Fields Academy Shared Graduate Course: Numerical Algebraic Geometry
Description
Instructor: Professor Taylor Brysiewicz, Western University
Course Description
Numerical algebraic geometry is a computational paradigm for studying algebraic varieties, i.e., solutions to polynomial systems. Algorithms in numerical algebraic geometry work by manipulating numerical approximations of points on varieties. In this sense, numerical algebraic geometry is the geometry of computational algebraic geometry - contrary to algebraic approaches (e.g. Gröbner bases, regular chains, etc) which perform exact algebraic manipulations on polynomials.
Relaxing the need for exact computation offers enormous computational benefits - numerically, systems with millions of solutions can be efficiently and reliably solved on a personal laptop computer. Moreover, the algorithms for doing so are trivially parallelizable and can be certified, turning them into rigorous mathematical statements for which the computation is the proof.
This course is a broad introduction to the world of numerical algebraic geometry. Background in numerical analysis, algebraic geometry, or computation is helpful but ultimately not necessary.
Topics: The main topics in numerical algebraic geometry which will be covered in this course are
- Introduction to the main objects: Polynomial systems and varieties
- Homotopy Continuation: Newton, Euler, and numerical analysis
- Monodromy: a new way to solve
- Parametrized polynomial systems: zero-dimensional families of systems
- Witness sets: handling positive-dimensional varieties
- Verification: increasing the confidence in numerical computations
- Certification: extracting proofs from numerical computations
- Computation: Numerical algebraic geometry software and computational aspects of core algorithms
Additional Topics Time Permitting: Optimization via Hill-Climbing, Decomposable Branched Covers, Sparse Trace Tests, Connections to Tropical Geometry
(More logistical details are coming soon.)