# Math 602 schedule

Monday Wednesday Friday
August 26

No class
August 28

No class. Math classes numbered 500 and above do not meet before September 3.
August 30

No class. Math classes numbered 500 and above do not meet before September 3.
September 2

No class: Math classes numbered 500 and above do not meet before September 3.
September 4

In Lecture
• Definition of a group, basic examples
• Subgroups, homomorphisms, normal subgroups, quotient groups, exact sequences
• Permutation groups, operations of groups on sets
• The origin of group theory in Galois Theory
• The orbit formula

• D-F: 1.1, 1.2, 1.4, 1.6, 1.7, 2.1
• Lang: 1.1 - 1.5

September 6

No Class.
September 9

Lecture
• The orbit formula, continued
• The class equation
• Sylow theorems
• Symmetric groups and isometry groups

• D-F: 1.2, 1.3, 4.3, 4.5
• Lang: 1.5, 1.6,

September 11

Lecture
• Proof of the Sylow theorems
• Applications of the Sylow theorems

September13

Lecture (In room 4C8 of DRL labs, 9:00 - 10:30)
• Applications of the Sylow theorems, continued
• Semi-direct products

• D-F: 1.2, 1.3, 2.5, 3.4, 4.3, 4.5
• Lang: 1.3, 1.4, 1.5, 1.6

September 16

Lecture
• Topological groups
• Quotients of topological spaces by group actions
• Crystallographic groups

September 18

Lecture
• Schreier's Theorem
• Jordan Holder Theorem
• Classifying finite simple groups

• D-F: 2.5, 3.4, 4.3, 4.4, 4.5
• Lang: 1.3, 1.4, 1.5, 1.6

September 20

No class
Sept. 23

In Lecture
• Derived series, lower and upper central series
• Nilpotent groups

• D-F: 2.5, 3.4, 4.3, 4.4, 4.5
• Lang: 1.3, 1.4, 1.5, 1.6

Sept. 25

In Lecture:
• Nilpotent groups, continued
• The Frattini argument

Sept. 27

No class
Sept. 30

In Lecture
• Multiply transitive groups
• Blocks and primitive groups
• The group extension problem and H^2

• "Finite permutation groups" by H. Wielandt.
• D-F: 2.5, 3.4, 4.3, 4.4, 4.5 , 17.4
• Lang: 1.3, 1.4, 1.5, 1.6

Oct. 2

In Lecture:
• The group extension problem and H^2

• D-F: 17.4

Oct. 4

In Lecture: Room 4C8 of DRL labs, 9:00 - 10:30
• The group extension problem and H^2, continued
• Start of Category theory

• D-F: 17.4
• D-F: 2.5, 3.4, 4.3, 4.4, 4.5, 5.2, Appendix II, 6.3
• Lang: 1.3, 1.4, 1.5, 1.6, I.8, I.11, I.12

Oct. 7

In Lecture (room 4N30 of DRL):
• Category theory and applications
• Natural tranformations
• Automorphisms of functors

• D-F: Appendix II, 6.3
• Lang: I.11, I.12

Oct. 9

In Lecture:
• The category of rings
• Representations of groups
• Representability of the functor associated to representations of a group over a ring

• Lang, II.1 - II.3, Appendix 2.2.
• Dummit and Foote, 7.1 - 7.4, Appendix I.
• Dummit and Foote, chapter 18.1
• Pages 313 - 318 of This article by Lenstra and de Smit on deformation rings.

Oct. 11

No class
Oct. 14

In Lecture (room 4N30 of DRL):
• Products and coproducts
• Pullback and pushouts

• D-F: Appendix II, 6.3
• Lang: I.11, I.12

Oct. 16

In Lecture:
• Pullback and pushouts, continued
• Free groups, groups defined by generators and relations

• D-F: Appendix II, 6.3
• Lang: I.11, I.12

Oct. 18

No class
Oct. 21

In Lecture (room 4C8 of DRL):
• The Todd Coxeter algorithm

Oct. 23

No class
Oct. 25

In Lecture (room 4C8 of DRL from 9:00 to 10:30):
• Fundamental groups and the Van Kampen Theorem

Oct. 28

In Lecture (room 4C8 of DRL):
• Representable moduli problems
• Principal G-bundles
• Families of elliptic curves
• Coarse moduli spaces

Oct. 30

In Lecture (room 4C8 of DRL):
• More on the Todd Coxeter algorithm
• Vector bundles

Nov.1

No class
Nov. 4

In Lecture (room 4C8 of DRL):
• Another construction of vector bundles by group theory
• Cocyles and local trivializations

Nov. 6

In Lecture (room 4C8 of DRL):
• Direct and inverse limits in categories

• D-F: 5.2
• Lang: I.8, 1.10.

Nov. 8

No class
Nov. 11

No class
Nov. 13

In Lecture (room 4C8 of DRL):
• Abelian categories
• Injectives and projectiles
• Derived functors

• Section III.1 of Hartshorne's ``Algebraic Geometry".
• Lang, XX.6 - XX.7.
• Serre, Corps Locaux, on group cohomology

Nov. 15

In Lecture (room 4C8 of DRL at 9:00 a.m.):
• Derived functors, continued
• Cohomology of groups
• Presheaves and sheaves

• Section II.1, III.1 of Hartshorne's ``Algebraic Geometry".
• Lang, XX.6 - XX.7.
• Serre, Corps Locaux, on group cohomology

Nov. 18

No class
Nov. 20

No class
Nov. 22

No class
Nov. 25

No class
Nov. 27

No class
Nov. 29

No class
Dec. 2

In Lecture:
• Ext groups and Derived functors, continued
• Cohomology of groups

• Section II.1, III.1 of Hartshorne's ``Algebraic Geometry".
• Lang, XX.6 - XX.7.
• Serre, Corps Locaux, on group cohomology

Dec. 4

In Lecture (room 4C8 of DRL):
• Presheaves and sheaves
• Sheaf cohomology

• Section III.1 of Hartshorne's ``Algebraic Geometry".
• Lang, XX.6 - XX.7.

Dec. 6

In Lecture (room 4C8 of DRL at 9:00 a.m.):
• Betti homology and cohomology of simplicial complexes
• Ring theory: basic definitions
• Principal ideal domains
• Maximal and prime ideals

• Lang, XII.6, II.5,
• DF, p. 232, 8.2, 8.3

Dec. 9

In Lecture:
• Spectrum of a ring
• Localization

• Lang, XII.6, II.5,
• DF, p. 232, 8.2, 8.3

Dec. 11

In Lecture (room 4C8 of DRL):
• Chinese remainder Theorem
• Euclidean rings
• Principal ideal domains

• Lang, XII.6, II.5,
• DF, p. 232, 8.2, 8.3

Dec. 13

In Lecture (room 4E18 of DRL at 9:00 a.m.):
• Principal ideal domains and Euclidean rings (continued)
• Valuations and discrete valuation rings
• Unique factorization domains

• Lang, XII.6, II.5,
• DF, p. 232, 8.2, 8.3