MATH 5020: Topics in Algebra
Description: Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory. With change of content, this course may be repeated to a maximum of twelve credits.
Prerequisites: MATH 5211.
MATH 5020 - Section 1: Enumerative and Algebraic Combinatorics (Roby)
Description: The course will give an introduction to enumerative and algebraic combinatorics at the graduate level. Topics will include: basic enumeration, generating functions, bijective proofs, q-analogues, sieve methods, theory of partially ordered sets, Moebius functions on posets, rational generating functions. Text: Enumerative Combinatorics 1 (2nd edition), by Richard Stanley.
Prerequisites: Math 5211 officially, but practically a student should have a solid back-ground in undergraduate abstract algebra, linear algebra, and analysis.
MATH 5020 - Section 2: Introduction to Commutative Algebra (Glaz)
Description: Commutative Algebra, the study of commutative rings and their modules, emerged as a definite area of mathematics at the beginning of the twentieth century. Its origins lie in the works of eminent mathematicians such as Kronecker, Dedekind, Hilbert, and Emy Noether who sought to develop a solid foundation for Number Theory. Later, the field was enriched by its relation to modern Algebraic Geometry, Topology, Homological Algebra, and Combinatorics. Today Commutative Algebra is a deep and beautiful area of study in its own right, which both draws on, and is applicable to, all the disciplines that contributed to its development. In this course we will study the fundamental notions and methods of research of commutative algebra. Topics will include: basic module and ideal notions and constructions (such as, prime ideals, zero-divisors, localizations, primary decomposition, inte- gral dependence, completions, and dimension theory), special types of rings (such as, valuation rings, Krull domains, Noetherian rings, Artinian rings, coherent rings), and homological al- gebra aspects of commutative algebra (such as, projectivity, flatness, grade, and factorization properties). Other topics may be introduced as time allows. We anticipate covering most of Chapters 1 - 11 from the main textbook, supplemented from the other texts. Textbooks: Main texbook: Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald. Supplementary texts: Commutative Algebra by Bourbaki, Commutative Coherent Rings by Sarah Glaz, Commutative Rings by Irving Kaplansky, and Commutative Ring Theory by Hideyuki Matsumura
Prerequisites: Math 5211 (more specifically, rings, ideals, and modules).
Sections: Fall 2014 on Storrs Campus
|05517||5020||001||Lecture||TuTh 2:00 PM-3:15 PM||MSB315||Roby, Thomas|
|09360||5020||002||Lecture||TuTh 12:30 PM-1:45 PM||MSB403||Glaz, Sarah|