M. Roth | Winter 2024
Mon: 1pm–2:30pm, Jeff 422
Wed: 11:30am–1pm, Jeff 422
Grading Scheme:
40% Homework
30% Take home exam
30% Written Final
Algebra is a subject with extremely concrete origins: it is the language of the everyday manipulation of symbols,
and the solution of equations. It is the language of computation in coordinates, of differential operators, and of
symmetries.
At the same time, it can be quite abstract; modern algebra is also the language of relations between objects,
sometimes divorced from the objects themselves.
This course is the second part of the graduate core algebra sequence. Its aim is to introduce
the algebraic concepts which are common knowledge for the working mathematician, and to make the
link between the abstract definitions and their concrete incarnations.
The emphasis in the course will be on
understanding ideas rather than speeding through the material. The following outline
of topics is therefore likely to change.
1. Galois Theory
| Jan | 8 |
Examples of fields, review of basics of field extensions.
|
| | 10 |
Bound on the order of automorphism group, separable and normal extensions.
|
| | 15 |
More about normal extensions, key lifting lemma.
|
| | 17 |
Galois extensions, idea of Galois correspondence.
|
| | 22 |
Artin's lemma, Fundamental theorem of Galois theory.
|
| | 24 |
Examples. Applications of Galois theory.
|
2. Basic Notions of Category Theory
| Jan. | 29 |
Categories, Functors. Definitions by diagrams.
|
| | 31 |
Universal properties. Products and Coproducts.
|
| Feb. | 5 |
Infinite products. Fibre products, push-outs.
|
| | 7 |
Adjoint functors. Preservation of products and coproducts.
|
3. Multilinear Algebra
A. Tensor Products
Homework 3 (Due: Mar. 23)
| Feb. | 12 |
Universal mapping properties, bifunctoriality.
|
| | 14 |
Tensor products of sums, tensor products of free modules.
|
| | 26 |
Right-exactness of tensor product. Examples
|
| | 28 |
Homomorphism identities. Associativity of the tensor product. Change of scalars.
|
B. Symmetric and Alternating Products
Homework 4 (Due: Apr. 6)
| Mar. | 4 |
Algebras, graded algebras, tensor product of algebras. Tensor algebra of a module.
|
| | 6 |
Basic functorial properties of the tensor algebra. Change of rings.
|
| | 11 |
Symmetric algebra, symmetric products, symmetric product and direct sums.
|
| | 13 |
Alternating products. Determinants. Comparison of sub- and quotient- constructions
|
4. Representations of finite groups
Homework 5 (Due: Apr. 20)
| Mar. | 18 |
Group representations. Examples. Preview of main theorems.
|
| | 21 |
Maschke's theorem. Schur's lemma. HomG and HomG.
|
| | 25 |
Inner product on representations. Class functions.
|
| | 27 |
Regular representation and its universal property. Proof of remaining theorems. Examples
|
| Apr. | 1 |
Using the orthogonality relations, representations of S4.
|
| | 3 |
Representations of the symmetric group, Isotypic components, Schur functors.
|