MathematicsPh.D. Qualifying Exams, Spring 2020
- Ph.D. qualifying exam - Analysis - Monday, January 13, 9:00am-12:00pm
- Ph.D. qualifying exam - Algebra - Wednesday, January 15, 9:00am-12:00pm
- M.S. comprehensive exam - basic topics - Monday, January 13, 9:00am-11:00am
- M.S. comprehensive exam - special topics - Wednesday, January 15, 9:00am-11:00am
Students wishing to take the mathematics M.S. or Ph.D. exams should contact Dr. Pei-Kee Lin by November 22, 2019.
Students wishing to take the comprehensive Master's exam should consult with their advisors who will make the necessary arrangements with Dr. Pei-Kee Lin.
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.
Ph.D. Qualifying Exam in Statistics
Comprehensive Master's Exam
The Master's Exam typically covers the following topics:
Applied Mathematics Concentration:
MATH 7350 – Real Variables I plus six additional hours of course work in one of the program's core categories and one additional course as described in the Graduate Catalog
MATH 7261 – Abstract Theory I, MATH 7350 – Real Variables I plus two additional courses as described in the Graduate Catalog
MATH 6636 - Intro Statistical Theory, MATH 7654 - Inference Theory plus two additional courses as described in the Graduate Catalog
Teaching of Mathematics Concentration:
Topics from four courses, each consisting of at least three credit hours of course work as described in the Graduate Catalog
Example Exams for the Applied Mathematics / Mathematics Concentration:
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 6411) - 2002U 2003S 2004S 2005F 2006S 2008S 2008F 2010S 2010U 2010F 2012S 2014S
Topology (Math 7411) - 2002S 2008F 2010S