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Graduate degree timelines

This page provides some information about when you should accomplish various milestones in your degree and what other events must occur along the way. It is for convenience only. Consult with the Graduate Chair to ensure you have all the information you need.

PhD timeline

The typical time to degree for a PhD here is 5 to 6 years and the maximum length of support is set to 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 what 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. Even if you fail, you’ll have a better idea of what they are about. By the end of the fall of your 3rd year, you need to have attempted at least 2 quals and by the end of the fall of your 4th year, you need to have passed 2 quals.

Once you pass your two qualifying exams, your top priority should be to find an adivsor; getting up to speed on the cutting edge of research, and then doing your own, takes a lot of time! Still, finding an advisor is an important decision and, while your interest in their research is a very important factor, personal chemistry should also be considered. If you’ve spent some of your early years talking to some professors, this important step should go more smoothly.

Your advisor will determine the format of your comprehensive exam. You must pass this exam by the end of your 4th year and you only have two tries. Once this is behind you, you can form a ‘thesis committee’.

Then, you do math.

Once you’ve done math, it’s time to write it up. The way in which you go about this will depend a lot on you and your advisor. 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.

MA timeline

The typical time to degree for an MA here is 2 to 3 years and the maximum length of support is set to 3 years.

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. Attending various seminars, talking to more advanced students about what they do, and talking to professors about what masters projects they might have in mind are a few ways you can go about accomplishing this.

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.

Prospective graduate students


Here in the Department of Mathematics at UH Mānoa, we offer both MA and PhD degrees in mathematics. We have about 20 faculty members with a wide range of interests who can mentor you in algebra, analysis, applied mathematics, combinatorics, geometry, logic, number theory, and topology. You would join a group of almost 40 graduate students learning math and pushing the boundaries of mathematical research. We offer introductory and advanced graduate courses, as well as regular ‘topics’ courses changing every year based on professors’ research interests and student demand. We have a vibrant faculty and grad student lounge that offers a lot of opportunity for talking with your fellow mathematicians.

For more information about faculty research, you can check out our research page.

Below are more details about the MA degree, the PhD degree, the application process, funding opportunities, and living in Hawai‘i.

Program summaries

PhD requirements

The PhD program in the Mathematics Department has four main components: coursework, qualifying exams, a comprehensive exam, and the writing and oral defense of a PhD thesis.

The goal of this program is to develop your abilities to the point where you can contribute original research in mathematics. To this end, we offer a wide range of basic graduate courses as well as advanced topics courses, and the faculty usually run a few seminars every semester. During your time here, you’ll work towards a broad understanding of graduate mathematics, develop your mathematical abilities and creativity, and reach the cutting edge of research in one of the many fields our faculty studies, all culminating in the writing and oral defense of your PhD thesis. Along the way, you’ll have a few milestones to reach: taking at least 10 courses (with some breadth requirements), passing two qualifying exams (out of four: algebra, analysis, applied mathematics, and topology), and completing a comprehensive exam with a faculty advisor.

For more details, see the full requirements page here.

MA requirements

The MA program in the Mathematics Department has two main components: coursework and the writing and oral presentation of a masters project.

As a student in this program, you’ll take both introductory and more advanced graduate level courses—at least 10 in total—with a lot of freedom to choose your own path. By about half way through the two to three year program, you should settle down on an advisor who can mentor you to the completion of a project that you’ll get to write up and present.

For more details, see the full requirements page here.

Funding opportunities

Graduate Assistantships are available at stipends which range from approximately \$17,500 to \$19,000 for the academic year, with waiver of tuition. At any given time, about three quarters of our graduate students are supported by teaching assistantships. Some faculty members also have grants that have funds to support students with Research Assistantships. Most graduate assistants teach recitation sections for pre-calculus and calculus courses though other options exist: tutoring, grading, teaching a class, or assisting a professor.

Living in Hawai‘i

The university campus is located on the Leeward side of O‘ahu at the mouth of the Mānoa Valley in Honolulu, about two miles from the beaches of Waikiki. The island offers an abundance of opportunity for outdoor activities from hiking to Mānoa Falls or the summit of Diamond Head to snorkeling in Hanauma Bay, and, as you might imagine, surfing. Moreover, Honolulu is a city of a million people with all sorts of cultural and social activities. You can find out about all this and more on the university’s website.

Application process

The typical requirement for admission to the graduate program is the completion of a standard undergraduate program in mathematics. The candidate will generally be expected to know linear algebra, the elements of abstract algebra, and elementary real analysis. A student whose degree has been awarded in some other field may be considered if they have had the appropriate background courses. Students should also have current GRE scores and are strongly encouraged to submit a personal statement describing their reasons for pursuing a graduate degree in math.

Applications for admission in the Fall semester are accepted from August 1 to February 15, and for the Spring semester from May 1 to October 1.

For some of the finer print, see our graduate application page.


Qualifying exams

Qualifying exams

This page contains specific information about each of the four qualifying exams (under the new system):

Algebra exam

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

  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

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
Analysis exam

The analysis qualifying exam covers topics in measure theory and real analysis.

