# The Qualifying Exam Syllabus

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**

- Group theory: Sylow theorems, p-groups, solvable groups, free groups.
- Rings and modules: tensor products, determinants, Jordan canonical form, PID’s, UFD’s, polynomials rings.
- Field theory: splitting fields, separable and inseparable extensions.
- Galois theory: Fundamental theorems of Galois theory, finite fields, cyclotomic fields.
- Representations of Finite Groups: character theory, induced representations, structure of the group ring.
- Basics of Lie groups and Lie algebras: exponential map, nilpotent and semi-simple Lie algebras and Lie groups.

**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**

- Affine and projective varieties; regular functions and maps; cones and projections
- Projective space and Grassmannian
- Ideals of varieties; the Nullstellensatz
- Rational functions, rational maps and blowing up
- Dimension and degree of a variety; the Hilbert function and Hilbert polynomial
- Smooth and singular points of varieties; the Zariski tangent space; tangent cones; dual varieties
- Families of varieties (Chow varieties and Hilbert schemes)
- algebraic curves: genus; the genus formula for plane curves,
- the Riemann-Hurwitz formula. Riemann-Roch theorem.

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

**3) Complex Analysis**

- Holomorphic and meromorphic functions
- Conformal maps, linear fractional transformations, Schwarz’s lemma
- Complex integrals: Cauchy’s theorem, Cauchy integral formula, residues
- Harmonic functions: the mean value property; the reflection principle; Dirichlet’s problem
- Series and product developments: Laurent series, partial fractions expansions, and canonical products
- Special functions: the Gamma function, the zeta functions and elliptic functions
- basics of Riemann surfaces
- Riemann mapping theorem. Picard theorems.

**References**: Ahlfors: Complex Analysis (3rd edition)

**4) Algebraic Topology**

- Fundamental groups
- Covering spaces
- Higher homotopy groups.
- Fibrations and the long exact sequence of a fibration
- Singular homology and cohomology
- Relative homology
- CW complexes and the homology of CW complexes.
- Mayer-Vietoris
- Universal coefficient theorem
- Kunneth formula
- Poincare duality
- Lefschetz fixed point formula
- Hopf index theorem
- Cech cohomology and de Rham cohomology.
- Equivalence between singular, Cech and de Rham cohomology

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

**5) Differential Geometry**

- Basics of smooth manifolds: Inverse function theorem, implicit function theorem, submanifolds, integration on manifolds
- Basics of matrix Lie groups over R and C: The definitions of Gl(n), SU(n), SO(n), U(n), their manifold structures, Lie algebras, right and left invariant vector fields and differential forms, the exponential map.
- Definition of real and complex vector bundles, tangent and cotangent bundles, basic operations on bundles such as dual bundle, tensor products, exterior products, direct sums, pull-back bundles.
- Definition of differential forms, exterior product, exterior derivative, de Rham cohomology, behavior under pull-back.
- Metrics on vector bundles.
- Riemannian metrics, definition of a geodesic, existence and uniqueness of geodesics.
- Definition of a principal Lie group bundle for matrix groups.
- Associated vector bundles: Relation between principal bundles and vector bundles
- Definition of covariant derivative for a vector bundle and connection on a principal bundle. Relations between the two.
- Definition of curvature, flat connections, parallel transport.
- Definition of Levi-Civita connection and properties of the Riemann curvature tensor.

**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**

- Convergence theorems for integrals, Borel measure, Riesz representation theorem
- L
^{p}space, Duality of L^{p}space, Jensen inequality - Lebesgue differentiation theorem, Fubini theorem, Hilbert space
- Complex measures of bounded variation, Radon-Nikodym theorem.
- Fourier series, Fourier transform, convolution.
- Heat equation, Dirichlet problem, fundamental solutions
- Central limit theorem, law of large numbers, conditional probability and conditional expectation.
- Distributions, Sobolev embedding theorem.
- Maximum principle.

**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 L^{p} spaces. A good source of L^{p} 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