Algebraic Structures

This course provides an introduction to modern algebraic structures, particularly those developed since the 1960s. These structures are increasingly studied and applied in algebra, topology, and computational mathematics.

The course covers advanced and contemporary algebraic frameworks. Core topics include:

• Basic Category Theory: Categories, functors, natural transformations, adjunctions, and free constructions.
• Fundamentals of Homological Algebra: Chain complexes, resolutions, and homology.
• Lie Algebras and Their Universal Enveloping Algebras
• Hopf Algebras, with a focus on combinatorial Hopf algebras: symmetric algebras, tensor algebras, enveloping algebras, the Connes–Kreimer Hopf algebra, and other foundational Hopf structures.

Further topics may include:

• Advanced aspects of category theory
• Extended homological algebra: derived functors, homotopy, Ext and Tor functors
• Pre-Lie algebras arising from vector fields on ℝⁿ, free pre-Lie algebras, and their enveloping algebras
• Exact solutions to differential equations via Taylor expansion interpreted as a Butcher series for vector fields on ℝⁿ

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student will:

• Have insight into key developments in modern algebraic structures over the past 50–60 years
• Understand the fundamentals of category theory as a universal framework for structures in mathematics, computer science, and related fields
• Grasp the basics of homological algebra, a fundamental tool in algebra and topology
• Be familiar with advanced, rich, and compact algebraic structures such as universal enveloping algebras of Lie algebras and Hopf algebras for combinatorial structures.
• Understand definitions and essential properties, and how algebraic structures and methods provide universal tools for both theoretical and computational approaches in mathematics and, to some extent, computer science

Skills:

The student will:

• Be able to apply algebraic tools essential for solving problems and developing theory in algebra, topology, differential geometry, discrete mathematics, computational mathematics, and theoretical computer science
• Have solid experience and training in reasoning with abstract and general algebraic structures

General Competence:

The student will:

• Have insight into important algebraic theories and structures used in mathematical research
• Understand how various concrete problems and structures can be described using highly general mathematical tools
• Appreciate the value of abstract theoretical development as a universal language across many mathematical contexts