Homework and Syllabus
Topics in Algebra
MATH 8174 Spring 2022
| Date | Topics |
Reading |
Homework |
| L1 Monday
January 10 |
Introduction:
General overview, and a discussion of some of the main
areas of focus. |
The following .pdf has a
brief review of complex analysis in a single
variable. |
|
| L2 Wednesday
January 12 |
Discussion of working
groups: Start discussions about who wants to
work on what topics. |
||
| L3 Friday January 14 | Introduction
continued: Review of some linear and
multilinear algebra. |
You may want to
review an algebra text (e.g., Artin's Algebra,
Lang's Algebra, Dummit-Foote's Abstract
Algebra, Alufi's Algebra Chapter 0, etc.) |
Calum write up notes |
| Monday January 17 |
MLK
DAY |
NO CLASS |
MLK DAY |
| L4 Wednesday January 19 |
Introduction continued: More review of some linear and multilinear algebra. | Adrian write up notes |
|
| L5 Friday January 21 |
Introduction continued: More review of some linear and multilinear algebra. | Bob write up notes |
|
| L6 Monday
January 24 |
Hodge structures:
Complex and Hermitian Structures.
Complexification of almost complex vector spaces and
alternating products. |
Huybrechts Section 1.2 | Matt write up notes |
| L7 Wednesday
January 26 |
Hodge structures:
Complex and Hermitian Structures continued.
Hermitian forms revisited. |
Huybrechts Section 1.2 | Justin write up notes |
| L8 Friday January 28 |
Hodge
structures: Complex and Hermitian
Structures continued. Hermitian forms, symplectic
forms, and the Hodge star. |
Huybrechts Section 1.2 | Peter write up notes |
| L9 Monday January 31 |
Hodge
structures: Complex and Hermitian Structures.
The Hodge star continued, and intro to the Lefschetz
decomposition. |
Huybrechts Section 1.2 | Chris write up notes |
| Wednesday February 2 |
SNOW
DAY |
||
| L10 Friday February 4 |
Hodge
structures: Complex and Hermitian Structures.
Representations of sl_2(CC), and the Lefschetz
decomposition. |
Huybrechts Section 1.2 | Calum |
| L11 Monday February 7 |
Hodge
structures: Complex and Hermitian Structures.
The Hodge--Riemann bilinear relations. |
Huybrechts Section 1.2 | Adrian |
| L12 Wednesday February 9 |
WORK
IN GROUPS |
||
| L13 Friday February 11 |
WORK
IN GROUPS |
||
| L14 Monday February 14 |
Hodge structures: Basic definitions, polarized Hodge structures. | Peters--Steenbrink
Section 2.1 You may also want to take a look at Huybrechts Sections 1.2 and 3.C, and Henry Fontana's senior thesis (I am updating these notes and will post the revisions in the next week or so). |
Bob |
| L15 Wednesday February 16 |
Hodge structures: Hodge structures via Hodge Filtrations. | Peters--Steenbrink Section 2.2 | Matt |
| L16 Friday February 18 |
Hodge
structures: Hodge structures via
representations of the torus. |
Peters--Steenbrink Section 2.3 | Justin |
| L17
Monday February 21 |
Hodge structures: Mumford--Tate Groups. | Peters--Steenbrink Section 2.3 | Peter |
| L18
Wednesday February 23 |
Introduction to
homological algebra, and Hodge structures: Complexes
of modules, some basic ideas motivating derived
categories, Hodge complexes. |
Peters--Steenbrink Section 2.3 | Chris |
| L19 Friday February 25 |
Introduction
to sheaves Introduction to sheaves of abelian
groups. |
Hartshorne Chapter 2
Section 1 |
Calum |
| L20 Monday February 28 |
Introduction
to homological algebra: Derived functors. |
Weibel, and
Peters--Steenbrink appendix. |
Adrian |
| L21 Wednesday March 2 |
