By Casazza P. G.

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This text discusses a few tools of describing and relating mathematical gadgets and of regularly and unambiguously signaling the logical constitution of mathematical arguments.

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3 Let M = M(E) be the oriented matroid of a vector configuration in W. Show that we get all the covectors of M from the linear hyperplanes in R', by considering which vectors of E lie "on the positive side" respectively "on the negative side" of the hyperplane. 4 Consider the vertex set E of a regular 3-polytope (the icosahedron, say). What is the relationship between the oriented matroids that are obtained if E is considered as (a) an affine point configuration (of rank 4), (b) a vector configuration (of rank 3)?

14 Prove that the little and the big oriented matroids of an allowable sequence are oriented matroids. 4 is equivalent to the statement that the intersection of the interiors of the hemispheres Se e E X, is empty. 16 Let t1 < t2 < ... , ... , t2 -1) E A"'-1 i = 1, ... , n}. Determine the : chirotope, all circuits and all cocircuits of M. 17 Using oriented matroid duality, list all combinatorial types of simplicial 4-polytopes with seven vertices. In each case, determine the number of edges, triangles and facets.

5. Realizability 23 Note that it is easy to determine the facet triangles from this list: they are exactly the complements of positive (or negative) cocircuits. A realization of RS(8) is a labeled subset {xi, ... , sign of determinant) equals X(i, j, k, l) for all 1 < i < j < k < 1 < 8. We interpret the xi as homogeneous coordinate vectors of points in affine 3-space, and we list the xi as the column vectors of a real 4 x 8-matrix X. We will now show that the above sign patterns describe an abstract configuration which cannot be globally realized.

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Complemented subspaces of A, H1, and H by Casazza P. G.

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