The University of Memphis, TN 38152-3240
Tel: (901) 678-2482 Fax: (901) 678-2480
|
Department of Mathematical Sciences The University of Memphis, TN 38152-3240 Tel: (901) 678-2482 Fax: (901) 678-2480 |
| Coordinators of Graduate Studies Dr. Anna Kaminska (Mathematics) kaminska@memphis.edu |
Dr. Paul Balister (Mathematics Exams) pbalistr@memphis.edu |
| Dr. Lih-Yuan Deng (Statistics) lihdeng@memphis.edu |
Dr. Lih-Yuan Deng (Statistics Exams) lihdeng@memphis.edu |
Mathematics
PhD Qualifying and Comprehensive Master's Exams, Spring 2008.
Students wishing to take the above Ph.D. exams should contact Dr. Paul Balister.
Students wishing to take the Masters' exams should contact Dr. Paul Balister.
Statistics
Students wishing to take the Statistics' exams should contact Dr. Lih-Yuan Deng.
| Mathematics Ph.D. | Mathematics Masters |
| Statistics Ph.D. | Statistics Masters |
| Mathematics Masters for Teachers | |
| Note: You must have the Adobe Acrobat® Reader for the PDF files. If you do not have the reader installed on your system then to download the free Acrobat® Reader 5.0 click here. Adobe Acrobat® Reader |
Consists of three core topics - Abstract Algebra (3 hours), Real Analysis (3 hours), Topology (1.5 hours) - and one optional topic chosen by the student (1.5 hours).
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.
Main textbook:
D.S. Dummit and R.M. Foote,
Abstract Algebra, 2nd Ed. Chapters 0-14.
Alternative textbooks:
L.C. Grove,
Algebra, Chapters I-IV.
N. Jacobson,
Basic Algebra I, 2nd Ed. Chapters 0-4.
Example exams:
Fall 2000
Fall 2001
Spring 2002
Spring 2003
Spring 2004
Fall 2004
Spring 2005
Fall 2005
Fall 2006
Fall 2007
Background topics: algebras and sigma-algebras of sets, Lebesgue measure and integration on 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.
Main textbook:
H.L.Royden,
Real Analysis, Macmillan Publishing Company 1988.
Alternative textbooks:
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.
Example exams:
Spring 2003
Summer 2003
Spring 2004
Fall 2004
Spring 2005
Fall 2005
Spring 2006
Fall 2007
Background topics: Set Theory. Relations. Ordinal and cardinal numbers. Zorn's Lemma and the Axiom of Choice. Topological spaces, product and quotient spaces. Connected, locally connected, locally pathwise connected, and totally connected spaces. Compact and locally compact spaces. Compactifications. Product topology and the Tychonoff theorem. Baire spaces and the Baire category theorem. Separation and Countability Axioms. T0, T1, T2 spaces. Regular and completely regular spaces. Normal and completely normal spaces. The countability Axioms. The Urysohn's lemma and the Tietze's extension theorems. Metrizability and Paracompactness. The Urysohn's metrizationtheorem. Paracompactness. The Fundamental Group and Covering Spaces. Homotopy of paths. The Fundamental Group. The Fundamental Group of the circle. Covering spaces.
Main textbook:
C. Patty,
Foundations of Topology, PWS-KENT Publishing Co., Boston, MA, 1993, Chapters 1-4.
Alternative textbooks:
J. Munkres,
Topology: A First Course, Prentice Hall, 1975.
S. Willard,
General Topology, Addison-Wesley, 1970.
Example exams:
October 2001
Fall 2002
Spring 2003
Spring 2004
Fall 2004
Spring 2005
Fall 2005
Spring 2006
Fall 2006
Fall 2007
The optional topic may be chosen from, but not limited to, the following list of topics: Complex Analysis (MATH 7361), Differential Equations (MATH 7393 or 7395), Discrete Mathematics (MATH 7235 or 7236), Numerical Analysis (MATH 7721), and Optimization (MATH 7371 or 7391).
Example exams:
Calculus of Variations -
Spring 2005
Combinatorics -
Spring 2000
Fall 2000
Fall 2001
Spring 2003
Spring 2005
Fall 2005
Spring 2006
Complex Analysis -
Summer 2003
Fall 2005
Fall 2006
Fall 2007
Differential Equations -
Fall 2001
Spring 2003
Spring 2004
Interpolation and Function spaces -
Fall 2004
Linear Algebra -
Spring 2004
Local structure of Banach spaces -
Fall 2007
Mathematical Physics -
Spring 2004
P-adic Analysis -
Fall 2006
The Master's Exam in Mathematics consists of a 2 hour exam on Basic topics which is taken by all the students (covers Introduction to Real Analysis, Abstract Algebra, Linear Algebra, and Topology) and a second 2 hour exam which is either on Applied Mathematics or on two topics chosen by the student from a list of optional topics.
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.
Main textbook:
M.H. Protter and C.B. Morrey,
A First Course in Real Analysis, 2nd Ed., Springer 1991.
Alternative textbook:
R.G. Bartle,
The Elements of Real Analysis, 2nd Ed., John Wiley & Sons, 1976.
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.
Background topics: Linear transformations, polynomials, determinants, direct-sum decompositions, diagonalizable operators, rational and Jordan form, inner product spaces, Hermitian and normal operators, spectral theorem.
Main textbook:
K. Hoffman and R. Kunze,
Linear Algebra, second edition, Prentice Hall 1971.
Alternative textbook:
Sh. Axler,
Linear Algebra Done Right, Springer 1997.
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.
Main textbook:
C. Patty,
Foundations of Topology, PWS-KENT Publishing Co., Boston, MA, 1993, Chapters 1-4.
Alternative textbook:
S. Willard,
General Topology, Addison-Wesley, 1970.
Example Exams:
Core Topics -
Summer 2002
Spring 2003
Spring 2004
Fall 2005
Spring 2006
Spring 2008
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).
Example Exams:
Algebraic Theory I (MATH 7261) -
Spring 2002
Spring 2003
Fall 2005
Spring 2006
Algebraic Theory II (MATH 7262) -
Spring 2003
Fall 2005
Spring 2006
Complex Analysis (MATH 7361) -
Spring 2004
Real Variables I (MATH 7350) -
Spring 2003
Summer 2003
Spring 2004
Spring 2006
Topology (MATH 7411) -
Spring 2002
Example Exams:
Example 1
Example 2
Example 3
Example exams:
Past exams are available at: http://www.msci.memphis.edu/~dengl/sample-exams/
Example exams:
Past exams are available at: http://www.msci.memphis.edu/~dengl/sample-exams/
 |
SiteManager | |
Dept Email | |
Site Map | |
Homepage | |
University | |
Computer Science | |
Updated: April 11, 2008
 |
| Site Manager: Dr. R. A. Clapsadle | Copyright 2007 by the University of Memphis. Notice Statement | ||||||||||||