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Results tagged with gt.geometric-topology
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user 1358
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
6
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3
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Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\...
I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}_ …
- 22.8k
9
votes
2
answers
679
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Examples of automorphic forms over $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$
If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a sui …
- 22.8k
22
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3
answers
3k
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What is the "serious" name for the topograph (for a quadratic form)
One way to study (mixed signature) quadratic forms in two variables is to study the topograph. Looks like the signature doesn't matter: here is (1,1) and (1-,1).
The name is derived from τοποσ (Gree …
- 22.8k
3
votes
0
answers
143
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Seifert-Fibered 3-Manifolds and Rotation Numbers
I was trying to understand how the "ziggurats" come about in the paper by Calegari and Walker.
Motivating Question Given a free group F, and an element w of F, and given
values of the rotation …
- 22.8k
5
votes
1
answer
352
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Examples of Morse functions on unit tangent bundle of the sphere $T^1(S^2)$
In a sense, calculus is all about the study of critical points of functions on flat space $\mathbb{R}^N$ (e.g. here). Let's try a different venue, the unit tangent bundle of the sphere.
$$ T^1(S^2) …
- 22.8k
0
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0
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110
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Qualitative Solution of PDE on the 2-sphere (for weather prediction)
While I was watching the news last month I realized the weather report was basically a discussion of solutions to PDE. In particular, I was paying attention to the hurricane season (which is not yet o …
- 22.8k
2
votes
1
answer
219
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examples of surface diffeomorphism that exhibit heteroclinic bifurcation?
I was reading about horseshoes and heterclinic bifurcation but my knowledge of dynamical systems is really old fashioned.
as I understand the local stable manifold and the local unstable manifold in …
- 22.8k
0
votes
0
answers
57
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Mathematical Definition of $n$-Brouillin Zone [duplicate]
I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to …
- 22.8k
2
votes
0
answers
157
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Gaussian Integrals and Pseudo-Anosov Maps
The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated.
Here I take from: Asp …
- 22.8k
2
votes
2
answers
588
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Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)
In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff …
- 22.8k
9
votes
4
answers
1k
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Geometry of the space of circles in the Euclidean plane
We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula …
- 22.8k
1
vote
0
answers
121
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square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces
I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant: …
- 22.8k
11
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2
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Thurston-Cannon $S^2$-filling curves
I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers:
Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry
Gro …
- 22.8k
2
votes
1
answer
164
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the space of noncrossing partitions of S^1
A non-crossing partition of the set $\mathbb{Z}_{mn}=\{ 1, 2,\dots,m n\}=\bigcup X_i$ is a disjoint union of sets such that
if $ a < b \in X_1$ and $ c < d \in X_2$ then we can't have $a < c < b < …
- 22.8k
0
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1
answer
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Can vector fields be used to construct diffeomorphisms of the 2-sphere? [closed]
For some reason, today I want to understand better the group of diffeomorphisms of the 2-sphere, $S^2$.
After a few minutes I found this result by Smale in 1958.
The space $\Omega$ of all orien …
- 22.8k