Algebra studies the structure of sets with operations, such as integers with addition and multiplication, or vector spaces with linear maps. The abstract point of view, based on an axiomatic approach, reveals many deep ideas behind seemingly innocent structures—such as the arithmetic of counting numbers—and serves as an elegant organizing tool for the vast universe of modern algebra. Generations of brilliant minds have crystallized these ideas in the ideas in the concept of groups, rings, fields, modules and their quotient structures and homomorphisms. The main goal in MATH 850 is the study of advanced topics and applications of algebra.
Evaluation of Students
The students will be evaluated based on frequent homework assignments, midterm and final exam.
The graduate algebra course will build upon the two Modern Algebra courses and will not go back to the study of groups. Its starting point will be a 3 - 4 week review of ring and module theory. The instructor will be free to extend this review to other material if it is necessary. Based on this background we suggest to treat a well-established area of algebra in good detail. Here are a few suggestions:
1. Commutative Algebra and Algebraic Geometry:
Properties of prime and maximal ideals, radicals, primary ideals. Localization. Primary decomposition. Integral closure. Noetherian and Artinian rings. Varieties. Morphisms. Nullstellensatz. All these topics could be treated with a computational approach.
2. Homological Algebra:
Tensor products of modules. Exterior and symmetric products at least in the vector space setting. Complexes, exactness properies of hom and tensor. Projective and injective modules. Syzygies and free resolutions.
3. Representation Theory:
Representation theory of finite groups: Maschke's theorem, Schur's lemma. Tensor, exterior and symmetric product of vector spaces and representations. Representation theory of the symmetric group. Representation theory of the general linear group. The topics are not restricted to the above. For instance, algebraic number theory with a computational emphasis could be taught, as well as invariant theory.
Textbooks & Software
D. S. Dummit & R. M. Foote, Abstract Algebra, Wiley.
Submitted by: Matthias Beck, Joseph Gubeladze and Serkan Hosten
Date: 29 October 2006