Abstract: The A-groups of graphs, named after Atkin, are the key invariants in discrete homotopy theory. A natural problem is to identify the $\infty$-category obtained by localizing the category of graphs at maps inducing isomorphisms on A-groups. The discrete homotopy hypothesis (DHH), a major conjecture in the homotopy theory of graphs, asserts that the resulting $\infty$-category is equivalent to that of spaces.

What makes DHH interesting is that this conjecture is that it appears within reach, but not quite. On the one hand, there is substantial positive evidence, including Carranza-Kapulkin's recent work on a version of DHH for homotopy n-types. On the other hand, some basic questions, such as the existence of homotopy (co)limits of graphs, remain open.

In this talk, I will explain what additional ingredients are needed to deduce DHH from the evidence currently available. Remarkably, the existence of countable homotopy products would already suffice. In addition, I will sketch a possible approach for constructing such products.

Abstract: We show that the quotient of a complex projective toric variety by a finite group, generated by reflections of the associated lattice polytope, is another toric variety. This result generalizes results of Blume and of Gong on permutohedra, and positively answers a question of Horiguchi, Masuda, Shareshian, and Song. In a similar direction, we also study the topological quotient of the real points of a projective toric variety, and prove that the quotient is contractible in the permutohedron case.

Joint with Tao Gong and Connor Simpson.