Algebra is the study of operations (such as addition and multiplication) on objects (such as numbers, polynomials, and matrices). This course is an introduction to abstract algebra and its applications. Algebraic structure appears in problems in data analysis, modeling, and optimization. We will cover the abstract algebra theory required to extract insights in such contexts.
Part 1: Algebra for optimization. Algebraic structure in classical optimization problems behind data analysis tools such as regression and principal component analysis.
Part 2: Polynomial equations. The fundamental theorem of algebra and how to generalize it to multiple variables, via Gröbner bases and Bézout's theorem. Application to parameter estimation.
Part 3: Model invariants. Parametric and implicit descriptions and their uses in biological and physical models.
Part 4: Solving polynomial systems. Introduction to numerical algebraic geometry, and application to computer vision and chemical reaction networks.
Recommended preparation for the course is familiarity with linear algebra (at the level of Math 21b) and proofs (at the level of Math 22a, Math 101, or CS 20). Programming experience is helpful but not required.
Assessment. Class attendance (10%), homework (25%), quizzes (25%), final exam (40%).
See here for the course schedule.