This exam covers topics at the undergraduate level, most of which might be encountered in courses here such as Math 233, 235, 523, 545. Faculty members who teach these courses can recommend texts for review purposes. The emphasis is on understanding basic concepts, rather than performing routine computations. But exam questions often center on concrete examples of matrices, functions, series, etc.

• Vector spaces: subspaces, linear independence, basis, dimension.
• Linear transformations and matrices: kernel and image, rank and nullity, transpose.
• Linear operators: change of basis and similarity, trace and determinant, eigenvalues and eigenvectors, characteristic polynomial, diagonalizable operators.
• Inner product spaces: orthonormal basis, orthogonal complements and projections, orthogonal matrices, diagonalization of symmetric matrices.
• Functions of one real variable: continuity and uniform continuity, derivative and Mean Value Theorem, Riemann integral, improper integrals, Fundamental Theorem of Calculus.
• Sequences and series of numbers or functions: pointwise, uniform, absolute convergence; term-by-term differentiation and integration of series; Taylor’s Theorem with remainder.
• Functions of several variables: continuity, partial and directional derivatives, differentiability, maps from Rⁿ to R^m, Jacobian, implicit and inverse function theorems, chain rule.
• Extrema of functions of several variables: constrained extrema, Lagrange multipliers.
• Multiple and iterated integrals, change of variables formula.
• Vector calculus: gradient, divergence, curl; line and surface integrals; theorems of Green, Gauss, Stokes; conservative vector fields.

Below are the topics for the qualifying exam in Algebra

1. Group Theory
   • Group actions; counting with groups.
   • p-groups and Sylow theorems.
   • Composition series; Jordan-Holder theorem; solvable groups.
   • Automorphisms; semi-direct products.
   • Structure of finitely generated Abelian groups
2. Linear Algebra and Commutative Algebra.
   • Euclidean domain implies PID implies UFD.
   • Gauss Lemma; Eisenstein’s Criterion.
   • Exact sequences; isomorphism theorems for modules.
   • Free modules.
   • Hom and tensor product of vector space, Abelian groups, and modules; Restriction and extension of scalars.
   • Bilinear forms; symmetric and alternating forms; symmetric and exterior algebras.
   • Structure Theorem for finitely generated modules of a PID.
   • Rational canonical form.
   • Jordan canonical form.
   • Chain conditions; Noetherian rings and modules; Hilbert’s Basis Theorem.
   • Prime and maximal ideals.
   • Field of fractions.
   • Localization of rings and modules: exactness of localization; local rings; Nakayama’s Lemma.
   • Integral extensions.
   • Noether’s Normalization Lemma.
   • Integral closure.
   • Nullstellensatz.
   • Closed affine algebraic sets.
3. Field Theory and Galois Theory
   • Algebraic field extensions: finite extensions; degree of extensions; the minimal polynomial; adjoining roots of polynomials; the splitting field; algebraic closure.
   • Separable extensions.
   • Theorem of the primitive element.
   • Galois extensions: Fundamental Theorem of Galois Theory.
   • Finite fields and their Galois groups; Frobenius endomorphism.
   • Cyclotomic polynomial, Cyclotomic fields and their Galois groups.
   • Cyclic extensions.
   • Solvable extensions; Solving polynomial equations in radicals.
   • Transcendence degree.

References

• Dummit and Foote, Abstract Algebra
• Atiyah and MacDonald, Introduction to Commutative Algebra
• Lang, Algebra

• Lebesgue measure: Construction of Lebesgue measure on R and R^d. Measurable and non measurable sets; Cantor sets. Lebesgue and Borel measurable functions; Egorov theorem and Lusin Theorems. Construction and properties of Lebesgue integral; the space L¹ of integrable functions and its completeness; comparison with Riemann integral. Fubini-Tonelli theorem in R^d. Modes of convergence: convergence almost everywhere, convergence in measure, convergence in L¹.
• Integration and differentiation: Differentiation of the integral; Hardy-Littlewood maximal function; Lebesgue differentiation theorem. Functions of bounded variation; absolutely continuous functions; the fundamental theorem of calculus.
• Hilbert spaces Abstract Hilbert spaces and examples; L² spaces; Bessel’s inequality and Parseval’s identity; Riemann-Lebesgue Lemma; Orthogonality; orthogonal projections. Linear transformations; linear functionals; Riesz representation theorem; adjoint transformations.
• Fourier analysis: Fourier transform in L¹ and L²; Fourier inversion formula. Fourier series; Dirichlet’s Theorem and Fejér’s Theorem
• General theory of measure and integration Measure spaces and σ-algebras. σ-finite measures. Caratheodory theorem and the construction of measures; outer measures and extension theorems. Integration theory. Product measures and Fubini-Tonelli theorem. Signed measure; Radon-Nikodym theorem; Borel measures on R and Lebesgue-Stieljes integral.
• Banach spaces and L^p spaces. Abstract Banach spaces and examples; completeness criterion. Convexity; L^p-norms; Schwarz, Hölder, Minkowski, and Jensen inequalities. L^p-spaces and their duals; Riesz-Fischer Theorem.

