Instructor: Professor Taylor Brysiewicz, Western University

Course Dates: September 10th - December 5th, 2024
Mid-Semester Break: October 14th - 18th, 2024
Lecture Times: Tuesdays, 11:30 AM - 12:30 PM (ET) | Thursdays, 10:30 AM - 12:30 PM (ET)
Office Hours: TBA (online & in person at MC 114, Western University)

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 40 students

Format: Hybrid synchronous delivery

• In Person (for UWO students)
• Online via Zoom (for non-UWO students)

Course Description

This course is a broad introduction to the world of numerical algebraic geometry.

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.

Tentative Schedule:

• Week 1: Polynomial systems and an introduction to numerical algebraic geometry
• Weeks 2/3: Univariate homotopy continuation
• Week 4: Multivariate homotopy continuation
• Weeks 5/6: Parameter homotopies
• Weeks 7/8: Mondoromy
• Weeks 9-11: Witness sets
• Week 12: Certification

Prerequisites:

• A course in linear algebra, a course in abstract algebra (or group theory), and basic programming skills.
• Background in algebraic geometry, complex analysis, numerical analysis, or topology will be helpful, but ultimately, is not required.

Auditing: Permission to audit the course, due to reasons which prohibit enrollment, will be considered. To be considered for auditing, please contact the professor directly.

Evaluation Method: The overall course grade will be calculated as listed below

• Assignments (8 out of 10): 60%
• Attendance: 20%
• Project: 20% = 5% (proposal) + 5% (draft 1) +10% (final version)

There are 5 due dates for assignments and each assignment has two parts, a “theory” part and an “applications/coding” part. This totals 10 assignments, worth 7.5% each. The lowest two scores are dropped. Assignment due dates are October 1, October 22, November 5, November 19, and December 5.

The project is to write a short paper (no more than 4 pages) summarizing a computational analysis of some polynomial system via techniques in numerical algebraic geometry. The student is free to choose the polynomial system, subject to approval. The project proposal (choosing a polynomial system) is due September 26; the first project draft is due November 14; the final version is due December 5.

The instructor for this course will NOT use Fields' learning management system, Moodle. For more information on course description, evaluation method (e.g., weight distribution), and course expectations, please refer to the course website: https://sites.google.com/view/taylorbrysiewicz/teaching/numericalalgebra....