Homological Algebra
Homology refers to various invariants of groups, rings, modules,
and spaces by which many of their properties may be expressed.
Homological algebra encompasses useful techniques for computing
and interpreting homology in algebraic contexts, a fundamental
subject underlying many parts of modern algebra, topology, and
geometry. In this introductory course, students will gain
familiarity with the basic terminology and methods of homological
algebra so that they may apply them in future courses, in their
research, and to understanding talks and papers in which homology
appears. The course will be a useful precursor to a commutative
algebra course (to be proposed by Laura Matusevich in Spring 2020).
Specific topics to be covered include: chain complexes, resolutions,
Ext and Tor, categories and functors, homological dimension,
bicomplexes, Kunneth Theorems, and spectral sequences. Further topics
will depend on the interests of the students, and may include group
cohomology, cohomology of tensor categories, or Hochschild homology
and cohomology (I am writing a book on the latter). Throughout the
course, general theory will be complemented by computations of
examples.