Abstract:

Abstract: Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Abstract: Here is a fun question: what has four points and the weak homotopy type of the circle? If you enjoyed this one, here is a generalization: what has 2n+2 points and the weak homotopy type of the n-sphere? If you solved this one too, then it might come as no surprise to you that finite spaces present all finite homotopy types, which is to say that for every finite simplicial complex there is a finite topological space weakly equivalent to it, and for every finite space, there is a finite simplicial complex weakly equivalent to it. This arguably surprising result was proven by McCord in 1966.

In joint work with Daniel Carranza (Johns Hopkins University), we revisit McCord's theorem through the lenses of abstract homotopy theory and give a new proof of his result. The key fact used in our proof is the fact that a pushout of two open embeddings is a homotopy pushout.

In the talk, I will present the new proof of McCord's theorem. No background in abstract homotopy theory will be assumed.