Syllabus for the Algebra Qualifying Examination
Groups. Group actions, Sylow theorems, $S_n$ and $A_n$, simple groups. Composition series, Jordan-Holder theorem; p-groups, solvable groups. Group extensions, semi-direct products, free groups, amalgamated products, group presentations. Symmetries of Platonic solids.
Sources: S. Lang’s book on Algebra, M. Artin’s book on Algebra.
Rings. Homomorphisms, ideals, prime and maximal ideals, radicals, Chinese Remainder Theorey. Examples: Rings of polynomials and power series (several variables). p-adic numbers. Fundamental theorem on symmetric functions. Quaternions. Rings of differential operators with polynomial coefficients. Principal ideal domains: examples, factorization theory. UFD’s: Gauss lemma, Eisenstein criterion.
Modules. Abelian groups, vector spaces, homomorphisms,kernels, images, submodules, quotients, direct sums and products, duals, cyclic modules, free modules, exact sequences. Finitely generated modules over PID’s: structure theorem.
Noetherian rings, Hilbert basis theorem. Localization: local rings, localization of rings and modules. Exact functors and localization. Nakayama’s lemma.
Categories: functors, equivalences of categories, characterizing object by universal properties, concepts of inverse/direct lemma, examples.
Tensor Products: basis (functoriality and exactness), extension of scalars, algebras, multilinear forms, exterior and symmetric powers, special case vector spaces. Determinant of a finite dimensional vector space and determinant of a linear operator.
Projective and injective modular: examples, resolutions, Hom, Tor, and Ext.
Linear Algebra. Rational and Jordan canonical form, minimal and characteristic polynomials, traces and determinants, eigenspaces and generalized eigenspaces, diagonalization, commuting matrices. Bilinear forms: orthogonal, symplectic, unitary groups. Spectral theorem over R,C.
A Source: I. M. Gelfand, Lectures on Linear Algebras - You should know everything discussed in the book.
Group Representations. Basic definitions, equivalence with the modules over group rings. Irreducible representations: finite abelian case, Schur’s lemma, Maschke’s theorem, characters, orthogonality relations, tensor/dual. Examples: representations of $S_4$, dihedral groups, groups of symmetries of cube and tetrahedron.
A Source: J. P. Serre, Linear Representations of Finite Groups, Chapters I and II.
Field Theory. Algebraic extensions: degree, minimal polynomials, adjoining a root. Existence and uniqueness of splitting fields, algebraic closure. Finite fields: classification, Frobenius automorphism, cyclicity of finite multiplicative subgroup of a field. Normal extensions, separable closures, perfect fields, primitive element theorem. Inseparable extensions. Galois theory: field embeddings and Galois groups, examples, fundamental theorem in finite case, cyclotomic extensions, norms and traces. Kummer theory, solvability by radicals via solvable groups. Examples. Infinite Galois theory: analogy with the fundamental group of a topological space, Krull topology, fundamental theorem.
A Source: S. Lang’s, Algebra.
Commutative Algebra. Hilbert’s Nullstellensatz. Integral/finite ring maps, characterizations, integrals closure, examples.
Noether normalization, transcendence degree, dimension for finitely generated algebras over a field. Artinian rings. Completion with examples - power series rings $Z_p$. Hensel’s lemma, Artin-Rees, flatness, Krull intersection theorem, $\hat Z = \prod_p Z_p$ as Galois group of a finite field.
A Source: Atiyah-Macdonald, Commutative Algebra.
Syllabus for Qualifying Examination in Topology
Cell complexes, simplicial complexes, manifolds.
Homotopy, homotopy equivalence, retracts, homotopy extension property.
Fundamental group: Seifert-Van Kampen, covering spaces and groups, lifting.
Fundamental groups and topological classification of 2d manifolds.
Eilenberg-MacLane spaces, knot complements.
Homology: simplicial, singular, and cellular homology with coefficients, relative homology, long exact sequence, Mayer-Vietoris sequence, excision, Euler characteristic, axioms for homology.
Applications - Lefschetz fixed point theorem, Brouwer fixed point theorem, Borsuk-Ulam theorem.
Cohomology: Simplicial, singular, and cellular cohomology with coefficients, universal coefficient theorem, ring structure, Kunneth formulae. Cohomology rings of surfaces, real and complex projective spaces, and complex Grassmannians.
Vector bundles and simplest operations with them. Tangent, cotangent bundles of a manifold. Differential forms, Stokes theorem, DeRham cohomology and DeRham theorem.
Orientations, degrees of maps. Poincare duality: simplicial and DeRham approach proofs.