Date: Wed, Nov 15, 2017
Time: 14:00 - 15:40
Venue: East Lower Hall 315
Title: Linear algebra without linear spaces: Ideas of Whitney and many others
Given a finite matrix, if we focus on the linear dependence relationship of its column vectors, we can eliminate many continuous parameters and distinguish a collection of subsets of those set of columns, thus seeing the combinatorial shadow of the matrix. Different viewpoints on the shadow lead us to different interesting objects, including matroids, oriented matroids and valuated matroids.
To have a good understanding of linear algebra and calculus, it is important to know what is an exterior algebra. The exterior algebra is the calculus for the join and meet of subspaces of a vector space and so is a linearization of the Boolean algebra of all subsets of a set. To understand a wide range of mathematical phenomena, it is important to develop dequantizations of classical mathematics; that is, we need to do linear algebra without linear spaces and we need to understand what is a matrix and how its combinatorial shadow will behave.
This talk is a brief introduction to matroid theory. We will tell you how the ideas of some mathematicians condense all trends of present day mathematics onto a single finite structure, a structure in the shadow of a matrix.
Slides: View slides