Graduate Exams

Exam Schedules


Ph.D. Qualifying Exams, Fall 2020
  • Ph.D. qualifying exam - Analysis - Monday, August 10, 9:00am-12:00pm
  • Ph.D. qualifying exam - Algebra - Wednesday, August 12, 9:00am-12:00pm
M.S. Comprehensive Exams, Fall 2020
  • M.S. comprehensive exam - basic topics - Monday, August 10, 9:00am-11:00am
  • M.S. comprehensive exam - special topics - Wednesday, August 12, 9:00am-11:00am

Note: that due to COVID-19, times and dates are subject to change, and students may be required to wear face masks during the exams.

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, 500 Internal Server Error

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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.

Example exams:   2000F   2001F   2002S   2003S   2004S   2004F   2005S   2005F   2006F   2007F   2008S   2008F   2009F   2010F   2011S   2011F   2012S   2012F   2013S   2013F   2014S   2016F   2017S

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.

Example exams:   2003S   2003U   2004S   2004F   2005S   2005F   2006S   2007F   2008F   2009F   2010S   2010F   2011S   2011F   2013S   2013F   2014S   2014F   2015S   2015F   2016F

Ph.D. Qualifying Exam in Statistics

Example exams for Exam I:
2001 2002 2003 2004 2005 2006 2007 2009 2010 2011 2013

Example exams for Exam II:
2001 2002 2003 2004 2005 2006 2007 2009 2010 2011 2013

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

Mathematics Concentration:
MATH 7261 – Abstract Theory I, MATH 7350 – Real Variables I plus two additional courses as described in the Graduate Catalog

Statistics Concentration:
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

Example exams for the Statistics Concentration:
  2001F   2002F   2003S   2003F   2004F   2005S   2005F   2006S   2006F   2007S   2007F   2008S   2008F   2009S   2009F   2010S   2010F   2011S   2011F   2012U   2013S

Example Exams for the Teaching of Mathematics Concentration:
2010S 2012S