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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).
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 …
- 25.7k
13
votes
2
answers
791
views
"C choose k" where C is topological space
One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum_{l \geq 0} \binom{ …
- 1,446
15
votes
1
answer
2k
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How to get 3-manifold, Knots from Number Fields
I'm reading a paper On the Torsion Jacquet-Langlands correspondence by Akshay Venkatesh and Frank Calegari.
Truthfully speaking I have no idea what Jacquet-Landlands is. I'm just trying to underst …
- 28.2k
9
votes
2
answers
679
views
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 …
6
votes
3
answers
867
views
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}_ …
- 5,259
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
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
views
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) …
- 60.6k
0
votes
0
answers
110
views
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
13
votes
1
answer
2k
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Is there a topograph for Pythagorean triples?
I have been reading Allen Hatcher's notes on quadratic forms. Naturally, we draw a picture encoding all the values of a quadratic form in a topograph. These are build by iterating the parallelogram …
- 1,228
5
votes
0
answers
271
views
deformed Gauss Bonnet formula?
I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss …
- 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 …
- 15.4k
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 …
- 437