Guidelines on Preparing for the Mathematics Preliminary Exams in Analysis

General background required:

1. Undergraduate Real Analysis, as covered in Math 25 and Math 125A,B. Prior to Fall 2006, these classes were Math 127 A, B, C.
2. Main concepts of Complex Analysis as covered in undergraduate courses (Math 185A). Minimum requirement: operations in the complex number system
3. Main concepts of topology as covered in undergraduate courses (Math 147). Topological spaces, bases, product topology, limit points, continuity, metric spaces, complete, connected, compact spaces
4. Graduate Analysis, as covered in Math 201 A,B,C

Specific topics:

1. Continuous functions
   Convergence of functions, spaces of continuous functions, approximation by polynomials, Arzela-Ascoli theorem
2. The Contraction Mapping Theorem in complete metric spaces.
3. Banach Spaces
   Bounded linear operators. Different notions of their convergence. Compact operators. Dual spaces. Finite dimensional Banach spaces.
4. Hilbert Spaces
   Orthogonality. Orthonormal bases. Parseval's identity.
5. Fourier Series
   Convolution. Young's inequality. Fourier series of differentiable functions. Sobolev embedding theorem.
6. Bounded Linear Operators on a Hilbert Space
   Orthogonal projections. The dual of a Hilbert space (Riesz representation). The adjoint of an operator. Self-adjoint and unitary operators. Weak convergence.
7. The spectral theory for compact, self-adjoint operators
   Spectrum. Compact operators. The spectral theorem. Hilbert-Schmidt operators. Functions of operators.
8. Linear Differential Operators and Green's Functions
   Unbounded operators. The adjoint of a differential operator. Green's functions. Weak derivatives. The Sturm-Liouville eigenvalue problem.
9. Distributions and the Fourier Transform
   The Schwartz space. Tempered distributions, operations on them and their convergence. The Fourier transform of test functions and of distributions. The Fourier transform on L1 and L2. The Poisson summation formula.

Guidelines on Preparing for the Mathematics Preliminary Exams in Algebra

The exam will include the material of first chapters of Vinberg's book (Chapters 1, 2, 4, 5, 6, 8), 9.1, 9.2, 9.3, first part of 9.5 (up to the definition of splitting field), 10.1, 10.2, 10.3, the beginning of 10.4, 11.1, 11.2, 11.4, Chapter 12. (Another exposition of the material in Chapters 11 and 12 was given in handouts.)

Typical problems:

• Find kernel and image of homomorphism
• Find all homomorphisms of G to G' (or all automorphisms of G or all endomorphisms of G).
• Find a system of generators of G. (In the above problems G can be a group or a ring.)
• Check if a subset of a group (of a ring) is a subgroup, a normal subgroup (a subring, an ideal)
• Describe the quotient of given group with respect to normal subgroup (or of given ring with respect to ideal)
• Find the normal form of a quadratic form.
• Is given quadratic (or Hermitian) form positive definite?
• Find eigenvectors and invariant subspaces of linear operator.
• A linear operator obeys a polynomial equation. Find possible Jordan normal forms of this operator.
• Find possible Jordan normal forms of an operator if the multiplicities of eigenvalues and numbers of linearly independent eigenvectors are known.
• Find exponent of a linear operator.
• An abelian group is specified by means of generators and relations. Find a decomposition of the group into a direct sum of cyclic groups.
• Find torsion submodule of a module.
• Find orbits and stabilizers of a group action.
• Is given action of a group transitive?
• Find the commutator subgroup G'of a group G and abelianization G/G' of G.
• Is G a direct product of subgroups A and B?
• Is G a semidirect product of subgroups A and B?
• Describe all semidirect products of N and H.
• Is a group G solvable? simple?
• Is linear representation reducible? completely reducible? equivalent to orthogonal (or unitary) representation?
• Find character of linear representation.
• Find missing entries in the table of character of finite group.
• Find character of dual representation, of tensor product of representations.
• Decompose a representation into direct sum of irreducible representations.
• Find all endomorphisms (or all automorphisms) of a representation decomposed into a sum of irreps.
• Describe all invariant subspaces of a representation decomposed into a sum of irreps.

Definitions:
Give definitions of group, subgroup, normal subgroup, coset, homomorphism, isomorphism, automorphism, endomorphism, kernel, image, quotient group, analogous notions for rings and modules, system of generators, cyclic groups and modules, Jordan normal form, tensor product, tensor algebra, symmetric tensors and symmetric algebra, antisymmetric tensors and Grassmann algebra, raising of a subscript and lowering of a superscript, group action, orbit, stabilizer, transitive action, direct and semidirect product of groups and subgroups. simple group, solvable group, splitting field of a polynomial, field expension, degree of field extension, linear representation, invariant subspace, irreducible and completely reducible representations, character, orthogonal group and orthogonal representation, unitary group and unitary representation, special linear group. Lie algebra, representation of Lie algebra, topological group, Lie group, tangent space, Lie algebra of Lie group, Lie algebras gl_n, sl_n, u_n, so_n.