Week

Date

Topics

HW(due dates in this table

overwrite dates in pdf files)

Week 1

1/29

Hermitian and Euclidean inner products, orthonormal sets in finite-dimensional vector spaces. Schur decomposition theorem.

1/31

Spectral theorem for unitary, self-adjoint and normal operators.

Week 2

2/5

Symmetric bilinear forms, quadratic forms, transformation to the canonical diagonal form, the law of inertia. Positive-definite quadratic forms, Sylvester’s criterion.

2/7

Gauss, Cholesky and Iwasawa decompositions. Skew-symmetric bilinear forms, symplectic basis. The Pfaffian.

Problem set 1

Week 3

2/12

Tensor algebra of a module, graded rings, Hilbert series. Tensor algebra as a Hopf algebra. Examples of Hopf algebras.

2/14

Symmetric algebra of a module, shuffle product, Heisenberg commutation relations and Weyl algebra.

Problem set 2

Week 4

2/19

Exterior algebra of a module, Koszul duality. Fermi-Dirac anticommutation relations and Clifford algebra. Determinants, Hodge star operator. Extra notes on multilinear algebra.

2/21

Symmetric and alternating tensors. Hom_R(- , -) and tensor products. Short five lemma.

Problem set 3

Week 5

2/26

The snake lemma. Categories and functors, examples. Products and coproducts.

2/28

Universal objects and free objects. Examples and applications.

Problem set 4

Week 6

3/5

Exact sequences of modules. Injective and projective modules. Hom_R(- , -), for R a commutative ring.

3/7

Midterm 1

Week 7

3/12

Cochain complexes and long exact sequence in cohomology. Projective resolution of an R-module and derived functors Ext and Tor.

3/14

The cohomology of groups; example of a finite cyclic group. Cross homomorphisms and

H¹(G,A); group extensions and H²(G,A).

Problem set 5

This assignment is optional

and will not be graded

Week 8

3/19

Spring recess Mon Mar18-Sun, Mar 24

3/21

Problem set 6.

This assignment is optional

and will not be graded.

Week 9

3/26

Field theory: field extensions, algebraic extensions.

3/28

Splitting fields, algebraic closures and algebraically closed fields.

Week 10

4/2

Separable and inseparable extensions.

4/4

Finite fields, cyclotomic polynomials and extensions.

Problem set 7

Week 11

4/9

The primitive element theorem. Galois theory: basic definitions, examples.

4/11

The fundamental theorem of Galois theory.

Problem set 8

Week 12

4/16

Examples. Finite fields. Linear independence of characters.

4/18

Hilbert’s Theorem 90. Cyclotomic extensions and abelian extensions over ℚ. More on Galois correspondence

Problem set 9

Week 13

4/23

Galois groups of polynomials, solvability in radicals.

4/25

Midterm 2

The exam covers material in §§13.1 - 13.2, §§13.4 - 13.6 and §§14.1 - 14.6 from Dummit and Foote.

Week 14

4/30

Integral extensions and closures, algebraic integers. Dedekind domains. Affine algebraic sets and Hilbert’s Nullstellensatz.

5/2

Representation theory of finite groups, examples, including the regular representation. Irreducible, indecomposable and completely reducible representations. Maschke’s theorem.

Problem set 10

Week 15

5/6

Basic properties of characters. Schur’s Lemma and orthogonality of characters. Decomposition of the regular representation.

5/9

The characters of irreducible representations as an orthonormal basis in the space of central functions. The second orthogonality relation for characters. Character tables, examples.

Problem set 11

Week 16

Review for the final exam

Week 17

5/17

Final Exam:

Friday, May 17, 11:15am-1:45pm