MTG 7397, Advanced Topics in Topology 2

Course Description

In this course we will introduce some powerful mathematical tools whose origins lie in algebraic topology but which have been useful in many other areas and have become objects of study themselves. Specifically, we will learn about Category Theory, Homological Algebra, and Spectral Sequences. Each of these topics provides a formal framework for organizing mathematical structures and computations, and is an important tool in modern mathematics.

Textbooks

  • Category Theory in Context, by Emily Riehl. Available by download from Riehl’s web page and there is an inexpensive paperback edition.
  • Homological Methods in Commutative Algebra, by Andrea Ferretti. Available from the UF Library via ProQuest.

Prerequisites

This course will be mostly self-contained. Students should be comfortable with abstract algebra.

Course topics

  • Category Theory: categories, functors, natural transformations, universal properties, limits, colimits, adjoints, Kan extensions, localization, symmetric monoidal categories
  • Homological Algebra: abelian categories, chain complexes, additive functors, projectives, injectives, derived functors, derived categories
  • Spectral Sequences: filtrations, bicomplexes, convergence, Grothendieck spectral sequence

Time

MWF Period 6, 12:50–1:40pm

Syllabus

Here is the pdf.

Please contact me if you have any questions and/or requests!