Let $(X, d)$ be a metric space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ a finite sample of $X$.

Q: How to infer the homology of $X$ from $\mathbb{X}_n$?

Let $(X, d)$ be a metric space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ a finite sample of $X$.

Q: How to infer the homology of $X$ from $\mathbb{X}_n$?

• A filtered simplicial complex $K_\bullet$ is a sequence of nested simplicial complexes \[K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \dots K_s = K\]

• A filtered simplicial complex $K_\bullet$ is a sequence of nested simplicial complexes \[K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \dots K_s = K\]

- $\check{\mathrm{C}}$ech filtration: For every $\varepsilon>0$, let $~\mathcal{U}_{\varepsilon} = \{B(x_i, \varepsilon)\}_{x_i\in \mathbb{X}_n}$ be a collection of open sets. Define \[\check C_{\varepsilon} = \mathcal{N}(\mathcal{U}_{\varepsilon})\]

- Vietoris Rips filtration: Let $\Delta_n$ be the $(n-1)$-simplex with vertices in $\mathbb{X}_n$. Define $$V_{\varepsilon} = \{\sigma \in \Delta_n: \mathrm{diam}(\sigma)<\varepsilon\}$$

• A filtered simplicial complex $K_\bullet$ is a sequence of nested simplicial complexes \[K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \dots K_s = K\]

- $\check{\mathrm{C}}$ech filtration: For every $\varepsilon>0$, let $~\mathcal{U}_{\varepsilon} = \{B(x_i, \varepsilon)\}_{x_i\in \mathbb{X}_n}$ be a collection of open sets. Define \[\check C_{\varepsilon} = \mathcal{N}(\mathcal{U}_{\varepsilon})\]

- Vietoris Rips filtration: Let $\Delta_n$ be the $(n-1)$-simplex with vertices in $\mathbb{X}_n$. Define $$V_{\varepsilon} = \{\sigma \in \Delta_n: \mathrm{diam}(\sigma)<\varepsilon\}$$

\[\check C_{\varepsilon}\subseteq V_\varepsilon \subseteq \check C_{2\varepsilon}, ~~\forall \varepsilon>0\]

• A filtration $\{K_i\}$ of simplicial complexes induces a persistence complex $\{C_i\}$ with $C_i = C_\bullet(K_i)$ and morphisms induced by the inclusions \[C_\bullet(K_0)\rightarrow C_\bullet(K_1) \rightarrow \dots\rightarrow C_\bullet(K_i) \rightarrow \dots\]
• The associated persistence module over a field $k$ at degree $j$ is defined as \[M_j = \bigoplus_{i\geq 0} H_j(C_i)\]

Theorem (Zomorodian, Carlsson, 2005). For every degree $j\geq 0$, there is a decomposition $$M_j = \left( \bigoplus_{s=1}^n \Sigma^{\alpha_s} k[x]\right) \oplus \left(\bigoplus_{l=1}^m \Sigma^{\gamma_l}k[x]/x^{n_l}k[x]\right)$$ where $\alpha_s, \gamma_j\in \mathbb{Z}$, $n_l< n_{l+1}$ y $\Sigma^{\alpha}$ denotes an $\alpha$-shift in the degree.

We can represent $M_j$ as a set of intervals $(\alpha_s, + \infty)$ and $(\gamma_l,\gamma_l+n_l)$, also known as barcode.

Theorem (Cohen-Steiner, Edelsbrunner, Harer, 2007). For any two precompact metric spaces $(X, d_X)$ and $(Y, d_Y)$, \[ d_b\Big(\mathrm{dgm}\big(\mathrm{Filt}(X, d_{X})\big),\mathrm{dgm}\big(\mathrm{Filt}(Y, d_{Y})\big)\Big)\leq 2 d_{GH}\big((X,d_{X}),(Y,d_{Y})\big). \]

• Phase transitions.

  N. Sale, J. Giansiracusa, and B. Lucini, Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology, 2021, ArXiv:2109.10960

• Development of new materials.

  Work of Vitaly Kurlin et. al. University of Liverpool

• Knotted structure of proteins.

  A. Barbensi, N. Yerolemou, O. Vipond, B.I. Mahler, P. Dabrowski-Tumanski and D. Goundaroulis. A topological selection of folding pathways from native states of knotted proteins. 2021.

• Biological networks.

  Work of Heather Harrington et. al. Oxford University

• Time series analysis.
• $\dots$

• A global continuous time dynamical system is a pair $$(\mathcal{M}, \phi)$$ where $\mathcal{M}$ is a topological space and $\phi \colon \mathbb{R} \times \mathcal{M} \to \mathcal{M}$ is a continuous map such that $\phi(0, p) = p$ and $\phi(s, \phi(𝑡, 𝑝)) = \phi(s + t, p)$ for all $p \in \mathcal{M}$ and all $t, s \in \mathbb{R}$.


• The typical examples arising from differential equations have as state space a smooth manifold $\mathcal{M}$ and the dynamics are given by the integral curves of a smooth vector field on $\mathcal{M}$.


• A set $A \subset \mathcal{M}$ is called an attractor if it satisfies the following conditions: it is compact, it is an invariant set (i.e. if $a \in A$ then $\phi(𝑡, 𝑎) \in 𝐴$ for all $𝑡 \geq 0$) and it has an open basin of attraction. Equivalently, there is an invariant open neighborhood $𝑈 \subset \mathcal{M}$ of $𝐴$ such that $\cap_{t\geq 0} \{\phi(𝑡, 𝑝) ∶ 𝑝 \in 𝑈\} = 𝐴$.

• A global continuous time dynamical system is a pair $$(\mathcal{M}, \phi)$$ where $\mathcal{M}$ is a topological space and $\phi \colon \mathbb{R} \times \mathcal{M} \to \mathcal{M}$ is a continuous map such that $\phi(0, p) = p$ and $\phi(s, \phi(𝑡, 𝑝)) = \phi(s + t, p)$ for all $p \in \mathcal{M}$ and all $t, s \in \mathbb{R}$.


• The typical examples arising from differential equations have as state space a smooth manifold $\mathcal{M}$ and the dynamics are given by the integral curves of a smooth vector field on $\mathcal{M}$.


Lorenz attractor \begin{equation}\begin{cases} \dot x = \sigma ( y - x ) , \\ \dot y = x(\rho-z)-y ,\\ \dot z = xy-\beta z \end{cases} \end{equation} with $(\sigma, \rho, \beta) = (10, 28, 8/3)$.

Theorem (Takens). Let $\mathcal{M}$ be a smooth, compact, Riemannian manifold. Let $\tau > 0$ be a real number and let $d ≥ 2 \mathrm{dim}(\mathcal{M})$ be an integer. Then, for generic $\phi \in C^2(\mathbb{R} \times \mathcal{M}, \mathcal{M})$ and $F\in C^2(\mathcal{M}, \mathbb{R})$ and for $\varphi_\bullet(𝑡)$ defined as above, the delay map \begin{align} \varphi~ \colon & ~~\mathcal{M} &\rightarrow & ~~\mathbb{R}^{d+1}\\ &~~p &\mapsto & ~~(\varphi_p(0), \varphi_p(𝜏), \varphi_p(2𝜏),\dots, \varphi_p(d\tau)) \end{align} is an embedding (i.e., $\varphi$ is injective and its derivative has full-rank everywhere).