A group-theoretic interpretation of Tutte's homotopy theory

Homotopy-Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it while also having a purely intensional interpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of...

Motivic Homotopy Theory of Group Scheme Actions

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic K-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.

A More General Type Theoretic Interpretation Of Constructive Set Theory

Some Problems Arising from Homotopy-theoretic Methods in Knot Theory

This is a partial list of some interesting questions that arose in the past decade or so from applications of homotopy-theoretic methods in knot and link theory. The ones I have in mind (because those are the ones I am familiar with) are manifold calculus of functors, cosimplicial spaces, and operad actions. (Some problems are also about configuration space integrals as they have proven to be a...

Geometric and Homotopy Theoretic Methods Innielsen Coincidence Theory

In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs ( f1, f2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC( f1, f2) (and MC( f1, f2), resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are ( f1, f2)...