The Earth is constantly being bombarded with radiation from outer space. In order to understand the effects of the radiation, imaging atmospheric Cherenkov telescopes $($IACTs$)$ are used to study high$-$energy gamma rays which enter the atmosphere.

A publicly available radiation dataset$1$ from the Max Planck Institute contains data from the major atmo$-$ spheric gamma imaging Cherenkov $($MAGIC$)$ telescope project. The MAGIC project dataset presents different types of radiation with corresponding characteristics. You are provided with three radiation datasets split into gamma radiation, hadron radiation, and an unknown $($either gamma or hadron$)$ radiation. Note that the pro$-$ vided datasets are small subsamples of the original datasets to reduce computational complexity, and you must not work with the original dataset.

$$ Discern the homology features of each dataset and describe the topological features which are formed during the persistent homology analysis. Present your homology results and explain the results in detail. $[15$ marks$]$

$$ Use persistent homology approaches to determine the type of unknown radiation. $[3$ marks$]$

$$ Can each of the three datasets be mapped to a known manifold based on the calculated homology features?

If so$,$ which manifold best describes the data?