Math 675R: Special Topics in Algebra.
Special Topics in Algebra.
Advanced topics in algebra drawn from pure and applied mathematics. Possible topics include: representation theory, Lie groups and Lie algebras, geometric group theory, Galois theory, algebraic number theory, computational algebra, Category theory, Grobner bases, algebraic geometry, algebraic combinatorics, cryptanalysis, finite group theory, modular forms, commutative algebra, homological algebra, group cohomology, character theory of finite groups, mathematical physics, ring theory.
Desired Learning Outcomes
Students should gain a familiarity with a particular area of algebra selected by the instructor.
Minimal learning outcomes
Minimal learning outcomes cannot be specified for a course in which topics will vary from year to year. However, regardless of the topic, successful students will know terminology, statements and approaches to problems undergoing active research, and major results in the area and techniques used to prove them. Students will demonstrate this knowledge by working suitable problems and developing their own proofs, by presenting and writing work inside and outside of class, and participating in other activities expected of more advanced graduate students.
Possible textbooks for this course include (but are not limited to):
Past topics chosen include:
- Representations of algebras and finite groups, using the book by Curtis and Reiner.
- Modular forms using a book by Kilford "Modular forms: a classical and computational introduction."
- Lie algebras, using a book by J Humphreys.
- Lie groups, using the book "Lie groups, Lie algberas and representations", by Brian C. Hall.