Questions tagged [coarse-geometry]
The coarse-geometry tag has no usage guidance.
17
questions with no upvoted or accepted answers
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Topology is to semi-decidability, coarse structures are to what?
There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like:
The monograph Synthetic Topology: of Data Types and Classical ...
- 1,191
8
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195
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Coarse quotient maps
Interesting connections and analogies have been observed between
non-linear geometry of Banach spaces and coarse geometry.
In the former subject, people have investigated the notion of
uniform (or ...
- 9,631
8
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181
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Can two random graphs be metrically embedded into one another?
Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal ...
- 3,301
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Coarsifying persistence modules
The context
Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying:
For all $t$ in $I$ but a closed discrete set of points $T$, there exists a ...
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5
votes
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108
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Why are coarse maps required to be proper?
In the context of coarse spaces, a map between coarse spaces $f:X\to Y$ is called coarse if it is bornologous (it maps controlled sets to controlled sets), and proper, in the sense that preimages of ...
- 2,129
4
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109
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Alternative uniformities on topological groups
Are there any interesting alternative uniformities defined on topological groups besides the usual four (left, right, and their meet/join)? I am curious because in the (sort of) dual setting of coarse ...
- 3,756
4
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152
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Ends of a negatively curved Riemannian manifold
Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...
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4
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153
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Ends of a metric space?
I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse ...
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3
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50
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Dependence of Roe algebra and coarse index on the Riemannian metric
Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$.
I ...
- 1,871
3
votes
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126
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Comparing the group convolution algebra with the equivariant Roe algebra
Let $G$ be a Lie group equipped with a left-invariant metric. Then $C_c(G)$ is a $*$-algebra of convolution operators on $L^2(G)$.
Let $\mathbb{C}[|G|]^G$ denote the $*$-subalgebra of bounded ...
- 1,871
2
votes
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127
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Algebra of finite width matrices
$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
- 4,436
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166
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Characterization of growth in terms of coarse algebraic topology
$$
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mbb}[1]{\mathbb{#1}}
\newcommand{\opn}[1]{\operatorname{#1}}
\DeclareMathOperator\cap{cap}
\def\sse{\subseteq}
$$
Coarse spaces
Let $X$ be a coarse ...
- 1,294
1
vote
0
answers
73
views
Groups without "almost equivariant" coarse embeddings
Let $X$ be a set. We say that $\psi:X\times X\to[0,\infty)$ is a CND (conditionally negative definite) kernel if there is a Hilbert space $\mathcal{H}$ and a map $f:X\to\mathcal{H}$ such that
\begin{...
- 81
1
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61
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Consequences of having unbounded points in a bornology
For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...
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1
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91
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Question about coarse fixed point property in large-scale geometry
I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...
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59
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Can the sequence of complete graphs coarsely embed into Hilbert space?
Basically the title. If I have the metric space which is the disjoint union of the sequence of complete graphs, and the usual graph metric, has it been shown that the metric space can be coarsely ...
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237
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The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation
Let $X$ be a coarse space, we define the following:
$D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$
$f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
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