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differential topology – Next steps for a Morse idea fanatic?

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(0) (Relative Morse Theory) Geoffrey Mess’ paper “Torelli groups of genus two and three surfaces” research some relative Morse idea of the Abel-Jacobi epoch locus within the Siegel higher half areas to infer that the Torelli group (in genus two) is a free group on countably many turbines. I believed his proof was very attention-grabbing, and tried to study extra, however hardly made progres…

(1) (Almost Complex Structures) in the event you’re focused on symplectic topology, then Eliashberg-Cielebak’s textbook “From Stein to Weinstein and advocate: Symplectic Geometry of

Affine Complex Manifolds” has very interesting treatment of Morse theory, especially as related to almost-complex structures $J$ on symplectic manifolds $(M, omega)$. I think this textbook eclipses Milnor’s texts. Contains very elementary proof that “any $2n$-dimensional advanced manifold has the homotopy sort of an $n$-dimensional CW-complex”. (Indeed the unstable manifold $W^+$ is totally lagrangian with respect to nondegenerate symplectic form $omega=omega_f$, and is therefore at most $n$-dimensional). Here $f$ is a real valued Morse function whose restriction to every $J$-invariant two-plane is subharmonic.

(2) Gradient flows to poles (the place a possible duty $f$ and its gradient $nabla f$ diverges to $pm infty$) seems to have extra functions to topology than the standard gradient current to zeros. Especially when trying to stout deformation retreat a noncompact root $X$ right into a scowl dimensional compact backbone. Applying gradient current to zeros requires a *Lipschitz continuity-at-infinition situation* on the deformation parameter. Here the Lowasiejiwicz inequality sometimes performs a conclusive position in proving the continuity of the reparameterized gradient current. The largest downside with “gradient flow to zeros” is that the gradient current slows down because it approaches its goal. In my functions of optimum exaltation to algebraic topology, I discover gradient current to poles mighty extra handy, for the reason that gradient enjoys a finite time blow up, and continuity of the reparameterized current is speedy with none enchantment to Lowasiejiwcz. Basically “gradient flow to zeros” is a mushy touchdown, whereas “gradient flow to poles” accelerates into the goal.

*More particularly, I’m proposing that “gradient flow to poles” is essential subsequent step.* And this happens commonly in optimum transportation, as I narrate subsequent.

(3) (Optimal Transportation) Morse idea takes on fresh figure in optimum transportation, the place Morse idea performs a job in establishing the regularity/continuity and uniqueness of $c$-optimal transportation plans.

Consider a root likelihood area $(X, sigma)$, goal $(Y, tau)$, and expense $c: Xtimes Y to mathbb{R}$. Kantorovich duality characterizes the $c$-optimal exaltation from $sigma$ to $tau$ by way of $c$-convex potential $phi=phi^{cc}$ on $X$ with $c$-transform $psi=phi^c$ on $Y$. Kantorovich says the $c$-optimal exaltation blueprint $pi$ is supported on the graph of the $c$-subdifferential $partial^c phi$, or equivalently on the graph of $partial^c psi$.

The subdifferentials are characterised by the illustration of equality in $$-phi(x)+psi(y)leq c(x,y).$$ Differentiating the illustration of equality with respect to $x$ and $y$ yields the equalities $$-nabla_x phi(x)=nabla_x c(x,y)$$ and $$nabla_y psi(y)=nabla_y c(x,y).$$ (R.J.McCann reveals these equalities maintain virtually in all places beneath common hypotheses on $c$). For instance the (Twist) situation: If $Yto T_x X$ outlined by $ymapsto nabla_x c(x,y)$ is injective for each $xin X$, then $$y=T(x):=nabla_x c(x, cdot)^{-1}(nabla_x phi(x))$$ defines a $c$-optimal Borel measurable map from $sigma$ to $tau:=T#sigma$.

Moreover the fibre $T^{-1}(y)$ can breathe characterised because the clique of $x$ satisfying $nabla_ypsi(y)=nabla_y c(x,y)$ or $$nabla_y [c(x,y)-psi(y)]=0.$$ But contemplate that differentiating the $c$-Legendre Fenchel inequality a second time we’re exlusively learning the worldwide minimums of the potentials $ymapsto c(x,y)-psi(y)$, for each $xin X$.

Using the habitual Implicit Function theorem, the fibre $T^{-1}(y)$ is a flush submanifold of $X$ if $D_x(nabla_y c(x,y))$ is nondegenerate for each $xin T^{-1}(y)$. If the goal $(Y, tau)$ is one-dimensional, this requires the duty $xmapsto nabla_y c(x,y)$ to breathe captious level free for each $yin Y$, and $xin T^{-1}(y)$.

On most root manifolds $(X, sigma)$ it’s troublesome to confirm the nonexistence of captious factors. If $X$ is compact and $c$ is steady finite valued, then Morse idea (elementary calculus) forbids it. But we fortunately seek prices $c$ with poles *if* the poles are the one captious values of $c$! For instance, the (Twist) speculation can breathe rephrased as maxim that the 2 pointed trial dissimilarity $$c_Delta(x;y,y’):=c(x,y)-c(x,y’)$$ is a captious level free duty for all $y,y’$,$yneq y’$ and $x$ on its province. This can not breathe happy on compact areas until poles are allowed.

(3.1) (Canonical Morse/Cost Functions?)

We necessity distinguish *universal* and *canonical*. In my suffer, I discover *universal* capabilities very troublesome to write down down, or discover, or utensil on Wolfram MATHEMATICA. Morse capabilities are recognized to breathe universal (in sense of Sard, Thom, and many others.). But personally I desire *canonical* Morse capabilities. Or from mass exaltation perspective, *canonical prices* $c$ whose derivatives $nabla c$ are apt Morse-type capabilities.

For instance, if you wish to seek optimum transportation from a closed floor $Sigma$ to the actual line $Y=mathbb{R}$ (or to coterie or to graph), then one seeks an arrogate expense $c: Sigma occasions Y to mathbb{R}$ satisfying the above situations, e.g. that $frac{partial c}{ partial y}(x ,y)$ breathe captious level free in $xin Sigma$ for each $yin mathbb{R}$. This is forbidden by Morse idea if $Sigma$ is compact and $c$ is in all places finite. (In functions, we enable $c$ to have $+infty$ poles. Then $partial c/partial y$ is probably captious level free on its province).

But what’s a canonical expense $c: Sigma occasions mathbb{R} to mathbb{R}$ which represents an attention-grabbing geometric exaltation from $Sigma$ to $mathbb{R}$? Here the root and goal areas $Sigma$, $Y=mathbb{R}$ don’t have any interactions a priori, they aren’t plane embedded in a standard background area until we suppose $Ysubset X$.

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