# Graduate Exams

## Exam Schedules

### Mathematics

**Ph.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 Exams, Spring 2020**

- 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.

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, *L _{p}*-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.

- 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