# Category theory I:

• Definitions, initial/final objects, (co)products.

# Groups:

• Notions: Definitions, normal subgroups, actions, nilpotent, solvable, simple, extensions, semi-direct product, composition series, presentation, Cayley graphs.
• Theorems: Isomorphism theorem, Lagrange, Jordan-Hölder, Sylow.
• Examples: $S_n$, $A_n$, $GL(n,p)$, $GL(n,R)$, $U_n$, free groups, $p$-groups.

# Rings and Modules:

• Notions: Definitions, ideals, integral domains, prime and maximal ideals, Noetherian rings, complexes, homology, exact sequences.
• Theorems: Isomorphism theorem, snake lemma, Hilbert's basis theorem.
• Examples: Polynomial rings, power series, $p$-adics.

# Factorization in rings:

• Notions: prime and irreducible elements, PID's, UFD's, Euclidean domains.
• Theorems: Polynomial rings over UFD's are UFD's, Chinese Remainder Theorem.
• Examples: irreducibilty in polynomial rings, Eisenstein's criterion, Gaussian integers.

# Linear Algebra I:

• Notions: bases and dimension, torsion, presentations and resolutions, characteristic polynomials, canonical forms.
• Theorems: classification of finitely generated modules over PID's, Cayley-Hamilton, Jordan canonical form.

# Fields:

• Notions: definition, simple extensions, finite and algebraic extensions, algebraic closure, splitting field, normal and separable extensions, Galois extensions, (transcendental extensions in 3rd quarter).
• Theorems: existence of algebraic closure, Galois correspondence.
• Examples: rational functions, finite fields, cyclotomic fields, applications of Galois theory (e.g. solvability of polynomials).

# Category theory II:

• Notions: Functors, natural transformation, exact functors, adjoint functors, limits, colimits (and filtered colimits), relation with universal properties.
• Theorems: Preservation of limits/colimits by adjoint functors.

# Linear Algebra II:

• Notions: Tensor product, extension of scalars, symmetric and exterior powers, flat modules, projective modules, injective modules, Tor, Ext.
• Theorems: A free module is projective. A projective module is flat. Existence and homotopy uniqueness of projective and injective resolutions of modules.

# Commutative algebra:

• Notions: Localization of rings and modules, integral homomorphism of rings, integral closure, normal domains, spectra of rings and Zariski topology, transcendental field exensions, Krull dimension.
• Theorems: Nakayama's Lemma, Going-up and Going-down, Noether normalization, Hilbert and Zariski Nullstellensatz, (transcendence degree equals Krull dimension for finitely generated algebras over a field).

# Homological algebra:

• Notions: Additive category, abelian category, complexes, homotopy category of complexes, derived category, derived functors and cohomology, spectral sequence.
• NOTE: At the discretion of the instructor, the Homological Algebra section, and possibly the final topics in the Commutative Algebra section, could be replaced by:
• Representation theory of finite groups (char. 0):
• Notions: Irreducible representation, group algebra, matrix coefficients, character, induction, Fourier transform, convolution.
• Theorems: Maschke, Schur's lemma, Frobenius's reciprocity, orthogonality relations, Plancherel.
• Examples: Abelian groups, $S_3$, $(S_4)$.

# Noncommutative algebra:

• Structure of semisimple rings, Jacobson radical, Wedderburn Theorem, division algebras.