## Notes of Leonid Polterovich’ Orsay lecture 03-04-2018

Persistence modules and barcodes in symplectic geometry and spectral geometry

1. Hamiltonian diffeomorphisms

Arnold:“Symplectic topology has the same relation to ordinary topology as Hamiltonian systems have to general dynamical systems”.

Hofer’s length on Hamiltonian diffeomorphism groups ${Ham}$. A path is determined by a path of normalized functions ${F_t}$. Its length is

$\displaystyle \int_0^1 \|F_t\|_\infty \,dt.$

This defines a kind of biinvariant Finsler structure on ${Ham}$. The corresponding distance is nondegenerate (Hofer, Polterovich, Lalonde-McDuff). It is essentially the only one (Buhovsky-Ostrover). Existence of this metric is remarkable. It is remiscent of commutator norms on finitely generated groups.

Autonomous Hamiltonian diffeos (generated by time-independent Hamiltonians ${F}$) correspond to 1-parameter subgroups of ${Ham}$. They admit roots of any degree. T-Such flows conserve energy. They are geodesics in Hofer’s metric (but not the only geodesics).

${Ham}$ has interesting algebraic properties. It is algebraically simple. Let ${Powers_k}$ be the set of Hamiltonian diffeos admitting a root of order ${k}$. Is ${Powers_k}$ metrically dense in ${Ham}$ ? I.e. is

$\displaystyle p_k(M):=\sup_{\phi\in Ham} d(\phi,Powers_k)$

finite? We conjecture that this is never the case.

Theorem 1 (Polterovich-Shelukin) Let ${\Sigma}$ be a closed surface of genus ${\geq 4}$. Let ${M}$ be an arbitrary closed manifold with ${\pi_2(M)=0}$. Then, for ${k}$ large,

$\displaystyle p_k(\Sigma\times M)=+\infty.$

This has been improved since by Jun Zhang, Polterovich-Shelukin-Stojisavljevic.

Our tools are Floer theory and persistence modules and their barcodes.

2. An example

In 2 dimensions, autonomous Hamiltonian flows are integrable, i.e. deterministic. Thus we look for chaotic Ham diffeos. Like in an eggbeater, combine two integrable diffeos performing mere shear motions, but on intersecting annuli. As soon as 1992, physicist Franjione-Ottini studied such linked twist maps. A parameter ${\lambda}$ (strength of shear motion) is introduced. As ${\lambda}$ tends to ${+\infty}$, we show that the distance of resulting diffeo ${\phi_\lambda}$ to ${Power_k}$ tends to infinity.

We study periodic orbits in special free homotopy classes. Handles are needed to separate periodic. Our example fails on the 2-sphere, as shown by Khanevsky.

Our invariant survives stabilization by dimension: product with identity does the job.

3. Motivation

3.1. Dynamics

In dynamics, it has been known for a long time that vectorfields generate few diffeomorphisms (Palis, Brin 1973). In ${Ham}$, non autonomous Ham. diffeos contain a ${C^\infty}$-dense open set (Salamon-Zehnder, Ginzburg-Gurel), for symplectically aspherical manifolds. Our methods upgrade ${C^\infty}$ open to Hofer-open, and make it quantitative.

3.2. Coarse geometry of ${Ham}$

Polterovich-Rosen: a ${C^\infty}$-generic Hamiltonian generates a nondistorted 1-parameter subgroup, distance to identity grows linearly.

In fact, the only other known behaviour is boundedness.

When I proved that ${Ham}$ has infinite diameter (for surfaces), Misha Kapovitch asked me wether ${Ham}$ did lie in a bounded neighborhood of a quasigeodesic. Our main theorem shows that this does not happen for ${\Sigma\times M}$.

3.3. Milnor’s constraint

In 1983, Milnor observed that if a diffeo is a square, ${\phi=\psi\circ\psi}$, the number of primitive geometrically distinct 2-periodic orbits is even. Indeed, ${\psi}$ induces a ${{\mathbb Z}/2}$ action of such orbits.

Find more restrictions on powers.

4. Barcodes

Edelsbrunner, Harer, Carlsson, in the context of topological data analysis. Has developped into a very abstract subject.

A barcode is a finite collection of intervals ${I_j}$ with multplicities ${m_j}$. The bottleneck distance is defined as follows. Erase intervals of length ${<\delta}$ and match the remaining intervals up to error ${\delta}$. Then infimize ${\delta}$.

Given a field ${\mathbb{F}}$, a persistence module is a pair ${(V,\pi)}$ of finite dimensional ${\mathbb{F}}$-vectorspaces ${V_t}$, ${t\in{\mathbb R}}$, and maps ${\pi_{st}:V_s\rightarrow V_t}$, ${V_s=0}$ for ${s << 0}$. Commuting diagrams. One assumes regularity: for all but a finite number of jump points in ${{\mathbb R}}$, ${\pi_{st}}$ are isomorphisms, together with semicontinuity at jump points.

Interval module is the tautological 1-dimensional persistence module supported on an interval.

