Mathematical Economics

We have been further occupied with the development of preparatory notions for the Brouwer's FPT. Those involved the topological notion of retraction. We have provided examples and counter examples. We have been examining the No Retraction Theorem-Borsuk's Lemma. We provided with a sketch of the proof using the concept of singular homology. Obviously this sketch of proof is out of the scope of the course. Using the latter theorem and the other preparatory notions we have proven the Brouwer Fixed Point Theorem. We have briefly examined an application in linear algebra, namely the Perron-Frobenius Theorem (the proof of which is an optional exercise). You can find notes for them here and here. (Notice that the construction of a retraction from a set in a Euclidean space to a non empty compact and convex subset of it involved the convexity of the Euclidean norm. A further detail is also used that essentially implies that if the objective function in the definition of the retraction has more than one minimizers then those would constitute a convex set that would essentially lie on the boundary of an Euclidean closed balls. But Euclidean spheres have only trivial convex subsets).