Fall 2024 Term

The purpose of the AARMS Advanced Course program is to give graduate students and upper year undergraduates from AARMS Member Universities an opportunity to take courses not offered at their home institution free of charge. In Fall 2024, the courses listed below will be offered in an exclusively online format.

Students may be able to claim credit for AARMS Advanced Courses from their home institution using mechanisms similar to the AARMS Summer School. Students hoping to receive academic credit for courses in this program are strongly encouraged to consult with their home institution about this process before the start of lectures.

Both students and instructors should consult the frequently asked questions.  Please send any addition questions about this program to Andrew Irwin.

Important dates

  • August 1, 2024: Application opens for AARMS Advanced courses (applications will be accepted on a rolling basis)
  • September 7, 2024: Application closes for AARMS Advanced courses (individual courses may close earlier without notice)
  • Early September, 2024: Classes begin

Registration

To register for an AARMS Advanced Course in Fall 2024 please contact the instructor directly.

Dr. Nathan Grieve
Phone: 902-585-1456
E-Mail: nathan.grieve@acadiau.ca
Webpage

A working introduction to Homological Algebra and its applications.

Nathan Grieve (Acadia University)

Homological algebra is a central tool for studying many flavours of Topology, Geometry, Number Theory, Algebra and Combinatorics. This course will develop the most basic foundations of homological algebra and will focus on several specific Applications. Among other features, these applications will serve to provide students with a survey and introduction to key principles in Algebraic and Combinatorial Topology, Algebraic Geometry and Commutative Algebra. This will include an introduction to the language of sheaves, Abelian categories, derived functors, free resolutions and syzygies. Another flavour of examples may include applications that are within the area of Topological Data Analysis and/or an introduction to the theory of Derived Categories. The intent is to make the course accessible to students of all backgrounds and the exact list of topics covered will reflect students’ backgrounds and interests. The intended audience for the course is graduate students of all levels. For example, these students may be working in some (possibly only tangentially) geometric area of Algebra, Number Theory, Combinatorics, Statistics and/or Computer, Network and/or Data Science. As a very modest pre-requisite, some familiarity with the most basic theory of modules over commutative rings may be helpful. The course may also be of interest to advanced undergraduate students. Selected textbooks (listed in no particular order) that will be of interest for the course include:

  • Weibel, C. A. An introduction to homological algebra;
  • Hartshorne, R. Algebraic Geometry;
  • Huybrechts, D. Fourier-Mukai transforms in algebraic geometry;
  • Miller, E. and Sturmfels, B. Combinatorial commutative algebra;
  • Carlsson, G. and Vejdemo-Johansson, M. Topological Data Analysis with Applications;

A main goal of the course will be to provide a working knowledge of some of the basic foundational theory together with a practical hands-on approach that is grounded in examples. This may include calculations with existing computer algebra packages and/or developing new scripts and/or advanced computational tools. Further, as needed, relevant background will be covered. Finally, students will be asked to complete an essay on topics related to the course as part of their final grade. These essays will be presented in a conference/research seminar type format.

Mark breakdown:

  • Assignments: 20%;
  • Final Exam: 20%;
  • Student presentations: 30%; and
  • Course Essay: 30%.