This course is an introduction to homological algebra. Rather than
regarding the subject as an abstract machine for proving
non‐constructive existence theorems, we present homological algebra as a
vast generalization of linear algebra: matrices are replaced by
appropriate sequences of linear maps. This perspective emphasizes
“thinking in terms of complexes” and developing effective computational
tools. We explore the fundamental structures and essential
constructions within homological algebra. A range of examples, many
explicitly illustrated via the
Macaulay2 software
system, will also enhance the learning experience.
Gregory G. Smith and Michael E. Stillman,
Complexes: an open source
package for computational homological algebra, distributed with the
Macaulay2 software system, 2022.
Assessment
The course grades will be computed as follows:
50% Homework
50% Project
Homework
Biweekly problem sets, each consisting of approximately 4 questions, are
available in the Portable Document Format (PDF) on the course website.
Solutions will be collected using Crowdmark software. The best 5 of 6
problem sets will determine your homework grade.
Project
Each student will independently learn a new theorem (interpreted broadly
to include constructions, important examples, or other homological
results) and communicate it in written form. Whenever possible, a
student will select their topic in consultation with their research
supervisor. The final short document must be typed and be available in
PDF.
Academic Integrity
Students are responsible for familiarizing themselves with all of the
regulations concerning academic integrity and for ensuring that their
assignments and their behaviour conform to the principles of academic
integrity. Students are encouraged to discuss problems, but should write up
the solutions individually. Students must explicitly acknowledge any
assistance including books, software, technology, websites, students,
friends, professors, references, etc.
Accommodations
The instructor is committed to achieving full accessibility for people
with disabilities. Part of this commitment includes arranging academic
accommodations for students with disabilities to ensure they have an
equitable opportunity to participate in all of their academic
activities. If you think you may need academic accommodations, then you
are strongly encouraged to contact both the instructor as soon as
possible.
Licensing
Materials generated by the instructor of this course may not be used
for commercial advantage or monetary compensation. Some material is
clearly copyrighted and may not be reproduced or retransmitted in any
form without express written consent. Other material, licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International
License, may be remix, adapt, or build upon it, as long as appropriate
credit is given and the new creation is distributed under the identical
terms.
