Descriptive statistics; Probability; Random variables; Special distributions; Estimation; Hypothesis testing; Normal distribution; Two-Sample Inference; Regression.

Review of vector calculus; Fourier series and Fourier transform; Calculus of functions of a complex variable.

Decision theory; estimation; confidence intervals; hypotheses testing; large-sample theory; efficiency of alternative statistical procedures.

Detailed examination of current topics in Mathematics.

Complex numbers and functions; exponential and trigonometric functions; infinite series and products; limits, continuity and derivatives of complex functions; Cauchy’s theorem; Taylor and Laurent series; conformal mapping.

Metric spaces and their topology; continuity, compactness and connectedness in a metric space; completion of a metric space; differentiation and Riemann integration; sequences and series of functions; uniform convergence; Ascoli-Arzela theorem; Stone-Weierstrass theorem; Banach fixed-point theorem and its 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.

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.

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.

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.