Graduate Exams

The Algebra Exam is a 3 hour written exam with 6 to 7 problems.

• Students with 4 essentially correct solutions and no significant errors earn a PhD pass.
• Students with 2 essentially correct solutions and significant progress toward a third earn a Masters pass.

Topics typically covered include:

Groups: Review of elementary group theory, isomorphism theorems, group actions, orbits, stabilizers, simplicity of An, Sylows theorems, direct prod- ucts and direct sums, semi-direct products and extensions of a group by an abelian group, Fundamental Theorem of Abelian Groups, solvable groups.

Fields: Relative dimensions, automorphisms, splitting fields, isomorphism extension theorem, sep- arable extensions, Galois correspondence, Funda- mental Theorem of Galois Theory, principal element theorem, traces and norms, radical extensions, finite fields, cyclotomic extensions, inseparable extensions, algebraic closure.

Commutative Algebra: Localization, integral extensions, unique factor- ization domains, Eisenstein criterion, principal ideal domains, Noetherian rings, Hilbert basis theorem, varieties, Zariski topology, Hilbert Nullstellen- satz.

Modules: Irreducible modules, torsion modules, free modules, projective modules, modules over PIDs, chain conditions, tensor products, exact se- quences. Noncommutative Rings: Artinian rings, Jacobson radical, Artin- Wedderburn theorem, Maschke’s theorem, Skolem- Noether theorem, divi- sion rings, Wedderburns theorem on finite division rings.

References:

• D. Rotman, An introduction to the theory of groups
• S. Lang, Algebra
• T. Hungerford, Algebra
• T.Y. Lam, Lectures on modules and rings
• M. Atiyah and I.G. MacDonald, Introduction to commutative algebra
• D. Dummitt and R. Foote, Abstract algebra

The (Real) Analysis exam typically has four questions. The passing requirements are:

• Students with 2 correct solutions earn a PhD Pass
• Students with 1 essentially correct solution in one problem and significant progress on another problem earn a Masters Pass

Topics typically covered include:

Measures: Sigma-rings, sigma fields. Set functions and measures. Outer measure. Construction of measures on Rn. Variation of signed measures. Hahn decomposition theorem. Absolute continuity. Mutually singular measures. Product measures. Regular measures. Measurable functions. Signed and complex measures.

Integration: Definition and basic properties of integrable functions over an abstract measure space. The Riemann integral and its relation to the Lebesgue integral. Lebesgue’s dominated convergence theorem and related results. Radon-Nikodym theorem. Fubini’s theorem. Convolution. The n-dimensional Lebesgue integral. Polar coordinates.

Convergence: Almost everywhere convergence, uniform convergence, almost uniform convergence, convergence in measure and in mean. Egoroff’s theorem. Lusin’s theorem.

Differentiation: Lebesgue differentiation theorem. Maximal function. Vitali covering lemma. Bounded variation. Absolutely continuous functions. Fundamental theorem of calculus.

Metric spaces: Topological properties, convergence, compactness, completeness, continuity of functions.

References

• G.B. Folland, Real Analysis: Modern techniques and their applications
• P. Halmos, Measure Theory
• W. Rudin, Real and Complex Analysis

The PDE exam has three questions. The passing requirements are:

• Students with 2 essentially correct solutions and some progress toward a third with no significant errors earn a PhD pass
• Students with 1 essentially correct solution and significant progress on another problem earn a Master pass

Topics typically covered include:

First order equations: Method of characteristics for fully nonlinear, quasilinear, and linear cases. The Cauchy problem.

Laplace equation: Harmonic and Subharmonic functions. Mean value property. Harnack principle. Maximum principle. Liouville’s theorem. Poisson formula. Green’s function.

Heat equation: Cauchy problem. Energy equality. Maximum principle. Nonhomogeneous heat equation. Backward uniqueness.

Wave equation: D’Alamert’s formula. Spherical means. Energy equality. Duhamel’s principle. Domain of dependence.

Sobolev spaces: Weak derivatives. Embedding theorems (Gagliardo-Nirenberg and Morrey). Rellich compactness theorem. Trace theorem. H-1 space. Rademacher’s theorem.

References:

• L.C. Evans, Partial Differential Equations
• G.B. Folland, Introduction to Partial Differential Equations
• F. John, Partial Differential Equations

The (Applied) Probability exam usually has 3, or sometimes 4, problems. On a 3-question exam:

• Students with 2 essentially correct solutions and some progress toward a third with no significant errors earn a PhD pass.
• Students with 1 essentially correct solution and significant progress toward a second, or 0 essentially correct but significant progress on all 3, earn a Masters pass.

Topics typically covered include:

Foundations of probability: Equally likely outcomes, principles of counting, permutations, combinations. Principle of inclusion/exclusion. Conditional probability. Independence. Random variables, distributions, joint distribu- tions (continuous and discrete), functions of random variables and vectors.

Properties of Random Variables: Expectation, moments, generating func- tions (for distributions of integer-valued random variables and general se- quences), moment generating functions, characteristic functions (excluding continuity theorem and Bochner’s theorem). Basic inequalities (Cauchy- Schwarz, Chebyshev, H ̈older, Jensen, Markov, power mean/Lyapunov).

Computations: Indicators. Covariance, correlation, covariance matrix. Conditional distribution and density, conditional expectation, conditional variance, the law of total variance. Sums of independent random variables, convolutions. Main families of discrete and continuous distributions (beta, binomial, Cauchy, exponential, gamma, geometric, negative binomial, nor- mal, Poisson, uniform) and relations among them. Multivariate normal distribution.

Limit Theorems: Convergence in probability, in Lp, and in distribution. Law of large numbers, Central Limit Theorem. Poisson approximation.

Special models: Simple random walk, reflection principle, gambler’s ruin.

References:

• G. R. Grimmett and D. R. Strizaker, Probability and Random Processes
• A. Klenke, Probability Theory, especially Chapters 2–8
• S. Ross, A First Course in Probability, especially Chapters 1–8
• A.N. Shiryayev, Probability, especially Chapters I and IV

The (Algebraic) Topology exam will have four questions. The passing requirements:

• Students with 2 correct solutions and significant progress in another problem earn a PhD Pass
• Students with 1 essentially correct solution in one problem and a significant progress in  another problem earn a Masters Pass

Topics typically covered include:

References:

• A. Hatcher: Algebraic Topology
• I.M. Singer and J.A. Thorpe: Lecture Notes on Elementary Topology and Geometry
• M.J. Greenberg and J.R. Harper: Lectures on Algebraic Topology
• J.W. Vick: Homology Theory
• W.S. Massey: Algebraic Topology: An Introduction
• I. Madsen and J. Tornehave: From Calculus to Cohomology