Math 237 Spectral Sequences - Spring 2020 at UCLA

Graduate topics course in algebraic topology (during a pandemic)
Syllabus

Format
Pre-recorded lectures with synchronous discussions during class periods.
Students gave final presentations in the last two weeks.

Video Lectures

Playlist of all videos

Motivation and setup
Video 1 - Intro and motivation
Video 2 - Graded rings and modules
Video 3 - Spectral sequence definition
Video 4 - 1st quadrant spectral sequence
Video 5 - Homology 1st quadrant spectral sequence
Video 6 - Unraveled exact couple (part 1)
Video 7 - Unraveled exact couple (part 2)
Video 8 - Unraveled exact couple (part 3)
Video 9 - Cohomology spectral sequence for a filtration
Video 10 - Unraveled exact couple (part 4)
Video 11 - Exact couple

Bockstein spectral sequence
Video 12 - Bockstein spectral sequence
Video 13 - Bockstein spectral sequence 1st application
Video 14 - Bockstein spectral sequence from a filtration

Serre spectral sequence
Video 15 - Serre spectral sequence
Video 16 - Serre spectral sequence sketch
Video 17 - Loop-path fibration
Video 18 - First computations (part 1)
Video 19 - First computations (part 2)
Video 20 - An application of the Serre spectral sequence
Video 21 - The Steenrod squares
Video 22 - The Steenrod algebra
Video 23 - Unstable modules over A
Video 24 - Cohomology of Eilenberg-MacLane spaces (part 1)
Video 25 - Cohomology of Eilenberg-MacLane spaces (part 2)
Video 26 - Cohomology of Eilenberg-MacLane spaces (part 3)
Video 27 - Cohomology of Eilenberg-MacLane spaces (part 4)
Video 28 - Cohomology of Eilenberg-MacLane spaces (part 5)

Stable homotopy groups
Video 29 - Calculating first stable stem
Video 47 - Stable stems are finite
Video 48 - Computing more stable stems (part 1)
Video 49 - Computing more stable stems (part 2)

Double complexes
Video 30 - Double complex
Video 31 - Double complex spectral sequence
Video 32 - Some applications (part 1)
Video 33 - Some applications (part 2)
Video 34 - Grothendieck spectral sequence
Video 35 - Base change
Video 36 - Construction of Grothendieck spectral sequence (part 1)
Video 37 - Construction of Grothendieck spectral sequence (part 2)

Convergence
Video 38 - Cohomology spectral sequence for a filtration (part 1)
Video 39 - Cohomology spectral sequence for a filtration (part 2)
Video 40 - Convergence (part 1)
Video 41 - Convergence (part 2)
Video 42 - Convergence (part 3)
Video 43 - Convergence (part 4)
Video 44 - Convergence (part 5)
Video 45 - Convergence (part 6)
Video 46 - Examples

Resources

Extra chapter on spectral sequences - Hatcher
Spectral sequences: friend or foe? - Vakil
Lecture notes in algebraic topology - Davis and Kirk
Cohomology operations and applications in homotopy theory - Mosher and Tangora
A user's guide to spectral sequences - McCleary
A user's guide to the Bockstein spectral sequence - Palmieri
Spectral sequences and an application - Orlich
Conditionally convergent spectral sequences - Boardman
An introduction to homological algebra - Weibel

Suggested Presentation Topics

Bockstein spectral sequence and H-spaces
Proof of cohomological Serre spectral sequence with algebra structure
Higher structure in Serre spectral sequence (e.g. coalgebras and/or Hopf algebras)
Proof of existence of Steenrod squares and properties
Computation of K(Z/2,n) with integral coefficients using Bockstein spectral sequence
Odd prime Steenrod operations
Proof of Thom isomorphism using Serre spectral sequence

Computations in and/or construction of:
The Serre spectral sequence with local coefficients
The Bökstedt spectral sequence
The Atiyah-Hirzebruch spectral sequence
The Künneth spectral sequence
The change-of-rings spectral sequence
The Cartan-Leray spectral sequence
The slice spectral sequence
The Bousfield-Kan spectral sequence
The homotopy fixed-point spectral sequence