Introduction to Derived Categories

Abstract

In the first talk, I will give a quick introduction to some basic notions in category theory that we need, with examples. Next I will introduce chain complexes and homologies, then homotopy category of chain complexes K(A). The talk should culminate at defining derived category D(A) as the category obtained from K(A) by "formally inverting" quasi-isomorphism.

In the second talk, I will begin with introducing "distinguished triangles" in D(A). Then we can talk about derived functors between derived categories. We will recover many familiar notions in homological algebra, such as Ext and Tor, as derived functors. Then I will talk about some applications (if time permit), with a focus on derived category of sheaves on some spaces (manifolds, algebraic varieties).

Biography

Qingjing Chen received his PhD from University of California, Santa Barbara in 2023, working on questions related to derived categories of algebraic varieties. After that he became a postdoc at YMSC in Tsinghua University from 2023 to 2024. He is currently a postdoc at Nanyang Technological University.

AToM@MAS

This talk is the first talk in a new seminar series we are organizing called AToM@MAS. This stands for The Algebra, Topology, and Mathematical Physics Seminar. This series is intended to complement the regular MAS Seminar Series by providing speakers with an opportunity to give supplementary introductory talks aimed at a wider mathematical audience, including researchers not working in the discipline in question. It also offers researchers at all career stages within SPMS a platform to present their work and hosts in-depth learning seminars on topics of interest within algebra, topology, mathematical physics and their interdisciplinary spaces