References

• Berberian, Introduction to Hilbert Spaces
• Folland, Real Analysis
• Gelbaum and Olmsted, Counterexamples in Analysis
• Halmos, Measure Theory
• Royden, Real Analysis
• Rudin, Principles of Mathematical Analysis
• Rudin, Real and Complex Analysis
• Stein and Shakarchi, Real Analysis
• Wheeden and Zygmund, Measure and Integral

Here are the topics for the Applied Mathematics qualifying exam:

Dynamical Systems

• Stability of Equilibria
• Omega-limit sets
• Floquet Theory
• Lyapunov Functions
• Bifurcation Theory
• Poincaré-Bendixson
• Index Theory
• Hamiltonian Systems

Applied Mathematics

• Dimensional Analysis
• Buckingham pi-theorem
• Random walks and continuum limits
• Regular and Singular Perturbation Theory
• Boundary layer theory
• Calculus of Variations
• Hyperbolic Conservation Laws
• Modeling with PDE

Sources

• James D. Meiss, Differential Dynamical Systems
• C. C. Lin, L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences
• J. David Logan, Applied Mathematics

Part A: Probability

• Probability axioms, conditional probability and independence and probability on countable sets.
• Construction of probability measures, in particular on R and R^d.
• Random variables and theory of integration.
• Independence random variables and sum of independent random variables.
• Convergence of random variables (almost sure, in probability, L^p) and laws of large numbers.
• Weak convergence and central limit theorems.
• Conditional expectation and martingales.

Part B: Stochastics

• Simulation of random variable and Monte-Carlo methods.
• Finite and countable state space Markov chains, stationary distribution, convergence, recurrence and transient behavior. Monte-Carlo Markov chains.
• Continuous time Markov chains, poisson processes and queueing processes.
• Martingales.
• Random walks and Brownian motion.

References (Probability)

• Cinlar, Probability and Stochastics
• Dudley, Real Analysis and Probability
• Durrett, Probability: Theory and Examples
• Jacod and Protter, Probability Essentials
• Resnick, Probability Path
• Rosenthal, A first look at rigorous probability
• Shiryaev, Probability

References (Stochastics)

• Lawler, Introduction to Stochastic Processes
• Resnick, Adventures in Stochastic processes
• Durrett, Essentials of Stochastic Processes
• Ross, Introduction to Probability Models
• Madras, Lectures on Monte-Carlo Methods
• Rubinstein and Kroese, Simulation and the Monte Carlo Method
• Levin, Peres and Wilmer, Markov chain and mixing times
• Hoel, Port and Stone, Introduction to Stochastic Processes
• Ross, Stochastic Processes
• Karlin and Taylor, A first course in Stochastic Processes

