Contingency mathematics in the areas of life and health insurance, annuities, and pensions from both the probabilistic and deterministic approaches. Survival distribution and life tables; life insurance; life annuities; net premiums; net premium reserves; multiple life functions; multiple decrement models; valuation theory for pension plans; the expense factor; and non-forfeiture benefits and dividends.

Basic principles of normed spaces. Normed and Banach spaces; Hilbert spaces; linear operators; dual spaces. Basic principles of functional analysis: Hahn-Banach theorem; open mapping theorem; uniform boundedness theorem, Krein-Milman theorem. Applications.

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.

Detailed examination of current topics in Mathematics.

Classification of second order partial equations; well posed problems; method of separation of variables and applications; wave equation: D?Alambert?s solution; Laplace equation: Poisson?s formula, maximum principle, boundary value and eigenvalue problems; heat equation: Cauchy problem, maximum principle.

Differential geometry of curves and surfaces in three-dimensional space; intrinsic geometry; geodesics; curvature; Gauss-Bonnett theorem.

Topological spaces, subspaces, continuous functions, base for a topology, separation axioms, compactness, locally compact spaces, connectedness, path connectedness, finite product spaces, set theory and Zorn?s lemma, infinite product spaces, quotient spaces, homotopic paths, the fundamental group,induced homomorphisms, covering spaces, applications of the index, homotopic maps, maps into the punctured plane, vector fields, the Jordan curve theorem.

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.

Finite-dimensional real and complex vector spaces, bases of a vector space, linear maps, dual spaces, quadratic forms, self-adjoint and unitary transformations, eigenvalue problem, canonical form of a linear transformation, tensors, and applications.

Fixed point iteration and Newton’s method for nonlinear equations, direct solution of linear systems and the least squares problem, symmetric positive definite and banded matrices, systems of nonlinear equations, the QR algorithm for the symmetric eigenvalue problem, Lagrange and Hermite polynomial interpolation, polynomial approximation in the infinity norm and the Chebyshev polynomials, approximation in the 2 norm and the orthogonal polynomials, numerical differentiation, Newton-Cotes and Gaussian quadratures for numerical integration.