Impact
~8k
people reached
 0 posts edited
 0 helpful flags
 78 votes cast
18h

comment 
Good book on Calculus of Variations
Surely Giaquinta&Hildebrandts represents the Bible, but there are more slender yet excellent references aroud. E.g., I have found very pleasant Bruce van Brunt's (amazon.com/CalculusVariationsUniversitextBruceBrunt/dp/…), since "Enlightening explanations and building sound heuristics and intuition based on carefully chosen (classical) examples and exercises; simplicity and clarity of the exposition. But there is a price to pay: The mathematics is approached rigorously, but the level of rigor and details may not satisfy the purists among the mathematicians." [MR review] 
Nov
6 
comment 
Is the Veronese variety “enough” to describe all the $SL(V)$orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
@DavidSpeyer Nice remark! However, me neither ever believed in this 'simple description' of the orbits: it is just an idealistic goal I used to clarify what I had in mind. What I'm really interested about is to get to know methods for producing invariant subsets out of the Veronese variety, which 'contain as few orbits as possible'. For instance, these infinite orbits of yours, where do they sit? Certainly outside the Veronese $v_4(\mathbb{P}^1)$, which is made of polynomials with quadruple roots. But how they interact with the tangent/osculating variety? Are they inside/outside/between them?? 
Nov
6 
revised 
Is the Veronese variety “enough” to describe all the $SL(V)$orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
Fixed a minor detail about the understanding of osculating varieties 
Nov
6 
asked  Is the Veronese variety “enough” to describe all the $SL(V)$orbits in $\mathbb{P}(\textrm{Sym}^dV)$? 
Oct
29 
comment 
Algorithm to generate hyperbolic metric on a compact surface
@Cusp Understood. So, there is a $4g$sided polygon $P_g$ in $\mathbb{H}$ such that the projection to $F$ is onetoone in the interior and identifies the sides. By joining in the middle the sides labeled by the same letter, you'll get $2g$ geodesics, and $\pi_1(F)$ should consist of discrete translations along them (I guess), which probably are easily described by the exponential map. I have played a lot with hyperbolic polygons in Wolfram Mathematica and I know there are a lot of packages for this purpose, though it was quicker for me to write simple routines from scratch! 
Oct
28 
comment 
Algorithm to generate hyperbolic metric on a compact surface
What do you mean by "a hyperbolic metric on $F$ corresponds to a discrete faithful representation of $\pi_1(F)$ into $PSL_{2}(\mathbb{R})$"? I'm not familiar with this correspondence: does it requires passing to the universal covering of $F$ and acting on it by $\pi_1(F)$? 
Oct
28 
comment 
geometric conditions on maps between manifolds inducing monomorphisms on cohomology
@JasonStarr I agree with you: one way to have $f^*$ injective, is to require it to admit a left inverse, which is the same as requiring $f$ to admit, up to homotopy, a right inverse, i.e., a section. To see it more geometrically, one may work with the graph $\Gamma_f$ of $f$ inside the cylinder $M\times N$: then if there exists a deformation of the cylinder which makes $\Gamma_f$ projecting onto the second factor, your $f^*$ is injective. 
Oct
26 
revised 
Maps between products of symmetric powers
Adjusted one part of my answer 
Oct
26 
answered  Maps between products of symmetric powers 
Oct
22 
comment 
Homotopy classes of homeomorphisms of a multiple pointed space
If there are no obstruction to the existence of a (non relative) isotopy connecting $h$ with the identity, then I guess that such an isotopy can be made into a relative to $P$ one. A relative isotopy is a special map from the cylinder $M\times [0,1]$ to $M$, sending each segment $\{p_i\}\times[0,1]$ to $p_i$, and I don't see why not a non relative one could not be suitably deformed to fulfill such a property. But this is just a guess. 
Oct
20 
accepted  Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$? 
Oct
19 
revised 
Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?
minor misprint fixed 
Oct
19 
comment 
Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?
I need time to figure out the complex case, but the rest is crystalclear! Do you have some quick intuition/reference about (null) geodesics in the real case? I would expect that a pair of curves $(P(t),\pi(t))$ defines a null geodesic iff $\pi(t)$ is somehow related to (one of) the osculating space(s) to $P(t)$, but without familiarity with this stuff I can just guess. I suspect that some classical constructions like tangent, osculating, secant, dual varieties, can be recast in geodesic terms (or higher order conditions): I'm sure much has been done, but I don't know exactly what to look for! 
Oct
19 
awarded  Nice Answer 
Oct
19 
asked  Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$? 
Sep
30 
comment 
what's the minimal embedding of orthogonal grassmannian
I posted a similar question, but in the Lagrangian context. I'm sure that minor modifications will allow you to fit the answer(s) I got to the orthogonal context you're interested in. See: mathoverflow.net/questions/209058/… 
Sep
30 
comment 
what's the minimal embedding of orthogonal grassmannian
I also believe the case $k=2r$ is an exceptional one. In fact, I asked the very same question posted above, but in the skewsymmetric (i.e., symplectic/Lagrangian) context, and the answer turned out to be a little bit trickier than the easy "half dimensional subspace" guess. See: mathoverflow.net/questions/209058/… 
Sep
25 
comment 
Relative invariants of $P\oplus P^*$
Let me get straight: $P$ is the Lie algebra ${\frak{sl}}(V)$ of traceless endomorphisms, and $S^k(P)$ is regarded as a subspace of the universal enveloping algebra $U(P)$, right? Then, since a nonzero Casimir commutes with $P$, it is fixed by the action of the Lie group $\mathrm{SL}(V)$ and then it spans an irreducible onedimensional submodule, did I get it correctly? So far, you have just spotted some onedimensional irreducible constituents in $S^k(P)$. But your way of understanding "prime" is not  I think  the one I meant in my question. 
Sep
23 
asked  Relative invariants of $P\oplus P^*$ 
Sep
23 
comment 
Can the conformal structure on the projective lightcone detect hyperplane sections?
@VítTuček: yes, you're right. I've contacted David Calderbank on this concern and he confirms that the above necessary condition wrote down by Willie Wong is also a necessary one... neat indeed! 