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”.
Already surfaces are difficult examples.
Hofer’s length on Hamiltonian diffeomorphism groups . A path is determined by a path of normalized functions . Its length is
This defines a kind of biinvariant Finsler structure on . 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 ) correspond to 1-parameter subgroups of . They admit roots of any degree. T-Such flows conserve energy. They are geodesics in Hofer’s metric (but not the only geodesics).
has interesting algebraic properties. It is algebraically simple. Let be the set of Hamiltonian diffeos admitting a root of order . Is metrically dense in ? I.e. is
finite? We conjecture that this is never the case.
Theorem 1 (Polterovich-Shelukin) Let be a closed surface of genus . Let be an arbitrary closed manifold with . Then, for large,
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 (strength of shear motion) is introduced. As tends to , we show that the distance of resulting diffeo to 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.
In dynamics, it has been known for a long time that vectorfields generate few diffeomorphisms (Palis, Brin 1973). In , non autonomous Ham. diffeos contain a -dense open set (Salamon-Zehnder, Ginzburg-Gurel), for symplectically aspherical manifolds. Our methods upgrade open to Hofer-open, and make it quantitative.
3.2. Coarse geometry of
Polterovich-Rosen: a -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 has infinite diameter (for surfaces), Misha Kapovitch asked me wether did lie in a bounded neighborhood of a quasigeodesic. Our main theorem shows that this does not happen for .
3.3. Milnor’s constraint
In 1983, Milnor observed that if a diffeo is a square, , the number of primitive geometrically distinct 2-periodic orbits is even. Indeed, induces a action of such orbits.
Find more restrictions on powers.
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 with multplicities . The bottleneck distance is defined as follows. Erase intervals of length and match the remaining intervals up to error . Then infimize .
Given a field , a persistence module is a pair of finite dimensional -vectorspaces , , and maps , for . Commuting diagrams. One assumes regularity: for all but a finite number of jump points in , 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
closed manifold, Morse function. Persistence module is
Inclusions induce persistence morphisms.
The following statement is a not that easy theorem.
Robustness: The map 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 be a Morse function on and a generic Riemannian metric. The Morse complex is spanned by the critical points of with value . The differential counts the number of gradient lines of connecting critical points,
4.3. Floer theory
Born in 1988. In symplectic topology, the role of is played by infinite dimensional manifold of contractible loops . Given a 1-periodic Hamiltonian , define action functional
where is an arbitrary disk spanning .
The original action functional has only one critical point of infinite index and coindex. The perturbation is much more interesting, since its critical points correspond to 1-periodic orbits of the Hamiltonian flow generated by .
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 , this is Floer homology.
Under certain asumptions (asperical, atoroidal,…), the corresponding persistence module depends only on the time 1 map .
One can extend the construction to noncontractible loops.
Theorem 2 (Polterovich-Shelukin) The map
Hence Lipschitz functions on barcodes yield numerical invariants of Ham diffeos. Powers give rise to representations on persistence modules. A Floer-Novikov variant has been developed by Usher-Zhang.
4.4. Representations on persistence modules
Since , acts on the Floer homology of , this is a action . Define be the -eigenspace of . There is a corresponding persistence module. If , then induces action on . Then on it. Thus the multiplicity of each bar in 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.
Question. Given a closed oriented surface, and a smooth Morse function on , how can one define the oscillation of ?
The Banach indicatrix is defined as follows. Let denote the number of connected components in . Let
Goes back to Kronrod and Vitushkin in the 1950’s. In the 1980’s, Yomdin rediscovered it, with the idea that if derivatives of are not too large, should be small.
Define the total length of the finite part of the barcode as
The following is an easy fact from surface topology.
Theorem 3 (PPS 2018)
Consider the Reeb graph of (space of connected components of fibers). descends to a function on it. Then is equal to the total variation of on its Reeb graph. The difference with 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 . Let be the Laplacian. For , consider
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 ) is for the -eigenfunction.
Theorem 4 (Polterovtch-Sodin 2006) For ,
1. For Euclidean domains, for Dirichlet boundary values, Alexandrov-Backelman-Pucci-Cabre 1995 show that
2. On the square 2-torus, satisfy , and barcodes have bars of length . This is sharp.
3. Say a critical value of is -significant if it is an endpoint of a bar of length . From our theorem, it follows that
Corollary 6 If , the number of -significant critical values of is .
4. Nicolaescu has shown that the expectation of the number of critical points of random linear combinations of eigenfunctions is .
has a number of critical points that tends to infinity, whereas for every fixed , the number of -significant critical values stays bounded.
7. Approximation theory
Let , be smooth functions on . In -norm, what is the best approximation of by functions of the form , ?
Let be the barcode of . Then is Diff-invariant. By robustness theorem,
Thus we get a lower bound from barcodes.
Question. Given a smooth function on the 2-sphere, find optimal approximation of by a function with 2 critical points.
If has 2 index 1 critical values , the lower bound is sharp.
7.2. Approximation with eigenfunctions and their images by changes of variables
From our theorem, it follows that
Corollary 7 Let be a Morse function on . Assume that is large compared to . Then
which, in turn, is the half of the average bar length.
Our goal is to prove that , imply that .
I do all computations on the square torus, for simplicity. Let denote the Hessian of function .
Equip the unit tangent bundle with the obvious metric . Look at a connected component of a regular fiber , parametrized by arclength. Let be its lift to via its field of normals. Since is not contractible, its length is .
the total length of the lift is , hence
by coarea formula, Cauchy-Schwartz (plus Bochner-Lichnerowicz in the general case).