Reply #29: A brief, yet merciless, introduction to Motivic Cohomology [View All]

Some of the most significant developments in mathematics in the past year stem from a breakthrough achieved by Vladimir Voevodsky, and from work by Voevodsky and Andrei Suslin which builds on this breakthrough. Providing bridges across different areas of mathematics, this work constitutes a significant step toward resolving some questions that had eluded mathematicians for several decades. Voevodsky has been invited to present a plenary lecture about his work at the International Congress of Mathematicians, the most important meeting in the mathematical world which takes place every four years and will next be held in Berlin in August 1998.
On its most general level, the work of Voevodsky provides a new link between two circles of ideas in mathematics: the algebraic and the topological. The term algebra as used here refers to a much broader and deeper field than that studied by high school students. What mathematicians mean by algebra is, roughly speaking, a theory for studying the general structure of sets endowed with algebraic operations, like addition and multiplication of the integers {...-3, -2, -1, 0,1, 2, 3,...}.

There are many different kinds of algebraic objects, the most basic one being an abelian group. An abelian group is a set together with an operation on the elements of the set, where the operation has all the properties of addition of the integers. More complicated structures, such as commutative rings and fields, arise when one considers more than one operation and how the operations interact. The simplest example of a ring is the integers, together with the two operations of addition and multiplication. If one introduces a third operation, division, one obtains the rational numbers, i.e., fractions, and such a structure is formalized in the notion of a field. Other examples of rings are the sets of polynomials (in any number of variables) whose coefficients belong to a given field. For any finite set of polynomials one can look at the common zeros of those polynomials. This set of zeros is called the algebraic variety defined by the polynomials. Adding more polynomials will often cut down the size of the common zero set, yielding subvarieties of the algebraic variety. Subvarieties are sometimes referred to as algebraic cycles on the variety.

Two algebraic objects are said to be isomorphic if they have the same size and structure. This means there must be a way to match the elements of the two objects in a one-to-one correspondence so that the matching uses up all the elements, and so that the matching preserves the algebraic structure. This kind of matching is called an isomorphism. Two algebraic objects that are isomorphic are, from the algebraic viewpoint, exactly the same.

And if you liked that, you can find the whole enchilada on the web at http://www.ams.org/new-in-math/mathnews/motivic.html

--bkl
From a seminar given at the Yogi Berra Institute for Linguistics, Mathematics, Logic, and a Clean Baseball Business.