Defined the homology 
H<sub>n</sub>(K) 
of a simplicial complex K (over a field). 
This is a vector space whose dimension equals the degree n Betti number:
<center> &beta;<sub>n</sub> = dim (H<sub>n</sub>(K)) .</center>

<br><br>
Explained that an inclusion of simplicial complexes L < K induces a linear homomorphism of vector spaces
<center>H<sub>n</sub>(L) ---> H<sub>n</sub>(K) .</center>

<br><br> Given a matrix of distances between n objects in a dataset, and given an increasing  sequence of thresholds, one can construct a sequence of inclusions of clique complexes. For each n this  sequence yields a sequence of linear homomorphisms of degree n homology vector spaces. It was explained how the sequence can be represented using barcodes and then used to extract meaningful geometric information about the dataset.

<br><br>
The following example from the R-TDA package was explained.
<br><br>library(TDA)
<br>Circle1 <- circleUnif(25)
<br>Circle2 <- circleUnif(25, r=1.2)
<br>Circle3 <- circleUnif(25, r=1.4)
<br>Circle4 <- circleUnif(25, r = 2) + 4.5
<br>Circle5 <- circleUnif(25, r = 2.2) + 4.5
<br>Circle6 <- circleUnif(25, r = 2.4) + 4.5
<br>Circles <- rbind(Circle1, Circle2, Circle3, Circle4, Circle5, Circle6)
<br>plot(Circles, pch = 16, xlab = "",ylab = "")
<br>DiagLim <- 4
<br>maxdimension <- 1
<br>Diag <- ripsDiag(Circles, maxdimension, DiagLim, printProgress = TRUE)
<br>par(mfrow = c(1, 1))
<br>plot(Diag[["diagram"]], barcode = TRUE)