Our comprehensive written exams cover material often taught in one of our foundational courses. Each instance of the associated course(s) covers a subset of the exam topics, so students generally must study additional topics independently to prepare for the exams.
Exams are offered twice per year, in August and in January the week before classes begin.
Revised Exam Guidelines
Our written exams serve the following important purposes:
 to ensure Ph.D. students have reached a satisfactory level of mastery for core areas of mathematics which relate to their field of study;
 to encourage Ph.D. students to study advanced topics independently and in depth to ensure they can pursue independent research;
 to ensure Ph.D. students are capable of expressing mathematical ideas clearly and precisely as required for eventual exposition of new results.
Our PhD Programs require completion of 3 exams at the PhD level.
Our Masters Programs require completion of 2 exams at the Masters level.
The information below should serve as a guide for students in preparing for these exams, but topics and exam structure may vary by semester. Similarly, past exams provide samples of previous exam problems but should not be considered a definitive resource for the scope of future 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, semidirect 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: Sigmarings, 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. RadonNikodym theorem. Fubini’s theorem. Convolution. The ndimensional 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

NOTE: The passing benchmarks for this exam are currently being revised. Update coming Summer 2024.
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 (GagliardoNirenberg and Morrey). Rellich compactness theorem. Trace theorem. H1 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 3question 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 integervalued 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:
NOTE: This syllabus is currently being revised to focus on Topology more so than Differential Geometry. Updated syllabus coming Summer 2024.Differentiable manifolds: definition, submanifolds, smooth maps, tangent and cotangent bundles.
Differential forms: exterior algebra, integration, Stokes’ theorem, de Rham cohomology. Lie derivatives: of forms and vector fields.
Differential topology: regular values, Sard’s theorem, degree of a map, and index of a vector field.
“Classical” differential geometry: local theory of surfaces, 1st and 2nd fundamental forms, GaussBonnet formula.
Homotopy theory: definition of homotopy, homotopy equivalences, fundamental groups (change of base point, functoriality, Van Kampen theorem, examples such as the fundamental group of the circle), covering spaces (lifting properties, universal cover, regular (or Galois) covers, relation to π1), higher homotopy groups.
Singular homology theory: definition of the homology groups, functoriality, relative homology, excision, MayerVietoris sequences, reduced homology, connection between H1 and the fundamental group, homology of classical spaces.
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
Deprecated Exams
The Mathematics Department undertook a comprehensive review of our written exam policies during the 202324 academic year. Beginning Fall 2024, our new policies take effect. Students admitted prior to Fall 2024 may choose between following the program policies during their year of admission or following the new program policies. Below we include information on the pre2024 graduate exams.
Students admitted prior to Fall 2024 had the following requirements:
 AMAT PhD students required 4 exams at the PhD level
 MATH PhD students required 6 exams at the PhD level
 AMAT Masters students required 2 exams at the Masters level
 MATH Masters students required 4 exams at the Masters level
Note: under these policies, Algebra and Geo/Top counted as 2 exams each.

Most of the following topics are normally covered in the course Math 502a.
Direct Methods for Linear systems. Gaussian Elimination and LU Factorization, Banded Systems, Symmetric Matrices, Perturbation Theory and Error Analysis
Matrix Eigenvalue Problems. Canonical Forms, Perturbation Theory, Jacobi Methods, The Power Method (including Inverse and Rayleigh Quotient iterations), Eigenvalues of Condensed Matrices (including unitary elementary transformations, reduction to Hessenberg form, QR algorithm), Singular Value Decomposition (SVD)
Linear Least Squares Problems. The Method of Normal Equations, Least Squares and the SVD (including pseudoinverse solutions), Orthogonal Decompositions
Iterative Methods for Linear Systems. Stationary Iterative Methods (Jacobi and GaussSeidel), Successive Overrelaxtion Methods (including convergence analysis), The Conjugate Gradient Method
Preconditioned Methods
References:
 G. Dahlquist and A. Bjorck, Numerical Methods, SIAM, 2003
 L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM,1997
 J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997
 W. Cheney and D. Kincaid, Numerical Analysis, Brooks/Cole, 1996
 E.K. Blum, Numerical Analysis, AddisonWesley, 1972

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 integervalued 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

Foundations of probability: Axioms of probability, distribution func tion, generating σfields, Kolmogorov’s extension theorem. Principle of inclusion/exclusion. Conditional probability and independence. Ran dom variables, distributions, joint distributions (continuous and dis crete), functions of random variables and vectors.
Properties of Random Variables: Probability generating functions. Ex pectation, moments. Moment generating functions. Characteristic functions, inversion and continuity theorems. Basic inequalities (Cauchy Schwarz, Chebyshev, H ̈older, Jensen, Markov, power mean/Lyapunov).
Computations: Conditional distribution and density, conditional ex pectation given a σfield, conditional variance, the law of total vari ance. Main families of discrete and continuous distributions (binomial, Cauchy, exponential, gamma, geometric, normal, Poisson, uniform) and relations among them. Multivariate normal distribution. Sums of in dependent random variables, convolutions.
Limit Theorems: Modes of convergence (a.s., in probability, in Lp, and in distribution) and relations among them. Theorems of Slutsky and MannWald. Delta method. Convergence of expected values and mo ments. BorelCantelli lemmas. Weak and strong laws of large num bers, convergence of random series, Kolmogorov’s inequality. Weak convergence, tightness; HellyBray and Portmanteau theorems; multidi mensional weak convergence and characteristic functions. The classical Central Limit Theorem, Lindeberg’s condition. Poisson approximation.
References
 P. Billingsley, Probability and Measure
 L. Breiman, Probability
 K. L. Chung, A Course in Probability Theory
 R. Durrett, Probability: Theorem and Examples
 A. Klenke, Probability Theory, especially Chapters 1–8
 A. N. Shiryayev, Probability, especially Chapters II–IV

