberman, Applied partial differential equations with Fourier series and boundary value problems
• Olver, Introduction to partial differential equations

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Sample Exams (new format)

• Sample Quals (1 – 4)

Sample Exams (old format) These still have good problems to study from although the balance of material is different from the new format. 

• Sample Exam 1
• Sample Exam 2
• Sample Exam 3
• Sample Exam 4

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Previous Exams (current format)

• January 2026
• January 2025
• August 2024
• January 2024
• August 2023

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Algebra exam

The algebra qualifying exam covers several standard topics in abstract algebra.

Material

1. Group theory: basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan–Hölder theorem
2. Field and Galois theory: finite fields, separable and normal extensions, Fundamental theorem of Galois theory, applications (e.g. solvability by radicals, constructions by straightedge and compass, …), determining Galois groups
3. Ring theory: factorization in domains, simplicity of matrix algebras
4. Module theory: basics, projectivity, injectivity, tensor products, flatness, Noetherian property, exact sequences, commutative diagrams, structure theory of modules over a PID, consequences for canonical forms of matrices and other linear algebra
5. Language of category theory: objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma
6. Multilinear algebra: pairings, wedge products, symmetric products, multilinear forms over rings
7. Basic commutative algebra: local rings and localization, integral extensions, Hilbert Basis Theorem, Noether Normalization, Hilbert’s Nullstellensatz

Textbooks

The following textbooks are recommended:

• Dummit and Foote, Abstract algebra
• Lang, Algebra
• Rotman, Advanced Modern Algebra
• Hungerford, Algebra
• Jacobson, Basic Algebra I & II

Sample Exams

• Sample Exam
• Spring 2016 Exam
• Spring 2017 Exam
• Spring 2018 Exam
• Spring 20
• Fall 20
• Spring 21
• Fall 21

This page provides information about when you should accomplish various milestones in your degree and what other events must occur along the way. The target timeline is below. Consult with the Graduate Chair to ensure you have all the information you need.

• PhD timeline
• MA timeline

PhD process

The typical time to degree for a PhD here is 5 to 6 years and the maximum length of support is 6 years. In your first couple of years, you’ll be taking classes and studying to pass your qualifying exams. It’s also a good idea during that time to try to figure out which type of math you might do research in; attending various seminars, talking to more advanced students about what they do, and asking professors about their research are a few ways you can go about accomplishing this.  We encourage you to attempt qualifying as exams as soon as you can.   They can be taken multiple times without penalty.  Study sessions are organized during the summer to help you pass a qual in August.

Even before passing your  qualifying exams, a high priority is finding an advisor. Getting up to speed on the cutting edge of research, and then doing your own, takes a lot of time and is the main reason you are a PhD student! Finding an advisor is an important decision and, while your interest in their research is a very important factor, working style should also be considered.  See the page How to choose an Advisor for more info.

Once you have an advisor you’ll need to pass a comprehensive (or specialty) exam. Your advisor will determine the format of this exam. Usually the comprehensive exam is an oral presentation or a paper which  shows that you have knowledge of the forefront of an area and have a problem you are intending on working on for your thesis.  You should form a committee around the same time as you do the comprehensive exam.

Then, you do math.

Once you’ve done math, it’s time to write it up. Actually it’s a great idea to write as you go.  As you approach completion, make sure to discuss the timing with your advisor and your committee. You should contact the graduate chair at least a month ahead of defending so that they can ensure all the various forms and announcements go out in time.

A target timeline for Phd students is below. These are targets not deadlines.  It is meant to show what, in our experience, the schedule of a Ph.D should look like. It’s designed to make sure that you move through the process smoothly as possible, that you make good use of your time, and that you give yourself the best chance of successfully completing the degree and continuing on to further opportunities. Everyone’s experience with graduate school is different. If you fall being the schedule outlined in the timeline, we will meet with you and try to help you get back on track. If this doesn’t work, other options may be explored.

PhD Target Timeline

• Pass 3 600 level graduate courses by the end of the second semester.
• Pass one qualifying exam by the beginning of the second year.
• Pass second qualifying exam in January of the second year.
• Find an advisor by beginning of third year.
• Pass comprehensive (specialty exam) by middle of fourth year.
• Finish at the end of year 5.

 

 

MA process

The typical time to degree for an MA here is 2 to 3 years. We typically only fund MA students on a year by year basis, subject to availability of funds as well as good progress.

In your first year, you’ll be spending a lot of your time taking classes. As suggested above for PhD candidates, it’s also suggested that you try to figure out in what general area, and with whom, you’ll want to write your MA paper. See How to choose an advisor for more info.

By the end of the fall of your second year, you will need to have chosen an advisor. You will work with them to produce a masters paper and prepare an oral presentation. As you near completion, you should discuss timing with your advisor, form a masters committee, and talk with the graduate chair to ensure all the various forms and announcements go out in time.