  1. Measure theory on Euclidean space: the Borel σ-algebra, construction of Lebesgue measure on finite-dimensional Euclidean spaces.
  2. Functions: continuous functions, uniformly continuous functions, absolutely continuous functions, functions of bounded variation and rectifiable curves, Borel functions, measurable functions, simple functions, the relations between these classes, Lusin’s theorem.
  3. Integration: the Lebesgue integral, Fatou’s lemma, the monotone and dominated convergence theorems, applications to moving limits through integrals, Fubini’s theorem, definition and completeness of Lp spaces, the Lebesgue differentiation theorem.
  4. Convergence of functions: pointwise convergence, the supremum norm and uniform convergence, convergence in measure, convergence in Lp spaces, the relations between these notions, Egorov’s theorem.
  5. Inequalities: Chebyshev, Cauchy–Schwarz, Jensen, Minkowski (sum and integral forms), Hölder.
  6. Density: the Weierstrass theorem and density of polynomials in appropriate Lp spaces, convolution with approximate identities and density of smooth functions in appropriate Lp spaces.
  7. General measure theory: σ-algebras, outer measures, counting measure, product measures, Lp spaces of a general measure space, absolute continuity of measures and the Radon–Nikodym theorem.

The following textbooks are recommended:

  • Folland, Real analysis: Modern techniques and their applications
  • Royden, Real analysis
  • Rudin, Real and complex analysis
  • Stein and Shakarchi, Real analysis: Measure theory, integration, and Hilbert spaces
Sample Exams
Applied mathematics exam

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

  1. Basic dynamical systems concepts: definition of a dynamical system (continuous and discrete), equilibrium states, ω,α-limit sets, invariant sets, stability of equilibrium states and periodic solutions, population dynamics models; linear systems, stable, unstable, center spaces; non-linear systems and existence/uniqueness of solutions; linearization, topological equivalence/conjugacy, center manifold theory (applications: species competition models, SIR models, predator-prey models); some global nonlinear techniques (nullclines, Lyapunov function, applications: nonlinear pendulum, SIR models); limit cycles. Poincaré–Bendixson theory in $\mathbb R^2$ (applications: Van der Pol oscillator, predator-prey models with limit cycle, oscillating chemical reactions); stability of periodic solutions, Poincaré map.
  2. Bifurcation theory: family of systems, structural stability, definition of a bifurcation; Peixoto’s theorem, Morse–Smale systems; examples of one-parameter bifurcations of equilibrium states (application: laser phenomenon); genericity, transverse intersections, versal unfoldings (deformations) and codimension of a bifurcation (application: spruce budworm model (codimension-2 bifurcation)); the Hopf bifurcation (applications: oscillating chemical reactions, FitzHugh–Nagumo model); center manifold theory (for bifurcations); global bifurcations (homoclinic, heteroclinic).
  3. Introduction to chaos: examples of chaotic behavior (discrete logistic model, Duffing oscillator, Lorenz system, Henon map, Horseshoe map, symbolic dynamics), sensitivity to initial condition; more on logistic model (period doubling, Feigenbaum constant, dense periodic orbits and Sharkovskii’s theorem); strange attractors; Lyapunov exponents.
  4. Elements of partial differential equations: first order linear and quasilinear PDEs and the method of characteristics, second order linear PDEs and their classification, the Sturm–Liouville problem, Green’s functions and fundamental solutions, the Fourier transform, equilibrium solutions of time-dependent PDEs.
  5. Elements of applied linear algebra: eigenvalues, Rayleigh quotients, the Jordan normal form, singular value decomposition, Gram–Schmidt orthogonalization, convergence of finite difference schemes.

The following textbooks are recommended:

  • 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
  • Friedberg, Insel, Linear algebra
  • Roman, Advanced Linear algebra
  • Bleeker, Csordas, Basic partial differential equations
  • Evans, Partial differential equations
Sample Exams
Topology exam

The topology qualifying exam covers topics in algebraic topology.

  1. Some point set topological concepts: basic definitions, compactness, separation axiom, connectedness/path-connectedness, retractions, contractibility, quotient topologies.
  2. Homotopy theory: CW complexes, homotopic maps, properties of homotopy in CW complexes, homotopy equivalence, homotopy extension.
  3. Some group theory: free groups, free products, universal properties, presentations of groups.
  4. Brief overview of basic category-theoretic definitions: categories, functors, natural transformations, examples.
  5. The fundamental group: a brief discussion of homotopy groups and their functoriality, long exact sequences of homotopy groups, definition of the fundamental groupoid and the fundamental group, calculation of the fundamental group of a circle, winding numbers, the van Kampen theorem together with examples of fundamental group calculations, K(π, 1) spaces and their properties.
  6. Covering spaces: basic definitions, lifting properties, deck group actions, the Galois correspondence between covers and subgroups of the fundamental group.
  7. Brief overview of (co)homological algebra: (co)chain complexes, (co)chain maps, exact sequences, (co)homology, long exact sequences induced by short exact sequences.
  8. Homology: cellular and singular homology and their equivalence, reduced homology, relative homology, excision, Mayer–Vietoris sequences, the Künneth formula, examples, first homology and the fundamental group, homology with coefficients, definition of cohomology and calculation of examples.
  9. Applications of homology: orientability, degrees of mappings, Lefschetz fixed point theorem, Brouwer fixed point theorem, invariance of domain Borsuk–Ulam theorem.
  10. Structures on cohomology rings: universal coefficients theorem for homology/cohomology, the cup and cap products, calculating cohomology rings, intersection numbers, duality theorems.

The following textbooks are recommended:

  • Bredon, Topology and geometry
  • Hatcher, Algebraic topology
  • Spanier, Algebraic topology
Sample Exams