Implicit Function

Theorems

in

Geometry

and Dynamics



February 19–22, 2020


Schloß Rauischholzhausen




The winter school will present two incarnations of the implicit function theorem to master and PhD students: the implicit function theorem of Nash and Moser, and the implicit function theorem in scale calculus and polyfold-theory as developed by Hofer, Wysocki and Zehnder.

There will be two mini-courses on the theoretical framework of each incarnation which will be complemented by talks on classical and current applications to analysis and geometry.



Lecture courses



Massimiliano Berti

Nash-Moser implicit function theorems and applications in dynamics


In these lectures I will prove an abstract Nash-Moser implicit function theorem for a nonlinear operator acting in scales of Banach spaces. In contrast to the usual implicit function theorem the key feature is that the linearized operator admits and approximate inverse which in unbounded in a fixed Banach space. This abstract theorem is taylored to prove, in particular, the classical KAM (Kolmogorov-Arnold-Moser) result concerning the persistence of quasi-periodic solutions for sufficiently differentiable nearly integrable Hamiltonian systems. The proof of this result will be given in the talk by F. Giuliani. All the functional analytic preliminaries of smoothing operators, tame estimates and interpolation inequalities will also be proved.



Zhengyi Zhou

Implicit function theorems in scale-calculus and polyfold theory


Implicit function theorems in scale-calculus and more generally polyfold theory are at the heart of the applications to symplectic topology for which these theories were developed. At the same time, the notion of scale-derivative does not require continuity of the scale-differential in the operator topology as the base point varies, which prevents the obvious analogues in scale-calculus of classical implicit function theorems for Fréchet differentiable maps from holding. Much effort has gone into understanding under what conditions an implicit function theorem does hold, miraculously with success in all cases that are known to be relevant to symplectic topology. Hofer-Wysocki-Zehnder formulated the notion of a scale-Fredholm map and proved the implicit function theorem for scale-Fredholm sections of polyfold bundles. In joint work with Benjamin Filippenko and Katrin Wehrheim, we explore the boundary between success and failure of the implicit function theorem in scale-calculus, along the way justifying the assumption of a contraction germ property in the definition of scale-Fredholm. In this four lecture course, I will tell this story starting from the definitions of scale-calculus.


(pdf)



Additional talks



Pietro Baldi

A Nash–Moser implicit function theorem à la Hörmander and applications to PDEs


We present a recent version of a Nash–Moser implicit function theorem that is sharp regarding some regularity issues. At the base of the proof is the iteration scheme of Hörmander, very close to the original scheme by Nash, which we introduce in the talk. With respect to Hörmander’s version of Nash-Moser theorem, the new ingredient in our result is the use of orthogonality to estimate partial sums along the iteration, together with a dyadic block decomposition that imitates the classical Littlewood-Paley decomposition in Fourier Analysis. Some applications of the abstract result to control theory and Cauchy problems for PDEs will be also presented.

(pdf)



Gabriele Benedetti

Normal forms for vector fields and contact forms close to Zoll systems


I will discuss normal forms for vector fields and contact forms that are close to ones producing flows all of whose orbits are periodic with the same minimal period. For vector fields, this result is due to Bottkol, who used it to give lower bounds on the number of periodic orbits of the perturbed vector field. Its specialisation to contact forms was recently found together with Alberto Abbondandolo and has a number of interesting consequences: Sharp local systolic inequalities in Riemannian geometry, the perturbative case of a conjecture of Viterbo on volume/capacity inequalities for convex bodies and a symplectic non-squeezing result in the intermediate dimensions.



Lucas Dahinden

The Hölder homeomorphism problem for Subriemannian manifolds


Subriemannian manifolds have Hausdorff dimension larger than topological dimension. As a consequence, they are never bilipschitz equivalent to a Riemannian manifold. We can ask for which Hölder exponent there exists Hölder homeomorphism from a Euclidean space to a neighbourhood in a Subriemannian manifold. We don't know the precise answer to this question, even in the simplest examples. However, we can bound the Hölder exponent from above. For example for the standard contact three-space, the optimal exponent is somewhere between 1/2 and 2/3 (it is conjectured to be 1/2). I will present a construction due to Pansu to prove these upper bounds. It involves an implicit function theorem to guarantee an abundance of horizontal submanifolds in generic subriemannian manifolds.



