Additional Information

This course will be a survey of simplicial techniques in algebra and (algebraic/differential) geometry, both old and new. In the first part, we will review some of the classical topics in simplicial homological algebra (the Dold-Kan correspondence, the Eilenberg-Zilber theorem, canonical resolutions, Barr-Beck (co)triple (co)homology). In the second part, we will turn our attention to specific algebraic categories, such as simplicial groups and simplicial (commutative) algebras. (Topics may include the Kan loop group construction and an introduction to the Andre-Quillen homology). The third (main) part of the course will focus on more recent developments: we will try to give a gentle introduction to simplicial categories and DG categories which are the main technical tools in several areas of modern mathematics (derived algebraic geometry, modern representation theory and noncommutaive geometry). Depending on the audience preferences, topics may include a survey of \infinity-categories, simplicial presheaves on manifolds, homotopy theory of DG categories and derived Hall algebras. As for prerequisites, basic knowledge of homological algebra (at the level of MATH 6350), algebraic geometry and algebraic topology (at the level of MATH 6510) will be helpful.