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 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: Axioms of probability, σ-algebras, λ-systems, π- systems. Carath´eodory’s and Kolmogorov’s extension theorems, the π–λ theorem. Random variables, distributions, joint distributions, functions of random variables and vectors. Expected value and abstract integration. Fatou’s lemma, bounded, monotone and dominated convergence theorems. Product spaces and Fubini–Tonelli theorem. Independence.

Random variables: Expectation, variance, higher moments, probability generating function, moment generating function, characteristic function. Covariance, correlation, covariance matrix. Sums of independent random variables, convolutions. Main families of discrete and continuous distributions (beta, binomial, Cauchy, exponential, gamma, geometric, negative binomial, normal, Poisson, uniform) and relations among them. Multivariate normal distribution.

Limit theorems: Modes of convergence (a.s., in probability, in Lp, and in distribution) and relations among them. Portmanteau theorem. Scheff´e’s lemma. Strong and weak laws of large numbers. Series of independent random variables. Weak convergence and tightness. Central Limit Theorem. Poisson approximation. Theorems of Slutsky and Mann-Wald. Delta method. Convergence of expected values and higher moments, the moment problem. Convergence to types lemma. Multidimensional characteristic functions and weak convergence.

Additional topics: Borel–Cantelli lemmas. Kolmogorov and Hewitt–Savage 0–1 laws. Fundamental inequalities: Cauchy-Schwarz, Chebyshev, H¨older, Jensen, Markov, power mean/Lyapunov. Conditional probability, conditional distribution and density, conditional expectation given a σ-field, conditional variance and the law of total variance. The Poisson process. Infinitely divisible and stable laws. Principle of inclusion/exclusion. Indicator variables and counting.

References:

• R. Durrett, Probability: Theory and Examples, especially Chapters 1-3, section 4.1, and Appendix A (5th edition)
• P. Billingsley, Probability and Measure, especially Chapters 1-6
• O. Kallenberg, Foundations of Modern Probability, especially Chapters 1–8
• A. Klenke, Probability Theory, especially Chapters 1–8
• A.N. Shiryayev, Probability, especially Chapters II–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