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The Qualifying Exam

The questions on the Qualifying Exam (Quals) aim to test your ability to solve concrete problems by identifying and applying important theorems. They should not require great ingenuity. In any given year, the exam may not cover every topic on the syllabus, but it should cover a broadly representative set of Quals topics and over time all Quals topics should be examined.

The Qualifying Exam syllabus is divided into six areas. In each case, we suggest a book to more carefully define the syllabus. The examiners are asked to limit their questions to major Quals topics covered in these books. We have tried to choose books we think are good. However, there are many good books and others might better suit your needs.

Syllabus

1) Algebra

References: Dummit and Foote: Abstract Algebra, 2nd edition, except chapters 15, 16 and 17, Serre: Representations of Finite Groups (Sections 1-6). Fulton-Harris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics) Lie groups and algebras, Chapters 7-10.

2) Algebraic Geometry

References: Shafarevich: Basic Algebraic Geometry 1, 2nd edition, Harris: Algebraic Geometry: A First Course

3) Complex Analysis

References: Ahlfors: Complex Analysis (3rd edition)

4) Algebraic Topology

References: A. Hatcher: Algebraic Topology, W. Fulton: Algebraic Topology, E. Spanier: Algebraic Topology, Greenberg and Harper: Algebraic Topology: A First Course

5) Differential Geometry

References: Taubes: Differential geometry: Bundles, Connections, Metrics and Curvature Lee: Manifolds and Differential Geometry (Graduate Studies in Math 107, AMS), S. Kobayashi and K. Nomizu: Foundations of Differential Geometry

6) Real Analysis

References: Rudin: Real and complex analysis is a general reference but the following books have more useful techniques Stein and Shakarchi: Real analysis. Stein's book does not have Lp spaces. A good source of Lp spaces and convexity is Lieb-Loss: Analysis, Chapter 2. Fourier series: Stein and Shakarchi: Fourier Analysis. This book is very elementary but more than sufficient chapters 2 and 3 are Fourier series, chapter 5 is Fourier transform. Sobolev spaces: Evans: Partial Differential Equations. Chapter 5. Probability: Shiryayev: Probability. Feller: An Introduction To Probability Theory And Its Applications Durrett: Probability: Theory And Examples

Some old Exams

All links are to PDF files

Fall Spring
2012[S]
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2008[S]
2007
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2002
2001
1995
2011
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1996
Some old quals from 1990-2002 [PDF]
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