Outline of topics covered

(Meeting 1: August 26)

Overview

Structuralism in mathematics
Goals of the seminar

Background in number theory

Euclid and Diophantus
Various theorems and conjectures by Fermat
Euler's totient function
GCD and the Euclidean algorithm

Philosophical themes

Abstraction and abstractionism
Axiomatization
Totalities
Information-management in structuralist mathematics
Conceptual character of structurlist thought, geometric interpretation
The group representation "paradox"

(Meeting 2: September 6)

Overview (cont'd)

A (historically lite) discussion of the nineteenth century transformation of mathematics

Groups in number theory

Lagrange's theorem
Euler's theorem and Fermat's little theorem
Wilson's theorem
The Chinese remainder theorem and the structure of Z_m

Philosophical issues

Uniform treatment
Equivalence

(Meeting 3: September 9)

Background in algebra

Fields, vector spaces, independence, bases
Matrices and determinants
Rings and modules

Discussion of Conway's analysis of quadratic forms

Equivalence, primativeness, irrelevance of sign
Finding a representation that ignores irrelevant information

(Meeting 4: September 20)

Case study: the behavior of x^2 + y^2

Rings, fields, polynomials
x^2 = -1 mod p solvable iff p = 1 mod 4
Unique factorization in Z
The Gaussian integers, norms, and unique factorization
Sum of two squares theorem (3 proofs; discussion)
p = x^2 + y^2 solvable iff p = 1 mod 4
Numbers representable as sums of squares

Algebraic integers: Dedekind's "two imperfections" in the early theory of algebraic integers

(Meeting 5: September 23)

The algebraic integers

Algebraic extensions of Q (= quotients of Q[x] by irreducible polynomials)
The problem with choosing 1, alpha, alpha^2, ..., alpha^n as a basis
Integral bases, and tranformations between bases
The algebraic integers
The factorization problem; Dedekind's analysis of the "peculiar cases"

Euler's proof of Fermat's little theorem (discussion)

(Meeting 6: September 30)

The role of historical and cognitive data in the philosophy of mathematics (discussion)

Euler's proof of Fermat's little theorem vs. a "modern" version

The use of finite sets in the modern presentation, and de-emphasis of algorithmic details
The use of the group structure (eliminating repetitions, case distinctions, distractions)
Four ways of handling an equivalence relation: lifting operations to equivalence classes; using canonical representatives; using a "manner of speaking"; doing nothing

Discussion

Are the differences "merely" a matter of taste?
A distinction: structural advantages at the local (proof) and global (theory) levels

"De-geometrizing" Conway

Quadratic forms as functions
Automorphisms (of Z^2) at the source, and equivalence of forms
Automorphism (of Z) at the target, reflected in symmetries in the classification
Understanding behavior of quadratic forms via an action of SL_2(Z)
Notion of "primitive vectors" reflecting the co-basality condition
The generators, S and T
Well lemma: these generate SL_2
Tree property: they generate freely, subject only to T^2 = 1
(General strategy: inferring facts about a structure from a particular action)
Co-basality considerations reflected in the notion of a superbase?

(Meeting 7: October 7)

More historical background

From quadratic forms to quadratic number fields
Fermat's last theorem and cyclotomic number fields
Problems: failure of unique factorization, and determining the integers of, for example, Q(sqrt -3)

Algebraic background

Normal subgroups and kernels of group homomorphisms
Ideals and kernels of ring homomorphisms

Chapter 1 of Dedekind

Free modules, submodules, and bases
Z-modules in terms of lattices

(Meeting 8: October 14)

Algebraic background (cont'd)

Describing the group operation on G/H as set multiplication, or via representatives (discussion)
Ideals and kernels of ring homomorphisms (cont'd)

Group actions

Example: SL_2(Z) acting on Z x Z
Definition of a group action on X (homomorphism from G to Aut(X))
Case 1: X is a set, Aut(X) = permutations
Case 2: X is a vector space over k, Aut(X) = GL_n(k)
Case 3: X is a group

Dedekind, Chapter 2

Review of chapter 1
Inferring the behavior of ideal divisors in Z[sqrt -5]
Defining "omega is divisible by alpha^n", where alpha is the ideal (prime) square root of 2

(Meeting 9: October 21)

Group Actions (cont'd)

Conjugation: given a group G and element g, let f_g(x) = g^-1 x g
Each f_g is an automorphism => g -> f_g is a group action
The center of a group, Z(G)
Simple group = no normal subgroups other than {1}, G
The orbit of an element under a group action
Orbits under conjugation = conjugacy clases
Elements in a conjugacy class are indistinguishable in the language of group theory
Note: a normal subgroup is a union of conjugacy classes

