## Ph.D. Qualifying Exams

A Ph.D. student must pass two of the three qualifying exams, before becoming a Ph.D. candidate.

Upcoming Qualifying Exam Schedule for Fall, 2021

• Analysis: Wednesday, August 18, 2:00-5:00pm in SE 215

• Probability/Statistics:  Thursday, August 19, 2021, 2:00pm--5:00pm in SE 215

• Algebra:  Friday. August 20th, 2021, 2:00pm--5:00pm in SE 215

All the exams will be in-person format. Exam locations are to be determined.

### Topics covered in the Ph.D. Qualifying Exams

Algebra Exam: group theory, Sylow theorems, the structure of finitely generated abelian groups, ring theory, Euclidean rings, UFDs, polynomial rings, vector spaces, modules, linear transformations, eigenvalues, minimal polynomials, matrices of linear transformations, Galois theory, and finite fields.

Analysis Exam: the real numbers, metric space topology, uniform convergence, Arzela-Ascoli Theorem, differentiation and Riemann integration of single-variable functions, power series, Stone-Weierstrass Theorem, measure theory, Lebesgue integral, convergence theorems for the Lebesgue integral, absolute continuity, the Fundamental Theorem of Calculus.

Probability & Statistics Exam: Advanced topics in Probability and Statistics: Borel-Cantelli lemma, normal and Poisson distributions, Chi-square and exponential distributions, t and F distributions,  Markov and Chebyshev inequalities, convergence in distribution, in probability and almost surely, law of large numbers, central limit theorem, delta method, Slutsky lemma, LSE, MLE, BLUE, sufficient statistics, Cramer-Rao inequality, Fisher information matrix, hypothesis tests via likelihood ratio test and Bayes test.

The Probability & Statistics Exam will be divided into two parts. Total 3 hours.

Part 1   (Elementary part). This part consists of 5 elementary Probability and Stat questions. These will be the same (or very similar) questions that are given for Actuarial Exam. Students are expected to successfully complete at least 80% of these problems.

Part 2  (Challenging part). This part consists of 5 advanced problems from Probability Theory and Math Stat classes, including 2-3 proof problems from the  predetermined list  of about 20 basic well known facts in Probability and Statistics with fairly simple proofs (less than a page). Students are expected to successfully complete at least 70% of these problems.

Note: the syllabus in any particular section of the Introductory Abstract Algebra, Introductory Analysis, and Mathematical Probability/Statistics courses might differ slightly from the subject material listed above.

References:
Topics in Algebra, 2nd ed., by Herstein, Chapters 2-5, 6.1-6.3, and 7.
Algebra, 3rd ed., by Lang, Chapters 1-6.
Abstract Algebra, 3rd ed., by Dummit and Foote, Chapters 1-5, 7-9, and 13-14, excluding 9.6 and 14.9.

Introduction to Analysis
, by Maxwell Rosenlicht, Chapters 2-7.
Real Mathematical Analysis, by Charles Pugh, Chapters 1-4.
Real Analysis, 3rd ed., by H.L. Royden, Chapters 3, 4, 5.
Principles of Mathematical Analysis, 3rd ed., by Rudin, Chapters 2-8 and 11.
Measure and Integral, by Wheeden amd Zygmund, Chapters 3-5.

A Probability Path by Resnick
Probability theory by Shiryaev
Measure Theory and Probability Theory by Athreya and Lahiri
Mathematical Statistics by Bickel and Doksum
Statistical Inference by Casella