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answered Solving algebraic problems with topology
Feb
2
comment Hyperbolic 3-manifold groups acting on the plane
Good point, I misread the question as faithful.
Jan
29
comment What does the representation category of the knot group know?
Let's simplify to consider just the representation category of the fundamental group (this suffices to distinguish prime knots). Let me ask you whether this category can recover the finite quotients of the group? For some knots, such as the figure 8 knot, the finite quotients of the fundamental group determine the knot. Each representation factoring through a finite group $G$ should give a sub-category $Rep(G)$ . If one could recover the finite group from this sub-fusion category, then maybe one could recover the finite quotients from the representation category?
Jan
29
comment How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
@ChristianElsholtz, do you think the asymptotics with distinct denominators is comparable to allowing repeats like in your and Sandor's papers?
Jan
28
comment How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
See also: oeis.org/A006585
Jan
28
comment How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
See this paper with upper and lower bounds: link.springer.com/article/10.1134%2FS0001434614010295
Jan
15
comment Rational cohomology of the Rosenfeld projective planes
By frameable, do you mean parallelizable?
Jan
13
comment On the positive isotropic curvature in higher dimensions
I'll point out the trivial observation that since Micallef-Moore proved that such a manifold is homeomorphic to the sphere, one may conclude in certain dimensions that it is diffeomorphic to the standard sphere (e.g. dims. 5,6,12) since there is a unique smooth structure. So your question is interesting for $n\geq 7$. en.wikipedia.org/wiki/Exotic_sphere
Jan
12
answered In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?
Jan
12
comment Knot invariants which arise as the solution of differential equations along the knot
The Kontsevich integral is obtained by integrating certain differentials along level sets of the knot: mathworld.wolfram.com/KontsevichIntegral.html
Jan
5
comment Relations between some works by Deligne-Mostow and Thurston
I don't know Thurston's precise motivation, but I was at Davis when he was revising this paper (I don't have a copy of the original, which had fewer examples than the final version). Part of the motivation of Thurston came from finding parameterizations of the set of triangulations of a sphere where all vertices have degree ≤6≤6. In turn, I think this was motivated by his investigation of triangulations of 3-manifolds in which all the vertex links were sphere triangulations of this sort (see sciencedirect.com/science/article/pii/S0040938302001003).
Dec
29
comment In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?
On p. 21 of Polthier's thesis, equation (2.1), you'll find the explicit minimal surface equation in the upper half-space model. I just plugged the formula for an ellipsoid into this equation in Mathematica (I'm too lazy to do the computation), and found that the only solutions are for hemispheres.
Dec
26
awarded  Guru
Dec
24
awarded  Nice Answer
Dec
22
awarded  Enlightened
Dec
22
awarded  Nice Answer
Dec
19
comment Estimate for the first eigenvalue of the Laplacian
Presumably you mean minimal hypersurface? If so, it's false for immersed minimal hypersurfaces: consider the Clifford torus in $S^3$. Cyclic covers of this will be immersed with arbitrarily small first eigenvalue.
Dec
15
answered Volume form on a hyperbolic manifold with geodesic boundary
Dec
12
awarded  Nice Answer
Nov
30
answered What are the implications of the simple loop conjecture?