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Metric Spaces Of Non Positive Curvature

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry and analysis. In particular, we present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, we discuss topological function theory which provides a new insight on oscillation of functions. The material should be accessible to readers with a basic background in algebraic and differential topology.

Topological Persistence in Geometry and Analysis

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces $S^n, \mathbb{R} P^n, \mathbb{C} P^n, \mathbb{H} P^n,$ $n\geq 1.$ We discuss generalizations and give applications, in particular to $C^0$ symplectic topology. Our key methods, which are of independent interest, consist of a reinterpretation of the spectral norm via the asymptotic behavior of a family of cones of filtered morphisms, and a quantitative deformation argument for Floer persistence modules, that allows to excise a divisor.

Viterbo conjecture for Zoll symmetric spaces

We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $\gamma$-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of $\gamma$-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and $\gamma$-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances.

Ephemeral persistence modules and distance comparison

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. 
Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. 
In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

Barcodes and area-preserving homeomorphisms

Given a symplectic surface (Σ,ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ,ω),dH), where dH is the Hofer metric. The result stabili...

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humili\`ere-Leclercq-Seyfaddini's coisotropic rigidity on symplectic manifolds. Moreover, based on Shelukhin's norm defined on the contactomorphism group, we define a Chekanov type pseudo-metric on the orbit space of a fixed submanifold of a contact manifold. Moreover, we can show a dichotomy of (non-)degeneracy of this pseudo-metric when the dimension of this fixed submanifold is equal to the one for a Legendrian submanifold. This can be viewed as a contact topology analogue to Chekanov's dichotomy of (non-)degeneracy of Chekanov-Hofer's metric on the orbit space of a Lagrangian submanifold.

Chekanov's dichotomy in contact topology

Given a symplectic surface $(\Sigma, \omega)$ of genus $g \ge 4$, we show that the free group with two generators embeds into every asymptotic cone of $(\mathrm{Ham}(\Sigma, \omega), d_\mathrm{H})$, where $d_\mathrm{H}$ is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.

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Papers

Topological Persistence in Geometry and Analysis

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Chekanov's dichotomy in contact topology

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Chekanov's dichotomy in contact topology