## FET project TOPDRIM opened a new line of research using a branch of mathematics (topology) to explore massive amounts of data looking for hidden patterns of correlated information. WHAT IS TOPOLOGY.

We will introduce topology through an example. As most of you know, the highest number of people suffers from insomnia. The **fingerprint of insomnia** has not yet been discovered, but neuroscientists suppose to exist. Image to have available a massive amount of data - images, blood analysis, personal habits etc. – you need to extract from this data information, - hidden correlations patterns - to decrypt the insomnia fingerprint. You create a data set with data coming from healthy people and from patients suffering from insomnia. Topology allows you to study the **shape of data space** whence you realize that such space is the aggregation elements of increasing size, as shown in the above figure, which allows us to characterize the fingerprint of insomnia.

The topological data analysis returns shapes of data space. Data space has a shape and the shape represents the information carried by data.

The properties originating from the shapes are crucial to describe and understand the structural and functional properties of the analysed systems, the brain in our example. You can identify the **functional states** of the brain through which the brain processes its information. Then, you can identify, by composing feasible states, the **equivalence classes** that characterize different behaviours of people.

TOPDRIM had the goal to define computational, algebraic and probabilistic methods to construct effective topological shapes by analysizing large collection of data. TOPDRIM besides the expected outcomes, provided a methodological scheme (algorithms, automata, processes algebra). Who is familiar to the concept of Turing Machine can perhaps associate TOPDRIM to a machine able to construct algorithms. Some evidence can be found when we think an algorithm as automaton.