MATH 416: Survey of Algebra
|Description: This survey course begins with an overview of the classification of finite simple groups, focusing on theorems of Burnside, Sylow, and Jordan–Holder. Rings and algebras are studied next, from basic noncommutative theory to the study of modules over PIDs, with applications to the classification of abelian groups and matrix canonical forms. The survey concludes with another thread in the modern theory. Topic chosen from, e.g., category theory, Wedderburn–Artin theory, bilinear forms and the classical groups, homological algebra, theory of division rings, and representation theory.
|Outcomes: Upon completion of the course, students should demonstrate: facility with the standard proof techniques in abstract algebra; ability to work with the axioms governing algebraic structures (such as actions, morphisms and quotients, both abstractly and in specific examples); ability to formulate conjectures from worked examples, and analyze them critically by appealing to results from the course. Finally, students will recognize the common themes of classification and application uniting the topics in this survey.