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Algebra Definitions

Groups:

Homomorphism theorems, permutation groups, automorphisms, finitely generated Abelian groups, products of groups, group actions, Sylow theorems, p-groups, nilpotent groups, solvable groups, normal and subnormal series, Jordan-Hölder Theorem, special subgroups (e.g., commutator subgroup, Frattini subgroup, etc.).

Rings:

Matrix rings, polynomial rings, factor rings, endomorphism rings, rings of fractions, localization and local rings, prime ideals, maximal ideals, primary ideals, integral domains, Euclidean domains, principal ideal rings, unique factorization domains, Jacobson radical, chain conditions, modules, factor modules, irreducible modules, Artinian and Noetherian rings and modules, semisimplicity.

Fields:

Algebraic extensions, algebraic closures, normal extensions and splitting fields, separable and purely inseparable extensions, theorem of the primitive element, Galois theory, finite fields, cyclotomic extensions, cyclic extensions, radical extensions and solvability by radicals, transcendental extensions.

Linear Algebra:

Matrix theory, eigenvalues and eigenvectors, characteristic and minimal polynomials, diagonalization, canonical forms, linear transformations, vector spaces, bilinear forms, inner products, inner product spaces, duality, tensors.

Suggested Courses:

• MATH 61051 / 71051: Abstract Algebra I
• MATH 61052 / 71052: Abstract Algebra II

Suggested References:

• N. Jacobson, Lectures in Abstract Algebra, Vols. I, II, III, D. Van Nostrand Co
• N. Jacobson, Basic Algebra I and II, Freeman 
• S. MacLane and G. Birkhoff, Algebra, Chelsea
• T. Hungerford, Algebra, Springer-Verlag
• S. Lang, Algebra, Addison-Wesley
• M. Hall, The Theory of Groups, Macmillan
• J. Rose, A Course on Group Theory, Cambridge University Press
• J. Rotman, The Theory of Groups: An Introduction, Allyn and Bacon
• E. Artin, Galois Theory, University of Notre Dame Press
• Hoffman and Kunze, Linear Algebra, Prentice-Hall

This exam involves combining Complex Analysis with Real Analysis into one exam format.

Analysis

Preliminaries:

Review of basic notions of complex numbers, Argand diagram, etc. Review of basic necessary topological properties of C.

Basic Complex Variables Theory:

Including Cauchy-Riemann equations, power series, integral formulae and their applications, (e.g., maximum modulus theorem, Schwarz's lemma, etc.)

Singularities:

Analytic continuation, residues, and applications, (e.g., Rouche's theorem, argument principle, etc.)

Additional Topics:

Including infinite products and applications, Mobius transformations, conformal mapping.

Suggested Courses: 

• MATH 62151 / 72151: Functions of a Complex Variable I
• MATH 62152 / 72152: Functions of a Complex Variable II

Suggested References: 

• L. Ahlfors, Complex Analysis, McGraw Hill. 
• J. Conway, Functions of One Complex Variable, Springer-Verlag. 
• W. Rudin, Real and Complex Analysis, McGraw Hill.

Real Analysis

Preliminaries:

The student is expected to be familiar with those topics normally covered in a one-year, senior-level course in analysis. These topics include: real and complex numbers and their properties, properties of the metric space Rⁿ, continuity, differentiability, Riemann integration, Riemann-Stieltjes integration, sequences and series of functions.

Set Theory:

One-one functions, cardinal numbers, partial order, countability, Zorn's lemma.

Metric Spaces:

Open and closed sets, convergent sequences, functions and continuity, semi-continuity, separable spaces, complete spaces, compact spaces, Baire category theorem.

Measure Spaces:

Lebesgue outer measure, Borel sets, algebras, measurable sets and non-measurable sets, measure spaces.

Measurable Functions and Integration:

Properties, approximation by simple functions, Egoroff's theorem, monotone convergence theorem, convergence in measure, Fatou's lemma, Lebesgue dominated convergence theorem, signed measures, Lebesgue differentiation theorem, Hahn decomposition theorem, Radon-Nikodym theorem, Lebesgue decomposition theorem, Fubini's theorem, Tonelli's theorem, Stone-Weierstrass theorem, Ascoli-Arzela theorem.

L_p-Spaces:

Minkowski and Holder inequalities, Riesz representation theorem.

Banach Spaces:

Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem.

Hilbert Spaces:

Geometrical aspects, projection, Riesz-Fischer theorem.

Suggested Courses: 

• MATH 62051 / 72051:Functions of a Real Variable I
• MATH 62052 / 72052: Functions of a Real Variable II

Suggested References: 

• H. L. Royden, Real Analysis, MacMillan
• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill

Preliminaries:

The student is expected to be familiar with the major topics (at the advanced undergraduate, beginning graduate level) in linear algebra, advanced calculus, introductory partial differential equations, and introductory complex variables.

