The topology qualifying exam covers topics in algebraic topology.

Material

1. Some point set topological concepts: basic definitions, compactness, separation axiom, connectedness/path-connectedness, retractions, contractibility, quotient topologies.
2. Homotopy theory: CW complexes, homotopic maps, properties of homotopy in CW complexes, homotopy equivalence, homotopy extension.
3. Some group theory: free groups, free products, universal properties, presentations of groups.
4. Brief overview of basic category-theoretic definitions: categories, functors, natural transformations, examples.
5. The fundamental group: a brief discussion of homotopy groups and their functoriality, long exact sequences of homotopy groups, definition of the fundamental groupoid and the fundamental group, calculation of the fundamental group of a circle, winding numbers, the van Kampen theorem together with examples of fundamental group calculations, K(π, 1) spaces and their properties.
6. Covering spaces: basic definitions, lifting properties, deck group actions, the Galois correspondence between covers and subgroups of the fundamental group.
7. Brief overview of (co)homological algebra: (co)chain complexes, (co)chain maps, exact sequences, (co)homology, long exact sequences induced by short exact sequences.
8. Homology: cellular and singular homology and their equivalence, reduced homology, relative homology, excision, Mayer–Vietoris sequences, the Künneth formula, examples, first homology and the fundamental group, homology with coefficients, definition of cohomology and calculation of examples.
9. Applications of homology: orientability, degrees of mappings, Lefschetz fixed point theorem, Brouwer fixed point theorem, invariance of domain Borsuk–Ulam theorem.
10. Structures on cohomology rings: universal coefficients theorem for homology/cohomology, the cup and cap products, calculating cohomology rings, intersection numbers, duality theorems.

Textbooks

The following textbooks are recommended:

• Bredon, Topology and geometry
• Hatcher, Algebraic topology
• Spanier, Algebraic topology

Sample Exams

• Sample Exam 1
• Sample Exam 2
• Fall 2016 Exam
• Fall 2017 Exam

 

The following syllabus for the Analysis Qualifying Exam is based on the following references:

• Rudin’s Real and Complex Analysis 3rd edition
• Folland’s Real Analysis: Modern Techniques and Applications 2nd edition.

Material

Abstract Integration and positive Borel measures

• Rudin (Chapters 1 and 2), Folland (Chapter 1 and Sections 2.1, 2.2, 2.3, 2.5 and 2.6).

σ-algebras and Borel sets, measurable functions, simple functions, elementary properties of measures, integration of positive functions (including Lebesgue’s Monotone Convergence Theorem and Fatou’s Lemma), integration of complex functions (including Lebesgue’s Dominated Convergence Theorem), sets of measure zero and complete measures, Borel measures and their regularity properties, Lebesgue measure on finite-dimensional Euclidean spaces and linear transformations, continuity properties of measurable functions (including Lusin’s Theorem).  

L^p spaces

• Rudin (Chapter 3), Folland (Section 6.1).

  Convex functions and Jensen’s inequality, Hölder’s inequality and Minkowski’s inequality, completeness of L^p spaces, approximation by continuous function.  

Modes of Convergence

• Folland (Section 2.4).

Almost everywhere convergence, convergence in measure, convergence in L^p spaces, uniform convergence and Egorov’s theorem, the relations between modes of convergence.  

Differentiation

• Rudin (Chapter 7), Folland (Sections 3.4 and 3.5).

  Maximal inequalities, Lebesgue points and Lebesgue’s differentation theorem, Lebesgue’s density theorem, absolutely continuous functions and the fundamental theorem of calculus.

Integration on Product Spaces

• Rudin (Chapter 8), Folland (Section 2.5).

  Measurability on cartesian products, product measures, Tonelli’s theorem and Fubini’s theorem, completeness of product measures.  

Convolutions

• Folland (Sections 8.1 and 8.2).

  Continuity of translation in the L^p norm, basic properties of convolution, derivatives of convolutions and approximate identities, density of smooth functions in L^p spaces.

Complex Measures

• Rudin (Chapter 6), Folland (Sections 3.1, 3.2, 3.3 and 6.2).

  Total variation, absolute continuity of measures, the theorem of Lebesgue-Radon-Nikodym, bounded linear functionals on L^p spaces.  

Sample Exams

• Sample Exam 1
• Sample Exam 2
• Sample Exam 3
• Sample Exam 4
• Sample Exam 5
• Archived exams going back to 2016