Problems in arithmetic topology

Some problems in topology

Broadly speaking, we may say that analysis situs, or topology, deals with the properties of geometrical figures that remain invariant when the figures are subjected to arbitrary continuous transformations. There are, however, several distinct kinds of analysis situs, because there are several distinct ways of interpreting the physical notion of continuity in mathematical language. For example, ...

Open Problems in Topology

Problems in Topology

1. Background In computer graphics and image processing a scene is often represented as an array of 0's and 1's. The set of 1's represents the object or objects in the scene and the set of 0's represents the background. The array is usually two-dimensional, but three-dimensional image arrays are produced by reconstruction from projections in applications such as computer tomography and electron...

Topology and arithmetic of resultants, I

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in P. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen–Cohen–Mann–Milgr...

Arithmetic and topology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both f...