Filippo Giuliani

Proof of the KAM theorem by a Nash-Moser approach


In this talk we show a proof of the classical KAM theorem in the finite differentiable case by applying the Nash-Moser implicit function theorem presented by M. Berti in his course. More precisely, we prove the existence of Lagrangian invariant tori supporting quasi-periodic motions for small C^k-smooth perturbations (with k large) of a family of isochronous Hamiltonian systems.



Colin Guillarmou

Marked length spectrum local rigidity


We will explain how to show a local rigidity result for the marked length spectrum of Riemannian metrics with negative curvature, using a weak version of implicit function theorem.



Irene Seifert

Finding periodic delay orbits via the Polyfold-IFT


A differential delay equation in R^n is of the form \dot{x}(t) = X_{t}(x(t-\tau)) with a vector field X on R^n and a delay parameter \tau. If there is a (suitably non-degenerate) periodic solution x without delay, intuitively one would expect existence of periodic solutions for sufficiently small positive delay. However, classical implicit function theorems fail to imply this, since the equation above is not smooth in the delay parameter. In my talk, I will show how to use the polyfold-IFT to overcome this problem in a natural set-up.​



Thibault Lefeuvre

Local rigidity of the boundary distance function


On a Riemannian manifold with boundary, the boundary distance function is the map assigning to each pair of points on the boundary, the Riemannian distance between the two points. It is conjectured that under some mild assumptions on the metric and/or on the topology of the manifold (that I will detail), one can reconstruct the metric up to isometries. This theoretical question naturally arises in seismic tomography, where one tries to reconstruct the inner structure of the crust from the knowledge of travel times of seismic P- and S-waves from the earthquake to seismic stations. I will explain some partial results on this question, involving (in two different steps) an implicit function theorem. Some tools will be reinvested in a subsequent talk by C. Guillarmou in order to deal with a rather similar problem on closed manifolds.



Dingyu Yang

Chain level polyfold theory, infinity structure and relation to derived geometry


Polyfold theory is a super-car driving to the promised land of virtual transversality, but its build is intricate albeit beautiful, and its potential still under-explored. Smoothness is brand new and motivated from domain reparametrization phenomena in moduli problems, while retract local models are convenient to describe gluing and change of constraints. It’s a maximal (almost universal) framework every other virtual techniques can sit in, moreover all machineries of transversality of embedded techniques can be seen from the polyfold framework.
In this overview talk, we discuss polyfold theory of moduli spaces at chain levels when looking at infinity structures, where intersection homology, jet transversality (or via moduli thickening) can play a role, with applications to Legendrian SFT enriched by string topology, and with potential applications to stratified Lagrangian Floer theory and wall-crossing in algebra of infrared in chain level Landau-Ginzburg theory (where Floer type of theory bordered on complex Morse type).
If time permits, I will also talk a bit about relation of polyfold perturbation theory to perfect obstruction theory via intrinsic normal cone and in the derived geometry/dg-manifolds.​



Practical information




VENUE


Schloßpark 1

35085 Ebsdorfergrund

Germany



Arrival:

February, 19 – before lunch

Departure:

February, 22 – after lunch


Registration:

Please send an e-mail to Petra Kuhl.


Arrival:

February 19, before lunch

We will organise shuttle buses that will bring you up to the castle. The meeting point for the shuttle is in front of the main station Marburg (Lahn).


Departure:

February 22, after lunch

We will drive together at 1:45 pm with a bus to the main train station Marburg (Lahn), which will be reached by 2:20 pm at the latest.


Observe:

Lodging and meals will be provided by the castle. All local costs are covered by us. All lectures will take place at the castle.