Computational methods in algebraic geometry seek to understand the algebraic varieties (or schemes) associated to systems of polynomial equations. This can range from finding the solutions to a system of polynomials (when it consists only of points) to computing geometric and topological invariants of curves, surfaces, or higher dimensional algebraic varieties. This project would explore some current computational methods for understanding algebraic varieties, and their strengths and limitations. Depending on the interests of the student this could include Gröbner basis algorithms, polynomial homotopy continuation, geometric resolutions, or polyhedral combinatorics. Methods which obtain invariants such as the topological Euler characteristic or the multiplicity of a subvariety by performing several appropriate Gröbner basis or homotopy continuation computations could also be studied. As part of this project the student would have the opportunity to become familiar with some of these methods and apply them to problems in pure or applied mathematics; there would also be the opportunity to investigate how methods could be improved to better solve particular problems of interest. In addition to applications in algebraic geometry some other possible areas of application include algebraic statistics, optimization, systems biology, and robotics (depending on the interests of the student).