Introduction to homological algebra: Computing derived functors | Weibel, and Peters--Steenbrink appendix. | Bob |
| L22 Friday March 4 |
Introduction to homological algebra: Spectral sequences. | Weibel, and Peters--Steenbrink appendix. | Matt |
| L23
Monday March 7 |
Introduction to homological algebra: Spectral sequences continued, spectral sequence associated to a filtered complex, and a double complex. | Weibel, and Peters--Steenbrink appendix. | Justin |
| L24
Wednesday March 9 |
Introduction to homological algebra: Spectral sequences continued, examples, Hodge complexes of sheaves. | Weibel, and Peters--Steenbrink appendix. | Peter |
| L25 Friday March 11 | Mixed
Hodge structures: Basic definitions |
Peters--Steenbrink
3.1 |
Chris |
| L26 Monday March 14 |
Mixed Hodge structures: The Deligne splitting and applications. | Peters--Steenbrink 3.1 | Calum |
| L27 Wednesday March 16 |
Mixed Hodge structures: Mixed Hodge complexes, and mixed Hodge complexes of sheaves. | Peters--Steenbrink 3.3 | Adrian |
| L28 Friday March 18 |
Mixed Hodge structures: Hypercohomology of a mixed Hodge complex of sheaves, mapping cones, and applications to mixed Hodge structures. | Peters--Steenbrink 3.4 | Bob |
| March 21--25 | SPRING BREAK |
NO CLASS |
SPRING BREAK |
| L29 Monday March 28 |
Mixed Hodge structures: Extensions in abelian categories | Peters--Steenbrink
A.2.6 |
Matt |
| L30 Wednesday March 30 |
Mixed Hodge structures: Extensions in abelian categories continued | Peters--Steenbrink A.2.6 | Justin |
| L31 Friday April 1 |
Mixed Hodge structures: Extensions of mixed Hodge structures | Peters--Steenbrink 3.5 | Peter |
| L32 Monday April 4 |
Mixed Hodge structures: Intermediate Jacobians | Peters--Steenbrink
3.5 |
Chris |
| L33 Wednesday April 6 |
WORK IN GROUPS | ||
| L34 Friday April 8 |
WORK IN GROUPS | ||
| L35
Monday April 11 |
Review of differential
forms |
Bott--Tu, Differential Forms in
Algebraic Topology |
Calum |
| L36
Wednesday April 13 |
Vector bundles,
cotangent bundles, and differential forms |
Lee, Introduction to Smooth
Manifolds, Ch.10, Huybrechts. |
Adrian |
| L37 Friday April 15 |
D-modules:
Verdier duality |
Peters--Steenbrink
Ch. 13.1, Kashiwara--Schapira, Sheaves on
Manifolds, Ch.III 3.1-3 |
Bob |
| L38 Monday April 18 |
Kazdhan--Lusztig
polynomials and mixed Hodge structures |
Elias--Williamson, The
Hodge theory of Soergel bimodules, Ann.
Math. 2014. |
Chris, Justin, Matt
(Yano write up) |
| L39 Wednesday April 20 |
Simplicial
spaces and mixed Hodge structures |
Peters--Steenbrink,
Chapter 5 (Deligne's paper) |
Adrian and Bob (Yano
write up) |
| L40 Friday April 22 |
The
Hodge decomposition and PDE: Lecture 1 |
Huybrechts, Wells. |
Peter (Yano write up) |
| L41
Monday April 25 |
The Hodge
decomposition and PDE: Lecture 2 |
Peter (Yano write up)
|
|
| L42
Wednesday April 27 |
Torsors, principle bundles, connections, and infinite dimensional Lie groups | Calum (Yano write up) | |
| Friday April 29 | READING DAY |
NO CLASS |
READING DAY |
You may want to use LaTeX
for typing homework. If you have a mac,
one possible easy way to get started is with texshop;
you will want to download the MacTex
package. If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something
online. This site can help you find LaTeX symbols by
drawing: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cocalc.com
(formerly https://cloud.sagemath.com/)
or https://www.overleaf.com
(formerly sharelatex) for a cloud version.