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 finitedimensional
vector spaces. Schur decomposition theorem.


1/31

Spectral theorem for
unitary, selfadjoint and normal operators.


Week 2

2/5

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


2/7

Gauss, Cholesky and
Iwasawa decompositions. Skewsymmetric 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. FermiDirac 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
Rmodule and
derived functors Ext and Tor.


3/14

The
cohomology of groups; example of a finite cyclic group. Cross
homomorphisms and
H^{1}(G,A);
group extensions and H^{2}(G,A).

Problem
set 5
This
assignment is optional
and
will not be graded

Week 8

3/19

Spring recess Mon
Mar18Sun, 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,
11:15am1:45pm

