Syllabus for Pure Mathematics Comprehensive, Part 1
Matrices and systems of linear equations. Vector spaces
fields, subspaces, linear independence, basis, dimension. Determinants.
transformations, associated matrices, change of basis, dimension
vectorspaces. Eigenvalues, eigenspaces, diagonalization, Jordan
canonical form. Inner
product spaces, bilinear, quadratic and hermitian forms. Adjoint,
orthogonal and unitary operators. Diagonalization in Euclidean and
spaces. The spectral theorem.
References: Schaum's Outlines: Linear Algebra: Chapters 1--13.
Basic Real Analysis:
Real numbers: Infimum and supremum, limits of sequences, monotone
sequences, Cauchy sequences.
Continuity: limits of functions, continuous functions, the intermediate
value theorem, maxima and minima, uniform continuity, monotone
functions, inverse functions.
Differentiation: the derivative, mean value theorem, l’Hospital’s rule,
expansion with remainder.
Integration: Riemann integrals, the fundamental theorem of calculus,
Sequences of functions: pointwise and uniform convergence, continuity
and convergence, interchange of limit with derivatives and integrals,
Arzela-Ascoli theorem, Weierstrass and
Stone-Weierstrass approximation theorems. Differentiation of integrals
Infinite series: series of numbers and functions, absolute convergence,
Elementary functions: rigorous introduction of the exponential,
logarithmic, trigonometric and inverse trigonometric functions.
Functions of several variables: the derivative as a linear
theorem, the inverse and implicit function theorems.
Vector calculus: multiple integrals, path and surface integrals, change
of variables theorem for integrals, calculation of areas, volumes and
arc-lengths, the integral theorems of vector analysis (Green’s,
Stokes’, and Gauss’ theorems).
Metric spaces: basic topology, compactness, connectedness,
References: “Vector Calculus”, Marsden and Tromba
“Principles of Mathematical Analysis”, Walter Rudin
“Elementary Classical Analysis”, J. Marsden and M. Hoffman
Analytic functions, Cauchy-Riemann equations, entire
exponential, trigonometric, and logarithmic functions, Euler’s formula.
Line integrals, Cauchy’s theorem, Cauchy’s integral formula, power
representation and consequences, uniqueness theorem, mean value
maximum modulus principle, open mapping theorem.
Morera’s theorem, Liouville’s theorem and applications, meromorphic
Laurent expansions, residue theorem and applications, fractional linear
References: “Function Theory of One Complex Variable”, R. Greene and S.
Basic concepts, groups acting on sets, permutation groups, subgroups
(intersections, composita), order and index, quotient groups,
homomorphisms, kernel and image, center, normalizer, centralizer,
direct products. Sylow theorems.
Fundamental theorem on finite abelian groups. Simple groups, p-groups,
nilpotent groups, solvable groups.
Reference: Dummit--Foote: Chapters 1--6
Basic concepts, ideals (prime and maximal),
homomorphisms, Chinese remainder theorem, modular arithmetic. Integral
and fields of fractions. Unique factorization, principal ideal domains,
Euclidean domains. Polynomial rings, Gauss' Lemma, irreducibility,
criterion. Basic field theory: finite extensions, degree, and finite
References: Dummit--Foote: Chapters 7--9, Chapter 13.1, 13.2