Students should have a good background in linear algebra, including the basic canonical forms; these topics are covered in our undergraduate course Math 471.
Groups: Review of elementary group theory, isomorphism theorems, group actions, orbits, stabilizers, simplicity of An, Sylows theorems, direct prod ucts and direct sums, semidirect 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

Elementary properties of holomorphic functions: Power series representation, integral representation (Cauchy’s theorem for ”nice” domains). CauchyRiemann equations. Taylor series, Cauchy integral formula, classification of isolated singularities, meromorphic functions. Liouville’s theorem. The elementary holomorphic functions (rational functions, the exponential and logarithm functions, trigonometric functions, powers and roots).
The residue theorem and its applications: Evaluating integrals by the methods of residues, counting zeros and poles. Rouche’s theorem, open mapping theorem, inverse and implicit function theorems. Methods for computing residues. Harmonic functions: Mean value property and maximum principle for harmonic and analytic functions. Realization of a real harmonic function as the real part of an analytic function (construction of the conjugate harmonic function in a simply connected domain). Poisson integral formula. Schwarz’s lemma.
Limits of analytic functions: Properties carried over by uniform convergence of compact subsets, various hypotheses under which one may deduce uniform convergence on compact subsets, normal families. Conformal mapping: Local mapping properties of analytic functions, the elementary mappings (Mobius transformations, exp(z), log(z), etc.), Riemann mapping theorem.
Analytic continuation: Reflection across analytic boundaries (Schwarz reflection principle), conformal mapping of polygons to the disk, Picard’s theorem.
References:
 L.V. Ahlfors, Complex Analysis
 J.B. Conway, Functions of One Complex Variable
 W. Rudin, Real and Complex Analysis

Measures: Sigmarings, 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. RadonNikodym theorem. Fubini’s theorem. Convolution. The ndimensional 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

Differentiable manifolds: definition, submanifolds, smooth maps, tangent and cotangent bundles.
Differential forms: exterior algebra, integration, Stokes’ theorem, de Rham cohomology. Lie derivatives: of forms and vector fields.
Differential topology: regular values, Sard’s theorem, degree of a map, and index of a vector field.
“Classical” differential geometry: local theory of surfaces, 1st and 2nd fundamental forms, GaussBonnet formula.
Homotopy theory: definition of homotopy, homotopy equivalences, fundamental groups (change of base point, functoriality, Van Kampen theorem, examples such as the fundamental group of the circle), covering spaces (lifting properties, universal cover, regular (or Galois) covers, relation to π1), higher homotopy groups.
Singular homology theory: definition of the homology groups, functoriality, relative homology, excision, MayerVietoris sequences, reduced homology, connection between H1 and the fundamental group, homology of classical spaces.
References:
 M. Berger and B. Gostiaux: Differential Geometry: Manifolds Curves and Surfaces
 A. Hatcher: Algebraic Topology
 I.M. Singer and J.A. Thorpe: Lecture Notes on Elementary Topology and Geometry
 H. Hopf: Differential Geometry in the Large, Springer Lecture Notes in Mathematics, V. 1000
 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

Distributions: Parametric models, families of discrete and continuous distributions, exponential families, multivariate normal distribution, derived distributions from normal samples including t, chisquared, and F; mixtures.
Probability: Jensen, correlation, Holder, Markov and Chebyshev inequalities; order statistics, quartiles, percentiles, probability integral transformation and its inverse, modes of convergence, limit theorems, Slutsky theorems, delta method, variance stabilizing transformations.
Point estimation: method of moments, maximum likelihood, unbiased estimation, Bayes estimation, comparison of estimators, optimality, Fisher information, Cramer Rao inequality, asymptotic efficiency, sufficiency, completeness, Rao Blackwell and Lehman Scheffe theorems
References:
 G. Casella and R.L. Berger, Statistical Inference
 T.S. Ferguson, A Course in Large Sample Theory
 E.L. Lehmann, Theory of Point Estimation

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 (GagliardoNirenberg and Morrey). Rellich compactness theorem. Trace theorem. H1 space. Rademacher’s theorem.
References:
 L.C. Evans, Partial Differential Equations
 G.B. Folland, Introduction to Partial Differential Equations
 F. John, Partial Differential Equations

Existence, uniqueness and dependence of initial data. Continuation of solutions. Linear systems, periodic linear systems, Floquet’s Theorem, stability of critical points, and peri odic orbits. GrobmanHartman theorem. Lyapunov functions. Two dimensional systems, classification of elementary critical points, Poincar ́eBendixson Theorem. Invariant sets and manifolds. Stable Manifold Theorem.
References:
 L. Barreira and C. Valls: Ordinary Differential Equations: Qualitative Theory (main reference)
 E.A. Coddington and N. Levinson: Theory of Ordinary Differential Equations
 J. Hale: Ordinary Differential Equations
 C. Chicone: Ordinary Differential Equations with Applications
 P. Hartman: Ordinary Differential Equations