Structure theorem: every persistence module is associated with a unique barcode, as the sum of intervals modules of the intervals of the barcode.

4.1. Original example: Morse theory

${X}$ closed manifold, ${f:X\rightarrow{\mathbb R}}$ Morse function. Persistence module is

$\displaystyle V_t(f):=H_*(\{f

Inclusions induce persistence morphisms.

The following statement is a not that easy theorem.

Robustness: The map ${(C^\infty(X),\|.\|_\infty)\rightarrow (\{barcodes\}, d_{bot})}$ is Lipschitz.

It follows that one can define critical points of merely continuous functions.

4.2. Morse homology

I need be more specific with the homology theory I use.

Let ${f}$ be a Morse function on ${X}$ and ${\rho}$ a generic Riemannian metric. The Morse complex ${C_t}$ is spanned by the critical points of ${f}$ with value ${f(x). The differential counts the number ${n(x,y)}$ of gradient lines of ${f}$ connecting critical points,

$\displaystyle dx=\sum n(x,y)y.$

4.3. Floer theory

Born in 1988. In symplectic topology, the role of ${X}$ is played by infinite dimensional manifold ${LM}$ of contractible loops ${z:S^1\rightarrow M}$. Given a 1-periodic Hamiltonian ${F(x,t)}$, define action functional

$\displaystyle A_F(z)=\int_0^1 F(z(t),t)\,dt-\int_D\omega,$

where ${D}$ is an arbitrary disk spanning ${z}$.

The original action functional ${\int_D\omega}$ has only one critical point of infinite index and coindex. The perturbation ${A_F}$ is much more interesting, since its critical points correspond to 1-periodic orbits of the Hamiltonian flow generated by ${F}$.

The solutions of the gradient equation are pseudoholomorphic cylinders. Therefore (Gromov 1985) they constitute a Fredholm problem. Although the gradient equation has no local (in time) solutions, the boundary value problem of gradient connections of critical points is well-posed. Therefore, one gets a well-defined complex, and homology groups ${HF}$, this is Floer homology.

Under certain asumptions (asperical, atoroidal,…), the corresponding persistence module ${V=HF(\{A_F depends only on the time 1 map ${\phi\in Ham(M,\omega)}$.

One can extend the construction to noncontractible loops.

Theorem 2 (Polterovich-Shelukin) The map

$\displaystyle (Ham,d_{Hofer})\rightarrow(\{barcodes\},d_{bot})$

is Lipschitz.

Hence Lipschitz functions on barcodes yield numerical invariants of Ham diffeos. Powers give rise to ${{\mathbb Z}/p}$ representations on persistence modules. A Floer-Novikov variant has been developed by Usher-Zhang.

4.4. Representations on persistence modules

Since ${\phi \phi^2 \phi^{-1}=\phi^2}$, ${\phi}$ acts on the Floer homology of ${\phi^2}$, this is a ${{\mathbb Z}/2}$ action ${T}$. Define ${L(\phi)\subset HF(\phi^2)}$ be the ${-1}$-eigenspace of ${T}$. There is a corresponding persistence module. If ${\phi=\psi^2}$, then ${\psi}$ induces action ${S}$ on ${L\psi^2)}$. Then ${S^2=T=-1}$ on it. Thus the multiplicity of each bar in ${L(\psi^2}$ is even. This is reminiscent of Milnor’s constraint.

Observation. Distance to full squares is controlled by stable multiplicity, i.e. parity of the dimension of eigenspaces.

5. Barcodes for eigenfunctions on surfaces

Joint work with Iosif Polterovich and Vukasin-Stojisavljevic. Elaborates on 2006 work with Misha Sodin.

5.1. Oscillation

Question. Given a closed oriented surface, and a smooth Morse function ${f}$ on ${M}$, how can one define the oscillation of ${f}$?

The Banach indicatrix is defined as follows. Let ${\beta(c,f)}$ denote the number of connected components in ${f^{-1}(c)}$. Let

$\displaystyle I(f)=\int_{\mathbb R} \beta(c,f)\,dc.$

Goes back to Kronrod and Vitushkin in the 1950’s. In the 1980’s, Yomdin rediscovered it, with the idea that if derivatives of ${f}$ are not too large, ${I(f)}$ should be small.

Define the total length of the finite part ${B_*(f)}$ of the barcode as

$\displaystyle \Phi(f):=\sum_{J\in B_*(f)}length(J).$

The following is an easy fact from surface topology.

Theorem 3 (PPS 2018)

$\displaystyle \max f -\min f +\Phi(f)\leq I(f).$

5.2. Example

Consider the Reeb graph of ${f}$ (space of connected components of fibers). ${f}$ descends to a function on it. Then ${f}$ is equal to the total variation of ${f}$ on its Reeb graph. The difference with ${\max f -\min f}$ can be seen on the barcode.

