# Qualifying exams

#### Qualifying exams

PhD students must pass  qualifying exams in any two out of the four subjects Algebra, Analysis, Applied Math, Topology.  Exams are typically offered twice per year in August and in January (right before the beginning of the term).  Exams are taken and graded anonymously. Students may take exams multiple times without penalty but are expected to pass them within their first 3 years to be on track in the program.

Specific information about each of the four qualifying exams is below:

##### Analysis exam

The analysis qualifying exam covers topics in measure theory and real analysis.

###### Material
1. Measure theory on Euclidean space: the Borel σ-algebra, construction of Lebesgue measure on finite-dimensional Euclidean spaces.
2. Functions: continuous functions, uniformly continuous functions, absolutely continuous functions, functions of bounded variation and rectifiable curves, Borel functions, measurable functions, simple functions, the relations between these classes, Lusin’s theorem.
3. Integration: the Lebesgue integral, Fatou’s lemma, the monotone and dominated convergence theorems, applications to moving limits through integrals, Fubini’s theorem, definition and completeness of Lp spaces, the Lebesgue differentiation theorem.
4. Convergence of functions: pointwise convergence, the supremum norm and uniform convergence, convergence in measure, convergence in Lp spaces, the relations between these notions, Egorov’s theorem.
5. Inequalities: Chebyshev, Cauchy–Schwarz, Jensen, Minkowski (sum and integral forms), Hölder.
6. Density: the Weierstrass theorem and density of polynomials in appropriate Lp spaces, convolution with approximate identities and density of smooth functions in appropriate Lp spaces.
7. General measure theory: σ-algebras, outer measures, counting measure, product measures, Lp spaces of a general measure space, absolute continuity of measures and the Radon–Nikodym theorem.
###### Textbooks

The following textbooks are recommended:

• Folland, Real analysis: Modern techniques and their applications
• Royden, Real analysis
• Rudin, Real and complex analysis
• Stein and Shakarchi, Real analysis: Measure theory, integration, and Hilbert spaces
##### Topology exam

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