Combinatorics/Algebra Seminar

September 25

Mitchell Ashburn: Isomorphisms of vector-matrix algebras.

Abstract:

We begin by considering anticommutative formed algebras, that is, anticommutative algebras with a distinguished bilinear form. Typically, we will consider bilinear forms that are symmetric, invariant, and possibly non-degenerate. The best examples of such formed algebras are Lie algebras along with their Killing forms. From these formed algebras, we then construct a family of non-associative algebras called vector-matrix algebras, which generalize Zorn's vector-matrix construction of the split octonions.

Isomorphisms of these vector-matrix algebras can be reduced to specific types of isotopies between their underlying formed algebras. Furthermore, beginning with one of these isotopies of formed algebras, we can actually construct the isomorphism between their vector-matrix algebras. We can then explore the properties of these isotopies, attempting to find under what conditions two formed algebras will produce isomorphic vector-matrix algebras. Specifically, we can say a bit about anticommutative algebras along with their Killing forms, before highlighting a few examples of specific Lie algebras with bilinear forms that have such isotopies.

September 18

Anna Romanowska (Warsaw University of Technology): Sparse barycentric coordinates for polytopes.

Abstract:

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc.

In this talk, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on the close relationship between affine and barycentric coordinates of points of polytopes, and a certain decomposition of each finite-dimensional polytope into a union of simplices of the same dimension. The talk is based on recent results obtained jointly with J.D.H. Smith and A. Zamojska-Dzienio.

September 11

Jonathan Smith: Barycentric algebras and their symmetry.

Abstract:

When algebra is applied to geometry, it is desirable for the symmetries of the algebraic structure to match the symmetries (local or global) of the geometric structure. Unfortunately, the neutral elements of traditional algebra structures such as groups and rings behave in a special, symmetry-breaking way. Thus, when attempting to apply rings to convex geometry, it often becomes necessary to restrict attention to convex cones in vector spaces.

In this talk, we will discuss the modeling of general convex sets using the non-traditional algebras known as barycentric algebras, which are based on the logic structure of open and closed unit intervals. Hitherto, there has been an asymmetry in the axiomatization of barycentric algebras. We will give a general introduction to barycentric algebras accessible at the advanced undergraduate level, and then present recent results which make their definition completely symmetric.

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