Special Topics course on Algebraic Computation and its Applications offered by Anton Leykin in Spring 2016.
Parts of Cox, Little, O'Shea, "Ideals, Varieties, and Algorithms" (electronic copy is available for free from the library), instructor's notes.
Prerequisite: abstract vector spaces or foundations of mathematical proof. (The prerequisites are intended only as an indication of mathematical maturity; the course will focus on ideas, _not_ the details of the proofs)
This course is an introduction to techniques of algebraic computation that found applications in a variety of areas both in and outside mathematics.
It delves into nonlinear algebra, whose basic problem is solving systems of polynomial equations in one and many variables.
Note that having "algebraic" in the name of the course does not imply that the techniques employed for solving are purely algebraic; most cutting-edge approaches are hybrid in nature and use both symbolic and numerical methods.
Several applications of algebraic computation to areas such as robotics, kinematics, computer vision, optimization, and chemical reaction networks will be highlighted in lectures and may become starting points for student projects.
* Solving (systems of) polynomial equations
* Polynomial homotopy continuation
* Ideals in polynomial rings
* Elimination theory and Groebner bases
* Polynomial models arising in applications