#### Qualifying exams

PhD students must pass qualifying exams in any two out of the four subjects Algebra, Analysis, Applied Math, Topology. Exams are typically offered twice per year in August and in January (right before the beginning of the term). Exams are taken and graded anonymously. Students may take exams multiple times without penalty but are expected to pass them within their first 3 years to be on track in the program.

Specific information about each of the four qualifying exams is below:

##### Algebra exam

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

###### Material

**Group theory:** basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan–Hölder theorem
**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
**Ring theory:** factorization in domains, simplicity of matrix algebras
**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
**Language of category theory:** objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma
**Multilinear algebra:** pairings, wedge products, symmetric products, multilinear forms over rings
**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

##### Analysis exam

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

###### Material

**Measure theory on Euclidean space:** the Borel σ-algebra, construction of Lebesgue measure on finite-dimensional Euclidean spaces.
**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.
**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 *L*^{p} spaces, the Lebesgue differentiation theorem.
**Convergence of functions:** pointwise convergence, the supremum norm and uniform convergence, convergence in measure, convergence in *L*^{p} spaces, the relations between these notions, Egorov’s theorem.
**Inequalities:** Chebyshev, Cauchy–Schwarz, Jensen, Minkowski (sum and integral forms), Hölder.
**Density:** the Weierstrass theorem and density of polynomials in appropriate *L*^{p} spaces, convolution with approximate identities and density of smooth functions in appropriate *L*^{p} spaces.
**General measure theory:** σ-algebras, outer measures, counting measure, product measures, *L*^{p} spaces of a general measure space, absolute continuity of measures and the Radon–Nikodym theorem.

###### Textbooks

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.

###### Material

**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.
**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).
**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.
**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.
**Elements of applied linear algebra:** eigenvalues, Rayleigh quotients, the Jordan normal form, singular value decomposition, Gram–Schmidt orthogonalization, convergence of finite difference schemes.

###### Textbooks

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.

###### Material

**Some point set topological concepts:** basic definitions, compactness, separation axiom, connectedness/path-connectedness, retractions, contractibility, quotient topologies.
**Homotopy theory:** CW complexes, homotopic maps, properties of homotopy in CW complexes, homotopy equivalence, homotopy extension.
**Some group theory:** free groups, free products, universal properties, presentations of groups.
**Brief overview of basic category-theoretic definitions:** categories, functors, natural transformations, examples.
**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.
**Covering spaces:** basic definitions, lifting properties, deck group actions, the Galois correspondence between covers and subgroups of the fundamental group.
**Brief overview of (co)homological algebra:** (co)chain complexes, (co)chain maps, exact sequences, (co)homology, long exact sequences induced by short exact sequences.
**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.
**Applications of homology:** orientability, degrees of mappings, Lefschetz fixed point theorem, Brouwer fixed point theorem, invariance of domain Borsuk–Ulam theorem.
**Structures on cohomology rings:** universal coefficients theorem for homology/cohomology, the cup and cap products, calculating cohomology rings, intersection numbers, duality theorems.

###### Textbooks

The following textbooks are recommended:

- Bredon,
*Topology and geometry*
- Hatcher,
*Algebraic topology*
- Spanier,
*Algebraic topology*

###### Sample Exams