Exam topics may include (but are not limited to):

Dimensional Analysis and Scaling:

Buckingham Pi Theorem, characteristic scales, well-scaled problems. Perturbation Methods and Asymptotic Expansions: asymptotic sequences and series, regular perturbations, Poincare'-Lindstedt method, singular perturbations, boundary layer analysis, WKB approximations, asymptotic expansions of integrals.

Calculus of Variations:

first and second variations, Euler-Lagrange equations, first integrals, isoperimetric problems.

Integral Equations and Green's Functions:

Volterra and Fredholm integral equations, degenerate kernels, Green's functions, Fredholm Alternative.

Partial Differential Equations:

well-posed problems, maximum principles, energy argument (Lyapunov functions), orthogonal expansions, Fourier Transforms, heat kernel.

Suggested Courses:

• MATH 41021 / 51021: Theory of Matrices
• MATH 42041 / 52041: Advanced Calculus
• MATH 42045 / 52045: Introduction to Partial Differential Equations
• MATH 42048 / 52048: Introduction to Complex Variables (for preliminary material)
• MATH 62041 / 72041: Methods of Applied Mathematics I (for core material)
• MATH 62042 / 72042: Methods of Applied Mathematics II (for core material)

Suggested References:

• James P. Keener, Principles of Applied Mathematics: Transformation and Approximation, 2nd ed., Westview Press, 2000
• C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM Classics, 1988
• J. David Logan, Applied Mathematics, 4th ed., Wiley, 2013

Preliminaries: 

The student is expected to be familiar with those topics normally covered in a one-year, senior-level course in numerical methods, including computer arithmetic, solving linear systems of equations (by direct methods), polynomial interpolation, numerical quadrature methods, linear least-squares data fitting, solving non-linear equations, and basic numerical methods for ODE initial-value problems.

Error Analysis: 

Floating-point arithmetic, roundoff-error analysis, mathematical conditioning. Interpolation: Lagrange formula, Neville's algorithm, Newton formula and divided differences, error in polynomial interpolation, Hermite interpolation, trigonometric interpolation, discrete Fourier analysis, fast Fourier transform, interpolation by spline functions.

Integration: 

Newton-Cotes formulas, Peano kernel theorem, Euler-Maclaurin summation formula, asymptotic expansions, extrapolation and Romberg integration, Gaussian quadrature, orthogonal polynomials.

Systems of Linear Equations: 

Gaussian elimination, LU-decomposition, Cholesky decomposition, backwards error analysis, matrix and vector norms and condition numbers.

Linear Least-Squares: 

Orthogonalization, Gram-Schmidt, Householder and Givens transformations, QR-factorization, condition of linear least-squares problems, pseudoinverse.

Eigenproblems: 

Matrix normal forms (Jordan, Schur), similarity reduction to tri-diagonal or Hessenberg forms, power method, inverse iteration, Rayleigh quotients, LR-method, QR-method, singular value decomposition.

Suggested Courses: 

• MATH/CS 42201 / 52201: Introduction to Numerical Computing I
• MATH/CS 42202 / 52202: Introduction to Numerical Computing II
• MATH 62251 / 72251: Numerical Analysis I
• MATH 62252 / 72252: Numerical Analysis II

Suggested References: 

• Conte and de Boor, Elementary Numerical Analysis: an Algorithmic Approach, McGraw-Hill
• Dahlquist and Bjorck, Numerical Methods, Prentice-Hall
• Golub and Van Loan, Matrix Computations, 3rd ed., Johns Hopkins
• Kahaner, Moler, and Nash, Numerical Methods and Software, Prentice-Hall
• Stewart, Introduction to Matrix Computations, Academic Press
• Stoer and Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer
• Trefethen and Bau, Numerical Linear Algebra, SIAM

Probability Theory: 

Distribution functions, random variables, expectation, independence, convergence concepts, law of large numbers, characteristic functions, the central limit theorem, conditional expectation, martingales, Brownian motion.

Suggested Courses: 

• MATH 60051 / 70051: Probability I
• MATH 60052 / 70052: Probability II

Suggested References: 

• P. Billingsley, Probability and Measure, John Wiley. 
• K. L. Chung, A Course in Probability Theory, Academic Press.

Statistics: 

Sufficient statistics, uniformly minimum variance unbiased estimates, maximum likelihood, method of moments, Bayes, minimax and least-square estimates, interval estimation, testing simple and composite hypotheses, Neyman-Pearson lemma, sequential tests, Wald's sequential probability ratio test.

Suggested Courses: 

• MATH 60061 / 70061: Mathematical Statistics I
• MATH 60062 / 70062: Mathematical Statistics II

Suggested References: 

• F. J. Bickel and K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Holden Day. 
• E. L. Lehmann, Theory of Point Estimation, Wiley Interscience. 
• E. L. Lehmann, Testing Statistical Hypotheses, Wiley Interscience.