Dedekind, Chapter 2 (cont'd)

Review from last time
The notion of an ideal
Addition, multiplication, and divisibility of ideals
The hard part: showing that if a divides b, then for some c, ac = b
The structure of ideals in Z[sqrt -5]
The norm of an ideal
Multiplication of ideals: [ma, m(b + theta)] [m'a', m'(b' + theta)] = [m''a, m''(b'' + theta)]
Corollaries: norm(m m') = norm(m) norm(m'), the product of an ideal and its conjugate

(Meeting 10: October 28)

Dedekind, Chapter 2 (cont'd)

Review from last time
"To contain is to divide"
Maximal implies prime
Unique factorization in Z[sqrt -5]
Discussion

Group actions (cont'd)

Orbits and stabilizers
Example: G acts on G (as a group) by conjugation
The class equation
Example: G acts on G (as a set) by right multplication

(Meeting 11: November 4)

Division rings (aka skew fields)

Definition (like a field, but multiplication not commutative)
Finite fields, Moore (1898): unique field of size p^n
The center of a division ring is a field
Example: Hamilton's quaternions; Frobenius: R, C, H the only division rings over R
Wedderburn (1905): every finite division ring is a field
Wedderburn's approach: use class equation, plus some number theory
Structuralist approach: use elementary properties, structural analysis of minimal finite dimensional division rings
Definition: K is a k-algebra (K is a k-vector space, with a multiplication compatible with k)
Finite field extensions: dim_k1(k2) dim_k2(k3) = dim_k1(k3)
Observation: if D is a minimal noncommutative division ring, a an elt not in the center, then C_D(a) is a field
The structure of a minimal division ring; easy to see that there is no 3-dimensional division ring over R

Aspects of structuralism (discussion)

Identifying structures, characterized axiomatically (e.g. groups, rings, fields, modules, vector spaces, division rings, ...)
Classifying structures (e.g. quadratic forms, finitely generated abelian groups, finite fields, finite simple groups, finite dimensional minimal division rings, ...)
Utilizing relationships to other structures (e.g. group of automorphism of a field; lattice of normal subgroups of a group; vector space structure of a field extension; group structure of an ideal; action of a group on a space)
Emphasis on mappings (e.g. normal subgroups and ideals as kernels of homomorphisms; automorphisms of a structure)
A collection of constructions (quotients, products, subobjects, lattice of subobjects,etc.)
Philosophical structuralism: de-emphasis on "elements," particulars of a construction

(Meeting 12: November 11)

Chapters 3 and 4 of Dedekind

Overview
Background from Galois theory
Algebraic integers

Characterizing the structure of minimal finite dimensional division rings (cont'd)

Review from last time
Lemma: any subring of a finite dimensional division ring containing the center is a field or division ring
A lemma from Galois theory
A hypothesis
Using the hypothesis: D has a basis of p^2 elements over k

(Meeting 13: November 18)

Dedekind's methodological claims (discussion)

"Fundamental properties"
Generality
Analogy with the rational integers
Simplicity

Wedderburn's theorem

Conjugacy classes of subfields under D*
Theorem: any two isomorphic fields are conjugate
Theorem: every automorphism arises from conjugation
Structural proof: specialize structure theory to finite case, immediate
Wedderburn's proof: minimal structural analysis, serious number theory

(Meeting 14: November 25)

Advantages of structuralist ideas in proving Wedderburn's theorem

Review of the proof
Continual confrontation of syntax and semantics
Structure types as organizing principles
Minimizing counting

Structures in the proof of Fermat's last theorem (James Cummings, guest)

Complex analysis (analytic and meromorphic functions, zeroes and poles)
Elliptic functions
Weierstrass' P function
Parameterizing the elliptic curves
A geometric interpretation of addition on C / (lattice of periodicity)
Modular forms
The Tanayama-Shimura conjecture

(Meeting 15: December 2)

Discussion

Edwards, and constructivity
Set-theoretic foundations vs. set-theoretic methodology
Ontological vs. methodological responses to Edwards
Stein: a modal structuralist?
"Conceptual" vs. structural
Calculation vs. "fundamental properties"

(Meeting 16: December 13)

Discussion

Scientific explanation (modal, epistemic, and ontic conceptions)
Mathematical explanation
Towards a theory of mathematical concepts
Towards a theory of mathematical methods
"Top down" vs. "bottom up" approaches to characterizing epistemic aims