• Day 1. Topological data analysis
• Day 2. Hands on: computational topology in action
• Day 3. Applications in dynamics

• Encodes topological spaces combinatorially for computational purposes.


• Encodes topological features in terms of mathematical quantities that are invariant to homotopy equivalences and easy to compute.

Simplicial complexes can be stored as a list of non-empty subsets of its vertices.

• The homology group $H_d$ characterizes the $d$-dimensional holes in a topological space.
• The $d$-th Betti number $b_d$ is the rank of the group $H_d$.

Q: How to algorithmically compute homology of simplicial complexes?

Q: How to algorithmically compute homology of simplicial complexes?

$\bullet$ Chain

$C = [v_0,v_1]+[v_1,v_2]+[v_2,v_3]+[v_3,v_4]+[v_4,v_5]+[v_5,v_0]$

Q: How to algorithmically compute homology of simplicial complexes?

$\bullet$ Chain

$C = [v_0,v_1]+[v_1,v_2]+[v_2,v_3]+[v_3,v_4]+[v_4,v_5]+[v_5,v_0]$

$\bullet$ Cycle

$\partial C = [v_1]-[v_0]+[v_2]-[v_1]+[v_3]-[v_2]+[v_4]-[v_3]+[v_5]\\-[v_4]+[v_0]-[v_5] = 0$

Q: How to algorithmically compute homology of simplicial complexes?

$\bullet$ Chain

$C = [v_0,v_1]+[v_1,v_2]+[v_2,v_3]+[v_3,v_4]+[v_4,v_5]+[v_5,v_0]$

$\bullet$ Cycle

$\partial C = [v_1]-[v_0]+[v_2]-[v_1]+[v_3]-[v_2]+[v_4]-[v_3]+[v_5] \\ -[v_4]+[v_0]-[v_5] = 0$

$\bullet$ Non boundary

Q: How to algorithmically compute homology of simplicial complexes?

• A $d$-chain is a formal sum of $d$-simplices.

$C = [v_0,v_1]+[v_1,v_2]+[v_2,v_3]+[v_3,v_4]+[v_4,v_5]+[v_5,v_0]$

• The boundary of a $d$-simplex is the alternate chain made of its $(d − 1)$-simplices.

  This operation extents linearly to chains of $d$-simplices.

$\partial_d[v_1,\dots,v_{d+1}] = \sum_{i=1}^{d+1}(-1)^i[v_1,\dots,v_{i−1},v_{i+1},\dots,v_{d+1}]$

• A $d$-cycle is a $d$-chain $C$ s.t. $\partial C = 0$.

$\partial C = [v_1]-[v_0]+[v_2]-[v_1]+[v_3]-[v_2]+[v_4]-[v_3]+[v_5]-[v_4]+[v_0]-[v_5] = 0$

• A $d$-cycle is a boundary if there exist a $(d+1)$-chain $C'$ such that $\partial (C') = C$.
• Homology in degree $d$ is the quotient of cycles over boundaries. \[H_d = \frac{Z_d}{B_d}\]

Q: How to algorithmically compute homology of simplicial complexes?

An element in $H_d$ is the equivalent class of a cycle modulo boundaries

Q: How to algorithmically compute homology of simplicial complexes?

Algorithm

1. Calculate boundary operator matrices.
2. Bring each matrix into Smith normal form (similar to Gaussian elimination).
3. The $d^{th}$ Betti number $b_d$ is computed as rank $Z_d$ - rank $B_d$, where
   • rank $Z_d$ is the number of zero columns of the boundary matrix of $\partial_d$,
   • rank $B_d$ is the number of non-zero rows of the boundary matrix of $\partial_{d+1}$.

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

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Let $(\mathbb{X}_n, d)$ be a finite set of points with a metric.

Given a parameter $\epsilon>0$, the Vietoris-Rips complex $V_{\epsilon}$ is defined as \[V_{\epsilon}(\mathbb{X}_n) := \{\sigma \subseteq \mathbb{X}_n\colon \forall u,v\in \sigma, d(u,v)\leq \epsilon\}\]

Equivalently, $V_\epsilon$ contains all simplices whose diameter is less than or equal to $\epsilon$.

Image from: Bastian Rieck. Topological Data Analysis for Machine Learning.

Let $(\mathbb{X}_n, d)$ be a finite set of points wiht a metric.

Given a parameter $\epsilon>0$, the Vietoris-Rips complex $V_{\epsilon}$ is defined as \[V_{\epsilon}(\mathbb{X}_n) := \{\sigma \subseteq \mathbb{X}_n\colon \forall u,v\in \sigma, d(u,v)\leq \epsilon\}\]

Equivalently, $V_\epsilon$ contains all simplices whose diameter is less than or equal to $\epsilon$.

Idea: Track the evolution of the homology of $V_\epsilon$ as the value of $\epsilon$ increases.

Image from: Bastian Rieck. Topological Data Analysis for Machine Learning.

