MathematicsPh.D. Qualifying Exams, Spring 2018
- Ph.D. qualifying exam - Analysis - Monday, January 8, 9:00am-12:00pm
- Ph.D. qualifying exam - Algebra - Wednesday, January 10, 9:00am-12:00pm
- M.S. comprehensive exam - basic topics - Monday, January 8, 9:00am-11:00am
- M.S. comprehensive exam - special topics - Wednesday, January 10, 9:00am-11:00am
Students wishing to take the mathematics M.S. or Ph.D. exams should contact Dr. Pei-Kee Lin by November 21, 2017.
Ph.D. Qualifying Exam in Mathematics
Consists of two core topics - Abstract Algebra (3 hours) and Real Analysis (3 hours).
Abstract Algebra, Math 7261-7262 (mandatory)
Background topics: Groups, subgroups, homomorphisms, Lagrange's Theorem, normal subgroups, quotient groups, isomorphism theorems, group actions, orbits, stabilizers, Cayley's Theorem, Sylow Theorems. Symmetric and Alternating groups, Solvable groups, Direct Products, Classification of Finite Abelian groups, Free groups, Group presentations. Rings, ideals, quotient rings, fields, Integral Domains, maximal and prime ideals, field of fractions, polynomial rings, Principal Ideal Domains, Euclidean Domains, Unique Factorization Domains, Gauss's Lemma, Eisenstein's Irreducibility Criterion, Chinese Remainder Theorem. Fields and field extensions. The Tower law. Algebraic and transcendental elements and extensions. Splitting field extensions. Algebraic closure. Normal and Separable extensions. Fundamental Theorem of Galois Theory. Finite fields. Cyclotomic extensions over Q. Solvability by radicals. Modules, direct sums, free modules and bases, torsion and torsion-free modules, finitely generated modules over a PID, tensor products (over commutative rings with 1), vector spaces, linear maps, dimension, matrices, minimal and characteristic polynomials, Cayley-Hamilton Theorem, Smith Normal Form, Rational Canonical Form, Jordan Normal Form.Example Textbooks:
- Serge Lang, Algebra 3rd Ed.;
- D.S. Dummit and R.M. Foote, Abstract Algebra, 2nd Ed, Chapters 0-14;
- L.C. Grove, Algebra, Chapters I-IV;
- N. Jacobson, Basic Algebra I, 2nd Ed. Chapters 0-4.
Real Analysis, Math 7350-7351 (mandatory)
Background topics: algebras and sigma-algebras of sets, Lebesgue measure and integration on the real line, differentiation and integration, Lp-spaces, metric spaces, linear operators in Banach spaces, Hahn-Banach theorem, closed graph theorem, general measure, signed measures, Radon-Nikodym theorem, product measure, Fubini and Tonelli theorems.Example textbooks:
- H.L. Royden, Real Analysis, Macmillan Publishing Company 1988 (3rd edition).
- H.L. Royden and P.M. Fitzpatrick, Real Analysis, Prentice Hall 2010 (4th edition).
- R.M. Duddley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics 1989 (2nd edition).
- S.K. Berberian, Fundamentals of Real Analysis, Springer-Verlag 1999.
- John N. McDonald and Neil A. Weiss, A Course in Real Analysis, Academic Press 1999.
- G.B. Folland, Real Analysis, Modern Techniques and their Applications, Wiley-Interscience 1999.
Comprehensive Master's Exam in Mathematics
The Master's Exam in Mathematics consists of two 2-hour exams. The first covers Introduction to Real Analysis (Math 6350, Math 6351), Abstract Algebra (Math 6261), Linear Algebra (Math 6242), and Topology (Math 6411). The second 2-hour exam is on two topics chosen by the student from a list of optional topics.
Introduction to Real Analysis, Math 6350 and 6351 (mandatory)
Background topics: The real number system, functions and sequences, limits, continuity, differentiation, Riemann-Stieltjes integration, functions defined by integrals, improper integrals, series of functions, differentiation of functions of several variables, Riemann integration of functions of several variables, implicit function theorem and Lagrange Multipliers.Example textbooks:
- M.H. Protter and C.B. Morrey, A First Course in Real Analysis, 2nd Ed., Springer 1991.
- R.G. Bartle, The Elements of Real Analysis, 2nd Ed., John Wiley & Sons, 1976.
Abstract Algebra, Math 6261 (mandatory)
Background Topics: Groups, subgroups, homomorphisms, Lagrange's Theorem, normal subgroups, quotient groups. Symmetric and Alternating groups. Rings, ideals, quotient rings, fields, Integral Domains. Polynomial rings.
Linear Algebra, Math 6242 (mandatory)
Background topics: Linear transformations, polynomials, determinants, direct-sum decompositions, diagonalizable operators, rational and Jordan form, inner product spaces, Hermitian and normal operators, spectral theorem.Example textbooks:
- K. Hoffman and R. Kunze, Linear Algebra, second edition, Prentice Hall 1971.
- Sh. Axler, Linear Algebra Done Right, Springer 1997.
Topology, Math 6411 (mandatory)
Background topics: Set Theory. Sets. Functions. Cartesian Products. Relations. Countable sets. Uncountable sets. Metric Spaces. Basic Concepts: Definitions and examples, continuous maps, uniformly continuous maps, and homeomorphisms. Convergence. Open and closed sets. Closure of a set and limit points. Dense subsets. Separable spaces. Complete metric spaces and uniform metric spaces. Topological spaces. Definitions and examples. Comparison of topologies. Bases. Closed sets, closures, and interiors of sets. The first and second axioms of countability. Continuous mappings and homeomorphisms. Product topology on X×Y. Connectedness. Connected spaces. Connected subsets of the reals. Pathwise and local connectedness. The Intermediate-value theorem. Connected components, the topologist's sine curve. Compactness. Compactness in metric spaces. Compact spaces. Countably compact, compact, sequentially compact and totally bounded. Continuous maps on compact spaces.Example textbooks:
- C. Patty, Foundations of Topology, PWS-KENT Publishing Co., Boston, MA, 1993, Chapters 1-4.
- S. Willard, General Topology, Addison-Wesley, 1970.
Two topics may be chosen from, but are not limited to, the following list: Algebraic Theory I (Math 7261), Algebraic Theory II (Math 7262), Combinatorics (Math 7235), Applied Graph Theory (Math 7236), Real Variables I (Math 7350), Real Variables II (Math 7351), Complex Analysis (Math 7361), Calculus of Variations (Math 7371), Topology (Math 7411), Modeling and Computation (Math 7721), Mathematical Methods for Physics (Math 7375).
Algebraic Theory I (Math 7261) - 2002S 2003S 2005F 2006S
Algebraic Theory II (Math 7262) - 2003S 2005F 2006S 2010S
Applied Graph Theory (Math 7236) - 2014S
Complex Analysis (Math 7361) - 2004S 2008F 2013S
Real Variables I (Math 7350) - 2003S 2003U 2004S 2006S 2010S 2012S 2013S
Topology (Math 7411) - 2002S 2008F 2010S