Finite geometries, transformations, Euclidean geometry, constructions, inverse geometry, projective geometry, non-Euclidean geometry.

This is a second course in the topics of pure mathematics, particularly those most commonly used in the study of Computing Science and related applications. It includes proof techniques, models of computation, formal languages, analysis of algorithms, trees and advanced general graph theory with applications, finite state and automata theory, encryption, and an elementary introduction to mathematical structures such as groups, rings, and fields.

Foundational mathematical concepts underpinning theoretical frameworks in data science that depend on linear algebra and multivariable calculus, with applications chosen from machine learning, statistical inference, and data assimilation. Possible topics include matrix decompositions, gradient and multivariate chain rule, Lagrange multipliers and constrained optimization, maximum likelihood, and Bayesian estimation.

This course covers numerical techniques for solving problems in applied mathematics, including error analysis, roots of equations, interpolation, numerical differentiation and integration, ordinary differential equations, matrix methods and selected topics from among: eigenvalues, approximation theory, non-linear systems, boundary-value problems, numerical solution of partial differential equations.

Sequences and induction; convergence of sequences and series; limits, continuity, and differentiability; Riemann integrals; sequences of functions and an introduction to topology.

First-order differential equations, linear differential equations, Laplace transforms, systems of differential equations, non-linear systems, series solutions, introduction to partial differential equations. Special emphasis is placed on applications to physics and engineering.

Functions of a complex variable, differentiation, analytic and elementary functions, Cauchy's theorem and contour integration, Taylor and Laurent series, residues and poles, and conformal mapping. Emphasis is placed on physical applications.

Introduction to principles of real analysis. Topics include rational numbers, completeness, sequences, limits, continuity, and implications to Calculus.

First-order differential equations, linear differential equations, Laplace transforms, systems of differential equations, non-linear systems, series solutions, applications in the physical, biological, social, and engineering sciences.

The mathematics of inferential statistics. Topics include probability distributions and densities, expectation, moment-generating functions, functions of random variables, limiting distributions, and the theory behind statistical methods such as estimation, hypothesis testing, regression and correlation.