A matroid is a set \(M\) with a collection of subsets \(C\) called
"circuits" which satisfy two properties:
1) No circuit is contained in another circuit,
2) If an element \(m\) lives in two circuits \(C_1\), \(C_2\), then the complement of m in the union of \(C_1\) and \(C_2\) contains a circuit.
This probably sounds a little weird.
The idea behind a matroid is to abstract the properties that are satisfied by the columns of a matrix. In particular, the columns are the elements of a matroid, and the circuits are minimal collections of columns which are linearly dependent. Many other objects in mathematics end up forming a matroid. For example, if we take any graph \(G\) then the set of edges form a matroid with circuits given by the simple cycles in the graph.
In this project we'll learn some basic matroid theory, and then set to work computing some recently defined and quite interesting invariants of matroids inspired from algebraic geometry.
1) No circuit is contained in another circuit,
2) If an element \(m\) lives in two circuits \(C_1\), \(C_2\), then the complement of m in the union of \(C_1\) and \(C_2\) contains a circuit.
This probably sounds a little weird.
The idea behind a matroid is to abstract the properties that are satisfied by the columns of a matrix. In particular, the columns are the elements of a matroid, and the circuits are minimal collections of columns which are linearly dependent. Many other objects in mathematics end up forming a matroid. For example, if we take any graph \(G\) then the set of edges form a matroid with circuits given by the simple cycles in the graph.
In this project we'll learn some basic matroid theory, and then set to work computing some recently defined and quite interesting invariants of matroids inspired from algebraic geometry.
Comments
Post a Comment