Advanced courses Fall 2024

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%.

Yorck Sommerhäuser (Memorial University)

Hopf algebras are algebras for which it is possible to form the tensor product of two representations. A typical example is the group algebra of a group. Hopf algebras play a role in remarkably many areas of mathematics, such as algebraic topology, Lie theory, or category theory, and also in physics, for example in the theory of exactly solvable models in statistical mechanics, conformal field theory, and the theory of renormalization.

The course will provide an introduction to the basics of Hopf algebra theory. Emphasis will be on results that generalize classical theorems from group theory, like Lagrange’s theorem, Cauchy’s theorem, and Maschke’s theorem. A prerequisite is a knowledge of multilinear algebra as contained in the first chapter of Werner Greub’s book on the topic. No prior knowledge of advanced algebra or group theory is required. During the class meetings, lecture notes will be distributed that cover the recently treated material.