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

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

Video of class (downloadable)
September 6

No Class.
September 9

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

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

Video of class (downloadable)
September 11

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

Associated Reading:
Video of class (downloadable)
September13

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

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

Video of class (downloadable)
September 16

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

Associated Reading:
Video of class (downloadable)
September 18

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

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

Video of class (downloadable)
September 20

No class
Sept. 23

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

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

Video of class (downloadable)
Sept. 25

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

Associated Reading:
Video of class (downloadable)
Sept. 27

No class
Sept. 30

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

Associated Reading:
  • "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

Video of class (downloadable)
Oct. 2

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

Associated Reading:
  • D-F: 17.4

Video of class (downloadable)
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

Associated Reading:
  • 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

Video of class (downloadable)
Oct. 7

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

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

Video of class (downloadable)
Oct. 9

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

Associated Reading:
  • 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.

Video of class (downloadable)
Oct. 11

No class
Oct. 14

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

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

Video of class (downloadable)
Oct. 16

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

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

Video of class (downloadable)
Oct. 18

No class
Oct. 21

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

Associated Reading:
Video of class (downloadable)
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

Associated Reading:
Video of class (downloadable)
Oct. 28

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

Associated Reading:
Video of class (downloadable)
Oct. 30

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

Associated Reading:
Todd Coxeter Recap Video (downloadable)
Vector bundle video (downloadable)
Nov.1

No class
Nov. 4

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

Associated Reading:
Video of class (downloadable)
Nov. 6

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

Associated Reading:
  • D-F: 5.2
  • Lang: I.8, 1.10.

Video of class (downloadable)
Nov. 8

No class
Nov. 11

No class
Nov. 13

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

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

Video of class (downloadable)
Nov. 15

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

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

Video of class (downloadable)
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

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

Video of class (downloadable)
Dec. 4

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

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

Video of class (downloadable)
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

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

Video of class (downloadable)
Dec. 9

In Lecture:
    • Spectrum of a ring
    • Localization

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

Video of class (downloadable)
Dec. 11

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

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

Video of class (downloadable)
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

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

Video of class (downloadable)

Last updated: 12/7/13
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