Milnor K-Theory    Math 9413a    Fall 2011

What has a Möbius strip to do with non-unique factorizations in rings of algebraic numbers? How many linearly independent tangent vector fields exist on the sphere Sn? For a field F, what is the smallest number of squares such that their sum is -1? What are the possible Galois groups over a given field? What are all of the possible division algebras over a given field? What is the connection between the distribution of primes and zeta functions?

Course Outline:

These are all natural, tantalizing questions discussed in cafes, on walks, at parties, and on soccer breaks. They are connected with ideas in K-theory.

Amazingly, some of these questions have been solved, and in some of the other questions, spectacular progress has been made. We are living in awesome mathematical times, and I would like to share them with you.

In this course only a basic, solid background in linear algebra is assumed, so that the members of this class will not fall from their chairs, surprised that vector spaces can be considered over all fields; not necessarily just real or complex numbers. Stress will be given to K0 and K2, as well as to basic Milnor K-theory. Connections with Witt rings of quadratic forms, representation rings, Brauer groups, and cohomology and Galois groups, will be explored. The Bloch-Kato conjecture and its various applications will be discussed.

Instructor:Ján Mináč
E-mail:minac@uwo.ca
Office:Middlesex College, room 131
Office Phone:519 661-2111 x86519.
Office Hours:Will be discussed in class.
Class times and  
location:To be decided in Department of Mathematics meeting, September 8, 2011.
Prerequisites:Good knowledge of undergraduate algebra.
Evaluation:Will be discussed in class.

Text:

An Algebraic Introduction to K-Theory, by Bruce A. Magurn, Cambridge University Press, 2009. This book will be available at the campus bookstore.

Scholastic offences:   Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following Web site: http://www.uwo.ca/univsec/handbook/appeals/scholastic_discipline_grad.pdf


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