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!

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.

Given a finite group $G$ and action $\sigma$ of $G$ on a finite set $S$, the $G-\sigma$ circulant ring is the ring of $S \times S$ matrices invariant under the induced action of $G$ on $S \times S$ and hence on $M_{S,S}$. We will see that this provides a generalization not only of circulant and $G$ circulant matrices but also the recently defined notion of a join of group rings. This provides a new approach to investigating an important family of matrices which have found many multidisciplinary applications. Several examples of generalizations to this context of properties of join rings will be given.

By work of Almkvist and Campbell, for a commutative ring R, the ring of big Witt vectors on R, W(R), is a completion of the Grothendieck ring of the category of endomorphisms of finitely generated projectives over R. On the other hand, the usual operations on Witt vectors, like Verschiebung and Frobenius can be defined at a categorical level. We extend the categorical operations of Verschiebung and Frobenius to the case where R is a non-commutative ring and show that these satisfy some expected properties. We also show that these play well with the iterated traces of endomorphisms, or ghost maps. Our work is inspired by the definition of Witt vectors in the non-commutative setting by Dotto, Krause, Nikolaus, and Patchkoria. This is Joint work with Agarwal, Campbell, Ponto, and Sun.

continuation from previous week