• A filtration of simplicial complexes is a sequence of nested simplicial complexes \[K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \dots \hookrightarrow K_s \]
• A filtration $\{K_i\}$ of simplicial complexes induces a sequence of homology groups $\{H_d(K_i)\}$ along with homomorphisms $f_d^{i,j}:H_d(K_i) \to H_d(K_j)$ induced by $K_i\subseteq K_j$ every time $i\leq j$ \[H_d(K_0) \rightarrow H_d(K_1) \rightarrow H_d(K_2) \rightarrow \dots \rightarrow H_d(K_s) .\]
• Given $i\leq j$, the persistent homology group $H_d^{i,j}$ is defined as \[H_d^{i,j} :=\frac{Z_d(K_i)}{B_d(K_j)\cap Z_d(K_i)}\] i.e., all the homology classes of $K_i$ that are still present in $K_j$.


• Persistent homology classes.
  • Birth in $K_i$: $C\in H_d(K_i)$, but $C\notin H_d^{i−1,i}$
  • Death in $K_j$: once $C$ is created in $K_i$, $C\in H_d^{i, j-1}$ but $C\notin H_d^{i, j}$

• Persistent homology classes.
  • Birth in $K_i$: $C\in H_d(K_i)$, but $C\notin H_d^{i−1,i}$
  • Death in $K_j$: once $C$ is created in $K_i$, $C\in H_d^{i, j-1}$ but $C\notin H_d^{i, j}$


• The birth and death of homology classes determine a barcode decomposition.

• Algorithm.
  As a consecuence of the Structure Theorem, the computation of persistent homology reduces to the algorithm for computation of regular homology.
  

  [Zomorodian, A.; Carlsson, G.. Computing persistent homology. Discrete Comput. Geom. 33 (2005), no. 2, 249--274.]



• Software.
  There are open access implementations to compute persistent homology in
  • R,
  • Julia,
  • C++,
  • Matlab and
  • Python (tomorrow).
  
  There is also a browser version of the software Ripser.

• Part of an environment-independent spatial coordinate system.
• Placed in the medial entorhinal cortex (mEC).
• Discovered by Edvard and May-Britt Moser in 2005.
  T. Hafting, M. Fyhn, S, Molden, M.B. Moser, E.I. Moser. Microstructure of a spatial map in the entorhinal cortex. Nature. 2005 Aug; 436(7052) 801-806.
• Awarded with Nobel Prize in Physiology or Medicine 2014.

• Neuroscience, neural activity (Day 2).

  [Curto, Carina. What can topology tell us about the neural code? Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 1, 63--78]
  [Gardner, R.J., Hermansen, E., Pachitariu, M. et al. Toroidal topology of population activity in grid cells. Nature 602, 123–128 (2022)]

• Chemistry, energy landscape of molecules (Day 2).

  [Martin, S., Thompson, A., Coutsias, E. A., & Watson, J. P. (2010). Topology of cyclo-octane energy landscape. The Journal of Chemical Physics, 132(23), 234115.]

• Proteins classification (Day 2).

  [Kovacev-Nikolic V, Bubenik P, Nikolić D, Heo G. Using persistent homology and dynamical distances to analyze protein binding. Stat Appl Genet Mol Biol. 2016 Mar;15(1):19-38.]
  [Cang Z, Mu L, Wu K, et al. (2015). A topological approach for protein classification. Computational and Mathematical Biophysics, 3(1)]
  [Xia, K. & Wei, G. W. (2014). Persistent homology analysis of protein structure, flexibility, and folding. International Journal for Numerical Methods in Biomedical Engineering, 30(8), 814–844]

• Dynamics and time series analysis (Day 3).

  [Perea, J A, Harer, J. Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis. Found Comput Math 15, 799–838 (2015).]
  [Perea J A, Topological Time Series Analysis, Notices of the American Mathematical Society, vol. 66, no. 5, pp. 686-694, May 2019.]
  [Fernandez X., Borghini E., Mindlin G., Groisman P. Intrinsic persistent homology via density-based metric learning, 2020. arXiv:2012.07621]
  [Fernandez X., Mateos D. Topology of epilepsy seizures. Work in progress, 2022.]

• Topological spaces can be encoded combinatorially as simplicial complexes.
• Homology groups characterise holes in topological spaces.
• The computation of homology groups boils down to linear algebra.

• Topological spaces can be encoded combinatorially as simplicial complexes.
• Homology groups characterise holes in topological spaces.
• The computation of homology groups boils down to linear algebra.


• Point clouds can be converted in filtrations of simplicial complexes.
• Persistent homology is the multi-scale equivalent of homology.
• The computation of persistent homology also boils down to linear algebra.

• Topological spaces can be encoded combinatorially as simplicial complexes.
• Homology groups characterise holes in topological spaces.
• The computation of homology groups boils down to linear algebra.


• Point clouds can be converted in filtrations of simplicial complexes.
• Persistent homology is the multi-scale equivalent of homology.
• The computation of persistent homology also boils down to linear algebra.


• Data sets can be viewed as samples of an (unknown) topological space.
• Persistent homology classes capture topological features in data.

• Computation of persistent homology in Python.
• Topology of real data sets.

GitHub ximenafernandez/biomat2022