6. Results on eigenfunctions

Main question. Bound oscillation via analytic properties of functions. Fix a Riemannian metric on surface ${M}$. Let ${\Delta}$ be the Laplacian. For ${\lambda>0}$, consider

$\displaystyle F_\lambda=\{f\in C^\infty(M)\,;\, \|f\|_2=1,\,\|\Delta f\|_2\leq \lambda\}.$

The study of the topology of eigenfunctions of the Laplacian has a long history. Richard Courant showed that the number of nodal domains (connected components of ${\phi\not=0}$) is ${\leq n+1}$ for the ${n}$-eigenfunction.

Theorem 4 (Polterovtch-Sodin 2006) For ${f\in F_\lambda}$,

$\displaystyle I(f)\leq k_g(\lambda+1).$

Corollary 5

$\displaystyle \max f-\min f+\Phi(f)\leq k_g(\lambda+1).$

6.1. Remarks

1. For Euclidean domains, for Dirichlet boundary values, Alexandrov-Backelman-Pucci-Cabre 1995 show that

$\displaystyle \max f -\min f \leq Const.\,\|\Delta f\|_2.$

2. On the square 2-torus, ${f_n(x,y)=\sin(nx)\sin(ny)}$ satisfy ${\|\Delta\|_2 \sim n^2}$, and barcodes ${B_*(f_n)}$ have ${\sim n^2}$ bars of length ${\sim 1}$. This is sharp.

3. Say a critical value ${\alpha}$ of ${f}$ is ${\delta}$-significant if it is an endpoint of a bar of length ${\geq \delta}$. From our theorem, it follows that

Corollary 6 If ${f\in F_\lambda}$, the number of ${\delta}$-significant critical values of ${f}$ is ${\leq k_g(\lambda+1)}$.

4. Nicolaescu has shown that the expectation of the number of critical points of random linear combinations of eigenfunctions is ${\sim\lambda}$.

6.2. Example

Let

$\displaystyle f_i(x)=\frac{1}{\sqrt{\pi(2+N_i^{-4})}}(1+\frac{1}{N_i^2}sin(N_i x))$

has a number of critical points that tends to infinity, whereas for every fixed ${\delta>0}$, the number of ${\delta}$-significant critical values stays bounded.

7. Approximation theory

Let ${f}$, ${g}$ be smooth functions on ${M}$. In ${C^0}$-norm, what is the best approximation of ${f}$ by functions of the form ${g\circ \phi}$, ${\phi\in Diff(M)}$?

Let ${B(f)}$ be the barcode of ${H_*(\{f. Then ${B(f)}$ is Diff-invariant. By robustness theorem,

$\displaystyle d_{bot}(B(f),B(g))=d_{bot}(B(f),B(g\circ\phi))\leq\|f-g\circ \phi\|_\infty.$

Thus we get a lower bound from barcodes.

7.1. Example

Question. Given a smooth function ${f}$ on the 2-sphere, find optimal approximation of ${f}$ by a function with 2 critical points.

If ${f}$ has 2 index 1 critical values ${a, the lower bound ${d_{bot}(\mathbb{F}(a,b),\emptyset)=\frac{b-a}{2}}$ is sharp.

7.2. Approximation with eigenfunctions and their images by changes of variables

From our theorem, it follows that

Corollary 7 Let ${f}$ be a Morse function on ${M}$. Assume that ${\Phi(f)}$ is large compared to ${\lambda}$. Then

$\displaystyle dist_{C^0}(f,F_\lambda\circ Diff(M))\geq \frac{\Phi(f)}{2|B_*(f)|},$

which, in turn, is the half of the average bar length.

8. Proof

Our goal is to prove that ${\|f\|_2=1}$, ${\|\Delta f\|_2\leq \lambda}$ imply that ${I(f)\leq k_g(\lambda+1)}$.

I do all computations on the square torus, for simplicity. Let ${H}$ denote the Hessian of function ${f}$.

Equip the unit tangent bundle ${ST^2}$ with the obvious metric ${dx^2+dy^2+d\phi^2}$. Look at a connected component ${\gamma}$ of a regular fiber ${f^{-1}(c)}$, parametrized by arclength. Let ${\tilde\gamma}$ be its lift to ${ST^2}$ via its field of normals. Since ${\tilde\gamma}$ is not contractible, its length is ${\geq 1}$.

Since

$\displaystyle length(\tilde\gamma)\leq \int_\gamma \sqrt{1+\frac{|H|^2}{|\nabla f|^2}}\,dt,$

the total length of the lift ${\widetilde{f^{-1}(c)}}$ is ${\geq\beta(c,f)}$, hence

$\displaystyle \begin{array}{rcl} I(f)&\leq &\int_{\mathbb R} \beta(c,f)\,dc\leq\int_{\mathbb R} L(c)\,dc\\ &\leq&\int_{\mathbb R} \int_\gamma \sqrt{1+\frac{|H|^2}{|\nabla f|^2}}\,dt\\ &\leq& k(\int_{M}(|\nabla f|^2+|H|^2))^{1/2}\\ &\leq & k(\|f\|_2+\|\Delta f\|_2), \end{array}$

by coarea formula, Cauchy-Schwartz (plus Bochner-Lichnerowicz in the general case).