• Topology and continuity: bases, order topology, subspace topology, product topology (infinite and finite), box topology, closed sets, limit points, Hausdorff spaces, homeomorphisms, Pasting Lemma, metric spaces, uniform topology, Uniform Limit Theorem, quotient topology, open and closed maps.
• Compactness and connectedness: connectedness of R, Intermediate Value Theorem, connectedness for products, path-connectedness, components and path-components, Tube Lemma, finite intersection property, uniform continuity, Heine-Borel Theorem, Lebesgue Number Lemma, sequential compactness, limit point compactness, local compactness, compactifications.
• Complete metric spaces: Cauchy sequences, equicontinuity, Ascoli Theorem (for Rⁿ), complete and totally bounded metric spaces.
• Definition and elementary properties of homotopy; homotopy equivalences; deformation retracts.
• The definition of the fundamental group π₁; functoriality under mappings and invariance under homotopy. The relation between π₁ at different base points. The fundamental group of a cartesian product.
• The path lifting/homotopy lifting lemmas, their proofs, and their use in proving that π₁(S¹)≅Z.
• The statement of the Seifert–van Kampen theorem, and its use in computing π₁ of various spaces, such as compact surfaces.
• Covering spaces; path and homotopy lifting theorems; classification of connected covers via subgroups of the fundamental group.
• Cell complexes, ∆-complexes and simplicial complexes, the classification of compact surfaces.
• Singular, simplicial and cellular homology; Hurewicz theorem relating H₁ and π₁; degree of maps between spheres (and connected orientable manifolds), induced homomorphisms, homotopy invariance; reduced homology; relative homology; long exact sequences of a pair, a triple, and the Mayer-Vietoris sequence; excision; Homology with coefficients, the universal coefficients theorem; Euler characteristic.
• Simplicial, singular and cellular cohomology; the cup product; Künneth theorems; orientations, the cap product and Poincaré duality

References

• Munkres, Topology: A First Course, Sections 1-7, 12-29, 43, 45
• Hatcher, Algebraic Topology, Chapters 0, 1, 2, 3.

• Probability density and distribution functions.
• Random variables and vectors, expectation, moments.
• Joint and marginal distributions.
• Conditional distributions and expectations.
• Transformations of random variables.
• Moment generating functions.
• Independence, laws of large numbers, central limit theorems.
• Special distributions (e.g., binomial, Poisson, normal, t, F, etc.).
• Basic combinatorics.

References

• Chung, Elementary Probability Theory
• Woodroofe, Probability with Applications
• Arnold, Mathematical Statistics
• Casella and Berger, Statistical Inference

• Simple and multiple linear regression; correlation.
• The use of dummy variables.
• Residuals analysis and diagnostics assessment of model assumptions.
• Model building/variable selection, regression models and methods in matrix form.
• Generalized linear models.
• Basic knowledge of weighted least squares, regression with correlated errors, and nonlinear regression.
• Programming in R or Python: functions, objects, data structures, flow control, input and output, debugging, logical design and abstraction, simulations, parallel data analyses, optimization, large data set handling, commenting and organizing code.

• Sampling distributions.
• Exponential families.
• Sufficiency and completeness.
• Estimation: maximum likelihood, method of moments, unbiasedness, efficiency, consistency.
• Bayes estimators.
• Interval estimation.
• Hypothesis testing: basic framework, UMP tests, likelihood ratio tests.

References

• Bickel and Doksum, Mathematical Statistics
• Mood, Graybill and Boes, Introduction to the Theory of Statistics
• Arnold, Mathematical Statistics
• Casella and Berger, Statistical Inference
• DeGroot, Probability and Statistics, 2nd ed.

This exam covers the theory and application of the linear model based mostly on the course content of Stat 607, 608 and 705. 

Topics include:

• Multivariate normal distribution
• Linear and quadratic forms and related distributions
• Linear model
  • estimation
    • estimability
    • least squares and perpendicular projection
    • maximum likelihood
    • sampling distribution
    • Gauss Markov
    • diagnostics and residuals
  • inference
    • general linear hypotheses
    • analysis of variance
    • confidence intervals

References

References for basic Statistics and Probability exams

• Ravishanker and Dey, A First Course in Linear Model Theory, Chapters 1,2,3, 5 and 6.
• Graybill, Theory and Application of the Linear Model, Chapters 3 and 4.
• Seber and Lee, Linear Regression Analysis, 2nd Edition, Chapters 1 and 2.

This exam covers materials on Probability and Mathematical Statistics based on course content in Stat 607, 608 and 725. Topics include:

Content of basic probability and statistics and statistical models; point estimation, set estimation and hypothesis testing from a frequentist’s, decision theoretic and Bayesian point of view; finite sample and asymptotic techniques in a variety of (parametric/semiparametric/non-parametric) statistical models.

References

References for basic Statistics and Probability exams

• Jun Shao, Mathematical Statistics (2nd Edition). Springer Series in Statistics.
• T. S. Ferguson, A Course in Large Sample Theory. Chapman & Hall/CRC
• A. W. van der Vaart, Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics