Search Results

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner tha …

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is …

Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I wou …

It is somewhat jarring to hear of people who "know nothing about the homology theories of topological spaces and their applications" but are "familiar with homological algebra, category theory, spectr …

Here are three vague theorems rolled up in one. Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) …

Let me answer the broad question first: depending on what you actually want to do, the barcode-type invariants extracted by topological data analysis could be quite useful in your work. And it doesn't …

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is know …

Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the op …

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question: Who was the first to prove the Nerve Theorem?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \h …

The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a …

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every possi …

Update: There is a now a new paper by Otter et al which went through all the trouble of comparing many software packages for performance, memory, ease-of-use etc.: http://arxiv.org/abs/1506.08903 Thi …

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a maxima …

I realize that this is an old question and it is likely that the OP has moved on, but let me summarize the state-of-the-art as it exists now. The given sizes (18 million by 15 million) are pretty h …