Mathematics

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.