Fall 2017 MATH 245A 001 LEC

General Theory of Algebraic Structures
Schedule: 
SectionDays/TimesLocationInstructorClass
001 LECMoWeFr 02:00PM - 02:59PMEvans 5George M Bergman32064
UnitsEnrollment Status
4Open
Additional Information: 

Prerequisites: The equivalent of one of Math H113, or 114, or 250A, or consent of the instructor.  (250A may be taken simultaneously.)

Syllabus: The theme of Math 245A, as I teach it, is universal constructions.  We begin with a well known case, free groups, which we will construct in three quite different ways, and show why these constructions come to the same thing.  We then examine a smorgasbord of other important universal constructions, noting similarities, differences, and general patterns.
     After that, we settle down to developing the tools needed to study the subject in a unified way:  Ordered sets and the axiom of choice, closure operators, category theory, and the general concept of a variety of algebras.  (We treat most of these, not merely as means to this goal, but as interesting landscapes worth lingering over.)  We find that the free group construction is a particular example of an adjoint functor (it is left adjoint to the underlying-set functor on groups), and eventually develop a marvellous result of Peter Freyd, characterizing the functors between varieties of algebras that have left adjoints.  We determine, in several cases, all such functors.

Office: 865 Evans Hall

Office Hours: TBA

Required Text: George M. Bergman, An Invitation to General Algebra and Universal Constructions, Springer Verlag, 2015; also available in a slightly different version here.

Recommended Reading: None.  (But some texts useful either for filling in background, or for pursuing some topics further, are listed in section 1.6 of the text.)

Homework and Grading: The text contains more interesting exercises than anyone could do! I will ask you to think briefly about each exercise, and select a few to work and hand in each week; more details at the first class meeting.  Grades will be based mainly on this homework.  Students wishing a reduced homework-load can enroll S/U.

Comments: My philosophy is that it does not make sense to spend the classroom time using the instructor and students as an animated copying machine.  Rather, the material that is typically delivered in a lecture should be put in duplicated notes which the students study, and class time should be devoted to the more human activities of discussing and clarifying the material, introducing some topics by the Socratic method, etc..  Such notes for Math 245, begun in Fall 1971 and reworked each time I taught the course, have evolved into the text we will use.
     There are difficulties with this way of teaching if a textbook leaves out motivation, examples, etc., that might be included in a lecture; but I have made it a point to include these in the notes.  Another problem is that doing the reading before class runs counter to the habits of many students.  To get around this, I require each student to submit, on each class day, a question about the day's reading.  If possible, these should be submitted by e-mail at least an hour before class, so that I can address some of the questions in class.