MATH808K-0101: Selected Topics in Algebra; ALGEBRAIC K-THEORY-Spring 2021 jmr

Algebraic K-theory is a blend of linear algebra, homological algebra, algebraic geometry, and algebraic topology.  It has important applications in number theory, geometric topology, algebraic geometry, ring theory, and functional analysis.  There is no way to cover all of these topics in one semester, but I hope to give an introduction to the subject that will explain some of the historical roots and motivation, and some of the key tools

The subject of algebraic K-theory evolved in the second half of the twentieth century, largely through the efforts of a number of remarkable mathematicians, notably Grothendieck, Bass, Borel, Milnor, Quillen, Suslin, and Voevodsky.  (This is a very incomplete list of contributors, but the fact that almost all of these were Fields Medal winners should give you an idea of the level of the mathematics.)

This is a graduate topics course and there will be no exams.  However, I will periodically assign problem sets.  If you are taking the course for credit, I expect you to at least make an effort to work on them.  Course meetings will be synchronous, on Zoom, MWF at 10, but I'll record the lectures in case you want to review any ones that you missed.  Please keep your camera on during class if at all possible; it's much easier lecturing if I can "see" the audience.  See the Zoom tab at the left.  There may be a few times during the semester when there will be a prerecorded video in place of a live session.  The lecture recordings will be in the Panopto Recordings tab on the left, and course notes will be in the Files tab.  The schedule of topics will (eventually) show up in the course calendar.

Textbook:

The K-Book Links to an external site. by Charles Weibel Links to an external site., published by the American Mathematical Society.

This is the most recent and complete comprehensive textbook on the subject.  It is basically a third-generation text.  The first- and second-generation books are still useful, so let me list them:

• First generation: H. Bass, Algebraic K-theory Links to an external site., W. A. Benjamin, Inc., New York-Amsterdam 1968.
• First generation: J. Milnor, Introduction to Algebraic K-Theory Links to an external site., Annals of Math. Studies, vol. 72, Princeton Univ. Press, 1971.
• Second generation: V. Srinivas, Algebraic K-theory Links to an external site., Birkhäuser, Boston, 1991.
  
• Second generation: J. Rosenberg (that's me), Algebraic K-theory and its Applications Links to an external site., Springer, New York, 1994, corrected second printing, 1996.

Please fill out the course evaluation questionnaire at CourseEvalUM during the evaluation period April 29 -- May 12.

Contact information:

Jonathan Rosenberg Links to an external site., office: 2114 of the Mathematics Building, and university phone extension:  55166. (Calling from outside the university, call 301-405-5166.) Since I don't expect to be on campus often, email is usually better.  I will be available on Zoom right after class or by appointment, and available by email for "electronic office hours" at any time at jmr@math.umd.edu.