Math 144: Model Theory and Algebra (Fall 2013)

Monday, Wednesday, Friday 11 am - 12 pm

Faculty: Nate Ackerman TA: ---
E-mail: nate AT math.harvard.edu



This is the approximate schedule of what should be covered in class with the associated reading.

Week
Date Reading
1
September 4, 2013
September 6, 2013
  • Introduction and History
  • Languages and Structures: Section 1.1
2
September 9, 2013
September 11, 2013
September 13, 2013
  • Theories: Section 1.2
  • Definable Sets: Section 1.3
    • Definable Sets
    • M^eq
  • Definable Sets: Section 1.3
    • Interpretations
    • Interpret Complex number in Reals
3
September 16, 2013
September 18, 2013
September 20, 2013
  • Compactness Theorem: Section 2.1
    • Compactness/Completeness
    • Non-standard models of arithmetic/set theory
    • Henkin constructions
  • Complete Theories: Section 2.2
    • Categoricity implies complete
    • Algebraicly Closed Fields
    • Vector Spaces
    • Equivalence Relations with properties
4
September 23, 2013
September 25, 2013
September 27, 2013
  • Complete Theories: Section 2.2
    • Injective polynomials from C^n to C^n
  • Elementary Chains
  • Lowenheim-Skolem Theorems: Section 2.3
    • Upwards
    • Elementary Embedding
5
September 30, 2013
October 2, 2013
October 4, 2013
  • Lowenheim-Skolem Theorems: Section 2.3
    • Downwards
    • Skolem Functions
  • Back and Forth: Section 2.4
    • Linear Orders
    • Random Graph
    • 0-1 Laws
  • EF Games
    • Scott's Isomorphism Theorem
    • Potential Isomorphism
6
October 7, 2013
October 9, 2013
October 11, 2013
  • Quantifier Elimination: Section  3.1
    • Dense Linear Orders
    • Divisible Abelian Groups
    • Presburg Arithmetic
  • Model Completeness
    • Model Completions
  • Algebraically Closed Fields: Section  3.2
    • Zariski Closed and Constructable Sets
    • Hilbert's Basis Theorem
    • Elimination of Imaginaries
7
October 16, 2013
October 18, 2013
  • Types: Section 4.1
    • Definitions/Basic Results
    • Stone Spaces
  • Examples
    • Dense Linear Orders
    • Algebraicly Closed Fields
  • Omitting Types and Prime Models: Section 4.2
    • Omitting Types Theorem
8
October 21, 2013
October 23, 2013
October 25, 2013
  • Omitting Types and Prime Models: Section 4.2
    • Prime and atomic models
    • Countable homogeneous models
    • Prime Model Extensions of omega-Stable Theories
  • Saturated and Homogeneous Models : Section 4.3
    • Countably Saturated Models
    • Existence of Saturated Models
9
October 28, 2013
October 30, 2013
November 1, 2013
  • Ultrapowers
    • Ultrafilters
      • Existence
      • Arrow's Impossibility Theorem
  • Ultraproducts
    • Saturation
    • Compactness of Topological Spaces
  • Monster model
  • Saturated and Homogeneous Models: Section 4.3
    • Homogeneous and Universal Models
    • Applications of Saturated Models
    • Vaught's Two Cardinal Theorem
  • Number of Countable Models: Section 4.4
    • aleph_0-Categorical Models
10
November 4, 2013
November 6, 2013
November 8, 2013
  • Partition Theorems: Section 5.1
    • Ramsey's Theorem
    • Connections to combinatorics
  • Indiscernibles: Section 5.2
    • Ehrenfeucht-Mostowski Models
    • Indiscernibles in Stable Theories
    • Applications of Erdos-Rado
11
November 11, 2013
November 13, 2013
November 15, 2013
  • A Many-Models Theorem: Section 5.3
  • Uncountably Categorical Theories: Section 6.1
12
November 18, 2013
November 20, 2013
November 22, 2013
  • Uncountably Categorical Theories: Section 6.1
    • Strongly Minimal Sets
    • Existence of Strongly Minimal Formulas
    • The Categoricity Theorem
  • Morley Rank: Section 6.2
    • Morley Rank
    • Monster Model
13
November 25, 2013
  • Morley Rank: Section 6.2
    • Morley Degree
    • Ranks of Types
    • Strongly Minimal Theories
    • Morley Rank in Strongly Minimal Theories
    • Morely Rank in Algebraically Closed Fields
14
December 2, 2013
December 4, 2013
Final Project Presentations

This page was created by Nate Ackerman, and last revised on August 3, 2013.