Skeleta of affine hypersurfaces

A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation T\u394 of its Newton polytope \u394, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S

Isoperimetric Inequalities using Varopoulos Transport

In this dissertation we obtain upper bounds of second order Dehn functions of lattices of the 3-dimensional geometries Nil and Sol using a variation of the Varopoulos transport argument and handle body diagrams by Buoncristiano, Roarke and Sanderson

Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes

For a log scheme locally of finite type over C, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over C, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over C, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite \'etale homotopy type of its infinite root stack