- Goal: Understand properties of shapes to distinguish it from other shapes (this turns out to be difficult by traditional methods)
- In 20th century, people started creating "machines" (e.g. functors) to take shape-like objects to groups/rings/fields/other algebraic structures
- Compare the algebraic structures inherent to the shapes rather than the shapes themselves, which is less hard
- Still some notion of collapse of info from going from shape to algebraic object

- Homology: Assign a vector space to a shape, with basis given by elements of a triangulation (triangles, edges, vertices, tetrahedra, etc)
- Each triangulation is called a simplicial complex
- Find a subspace (cycles) consisting of vectors/families of simplices whose boundaries cancel (e.g. two triangles with a shared edge have that edge cancel out)
- Equivalence: if a simplex is "filled in," their edges sum to 0

Given this:

Definition: Homology

The $k$-th **homology** is given as

$H_{k}(X) = \frac{\text{Cycles formed from k-simplices}}{\text{Boundaries of (k+1)-simplices}}$

Then, if we have a morphism of simplicial complexes $f: X\to Y$, then we gen a map $f_{*}: H_{k}(X)\to H_{k}(Y)$.

This can help us analyze the shape by taking some test space and mapping it to the shape, and seeing where the homology of the test space ends up as a subspace of the homology of the shape. In other words, we can test for certain structures in a shape.

- We have some real-world system and some number of sensors of that system into some sensor space (possibly metric), and hopefully the structure of the system is reflected in its image in the sensor space
- Use tools from algebraic topology to try to say something about the system using measurements in sensor space
- Vietoris-Rips complex $VR_{\epsilon}(X)$: if a bunch of points live in an $\epsilon$-ball, then it fills in the simplex from the points
- If we take points sampled from a topological space, then at scale $\epsilon$ we recover the original structure of the system
- If we vary $\epsilon$, we get an increasing family of complexes that encode the structure at a different scale
- Hope: This family will reflect the structure of the original system

- Use homology on VR complexes: $H_k (VR_{\epsilon}(X))\to H_k (VR_{\epsilon}(X))$ as we vary $k$ -> Persistent homology $PH_{k}(VR(X))$
- Persistent diagram: we have a whole bunch of points each representing a simplex, with x axis being when the simplex is "born" ($\beta$) and y axis being when it "dies" ($\delta$)
- Note that $\beta \leq \delta$
- Metric on persistent diagrams, check to see if two of them are "close" (possibly probability distributions (
**p. note: Wasserstein distance?**))

Guiding question: How does the point stuff represent the system?

Given two brain regions with strong connectivity between two regions, record neurons in the brain and encode signals, extract fast number of time series

If there is geometric structure encoded in the brain, we can get it back through this process of time series

Example: If we spin around a lot, we can kind of tell which direction we are facing -> How is this encoded in the brain?

If we have circular coordinate system in both regions, how is that information passed on? -> Use algebraic topology

- First ideas (Bower, Lesnick): At first, the only real way of using a continue map to match cycle samples, then only cycles with the same death time could be mapped
- Bootstraps (Reani, Bobrowski), 2022: Take subsamples $X_1$, $X_2$ of $X$, see if we see the same structure in $X_1$ and $X_2$, create maps $X_1\to X_1\bigcup X_2\gets X_2$, composite map $\sigma: X_1\to X_2$
- Confirming that a coherent structure exists in both $X_1$ and $X_2$
- Needs to be an ambient space $X$, but needs to have a coherent metric to be able to compare $X_1$ and $X_2$

- Wishlist for Chad Giusti
- Only use observable information -> relations, not functions (we can vary certain things without affecting the output)
- Robust to noise or resampling
- Comparable -> fits into a functional view of persistent homology

VR Complexes to Witness Complexes

Witness Complexes: $X, Y\subseteq M$ subsets of same metric space, instead of taking $\epsilon$-balls of every point, only take $\epsilon$-balls centered in points of $X$ of points in $Y$

Theorem: Dowker's Theorem

$H_{k}(W_{\epsilon}(X, Y)) \cong H_{k}(W_{\epsilon}(Y, X))$

This tells that the topological structure of the observer and the observed are basically the same.

Cycle Extension:

$\begin{CD} H_k(W(X, Y)) @>Dowker>\cong> H_k(W(Y, X)) \\ @ACycle ExtensionAA @VCycle ExtensionVV \\ H_k(VR(X)) @. H_k(VR(Y)) \\ @. @. \\ [\sigma] @>>> \text{Affine Subspace} [w(\sigma)] + B([\sigma]) \\ \end{CD}$

$VR(X) \cong W(X, X)$?

- Start with VR complex, do something with cycle extension (intersection of vector spaces) to move from VR to Witness (at the cost of being a multi-valued map)
- Then apply Dowker's theorem to flip X and Y and then run cycle extension backwards
- Functionally this "works", but might be the wrong language for it (not formaly correct)...
- We want to take the data and turn into a morphism