MATH245A
Overview
Subject: Math
Course Number: 245A
Department: Mathematics
Course Level: Graduate
Course Title
General Theory of Algebraic Structures
Course Description
Structures defined by operations and/or relations, and their homomorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence- and subalgebra-lattices, or category-theory and adjoint functors, or other aspects.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH249
Overview
Subject: Math
Course Number: 249
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Combinatorics
Course Description
(I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH250A
Overview
Subject: Math
Course Number: 250A
Department: Mathematics
Course Level: Graduate
Course Title
Groups, Rings, and Fields
Course Description
Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH250B
Overview
Subject: Math
Course Number: 250B
Department: Mathematics
Course Level: Graduate
Course Title
Commutative Algebra
Course Description
Development of the main tools of commutative and homological algebra applicable to algebraic geometry, number theory and combinatorics.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH251
Overview
Subject: Math
Course Number: 251
Department: Mathematics
Course Level: Graduate
Course Title
Ring Theory
Course Description
Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH252
Overview
Subject: Math
Course Number: 252
Department: Mathematics
Course Level: Graduate
Course Title
Representation Theory
Course Description
Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH253
Overview
Subject: Math
Course Number: 253
Department: Mathematics
Course Level: Graduate
Course Title
Homological Algebra
Course Description
Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH254A
Overview
Subject: Math
Course Number: 254A
Department: Mathematics
Course Level: Graduate
Course Title
Number Theory
Course Description
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH254B
Overview
Subject: Math
Course Number: 254B
Department: Mathematics
Course Level: Graduate
Course Title
Number Theory
Course Description
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH255
Overview
Subject: Math
Course Number: 255
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Curves
Course Description
Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH256A
Overview
Subject: Math
Course Number: 256A
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Geometry
Course Description
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH256B
Overview
Subject: Math
Course Number: 256B
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Geometry
Course Description
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH257
Overview
Subject: Math
Course Number: 257
Department: Mathematics
Course Level: Graduate
Course Title
Group Theory
Course Description
Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohomological methods.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH261A
Overview
Subject: Math
Course Number: 261A
Department: Mathematics
Course Level: Graduate
Course Title
Lie Groups
Course Description
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH261B
Overview
Subject: Math
Course Number: 261B
Department: Mathematics
Course Level: Graduate
Course Title
Lie Groups
Course Description
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH274
Overview
Subject: Math
Course Number: 274
Department: Mathematics
Course Level: Graduate
Course Title
Topics in Algebra
Course Description
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Units
Minimum Units: 4
Maximum Units: 4