Mathematics

Linear Algebra is one of the most widely used topics in the mathematical sciences. At lower levels students are taught standard techniques for basic linear algebra tasks including the solution of linear systems, finding eigenvalues/eigenvectors, and orthogonalization of bases. However, these techniques are usually computationally too intensive to be used for the large matrices encountered in practical applications. This course introduces students to these practical issues, and presents, analyzes, and applies algorithms for these tasks which are reliable and computationally efficient. The course includes significant lab work using an advanced programming language. The course studies three main topics: the solution of linear systems of equations, the solution of least squares problems and finding the eigenvectors and/or eigenvalues of a matrix.

This course addresses exploratory data analysis and graphs such as histograms, stem plots, measures of center and spread of a distribution, normal distribution, scatter plots, least squares regression (correlation), producing data (design of experiments, sampling design), probability (probability rules, random variables, probability distributions), and statistical inference (confidence intervals, tests of significance, nonparametric methods, categorical or count data). 

This course covers ordinary differential equations and partial differential equations. Topics include: first-order differential equations; second-order linear differential equations; linear systems of differential equations; nonlinear systems and stability; method of separation of variables; Sturm-Liouville Problems; Inhomogeneous Problems.

Pre-requisites: Calculus I, Calculus II, and Linear Algebra.

This course provides an introduction to programming and numerical methods. Topics covered include computer arithmetic and computational errors, solution of nonlinear equations, systems of linear equations, numerical integration and differentiation, interpolation and approximation, initial value problems for ordinary differential equations. This course will cover basic concepts of numerical optimization and applications such as for instance numerical optimization in the rapidly evolving domain of machine learning.

The course covers elementary programming concepts (arithmetic expressions, for-loops, logical expressions, if-statements, functions and classes) that are closely connected to mathematical/technical problems and examples, as well as mathematical manipulations (setting up matrices, solving linear problems, solving differential equations, finding roots). 

This course will study the design and analysis of exact and approximation algorithms for various optimization problems using advanced techniques such as combinatorial methods, probabilistic methods, linear programming, semidefinite programming, and spectral methods. The course also covers spectral algorithms and related convex programs, SDP duality, multiplicative weight update, graph spectrum, eigenvalue interlacing, Cheeger–Alon–Milman inequality, random walks, local graph partitioning, expanders, Laplacian solver, effective resistance, sparsification, matrix scaling, abstract simplicial complex, and random spanning trees.