Projects

The concept of "information" has acquired more and more importance during the last years, in many fields of science. In Physics, namely, a new interpretation of quantum theory is fundamentally based on information, and ssome researchers are focusing on it in order to approach the problem of establishing a coherent Quantum Gravity Theory.
"Information" is actually a term from engineering: it was first introduced, in its mathematical formalization, by Claude E. Shannon [1948] while trying to define a mathematical theory of communication. His definitions and his remarkable results led to the foundation of many fields of Communication Engineering.
However, information has a great theoretical potential. This was noted by other mathematicians, who thoroughly analysed "information measures" and "entropy functions" generalising Shannon's entropy. Anyway, to the best of my knowledge, there is no literature about potential theoretical applications of the Information Theory, which may be outstanding.
I attempted to deal with a simple logical system with the tool of Information Theory, and a result I stated is that axioms are determined by their "load" of information, i.e. a statement is an axiom in a given system if and only if its information measure is a given value.
Another interesting, although easy result is that Shannon's entropy of a system does not depend on the number of axioms, but only on how many nontrivial deductions have been done.
Notwithstanding the interest such statements may arise, they are still very limited in applications. One of the paths I think I will walk in the future is using information to clearly define case-defined functions. In what do they differ from "simple" ones? And what properties can we prove, in a certain environment, if we exclude such too "strange" functions?

What is disorder?
The problem, of course, arises when considering infinite sets: in finite contexts there is no disorder.
What is called "disorder" is, actually, the possibility to find sets with no defining property at all. This deeply differs from the analytic definition of "chaos": a chaotic behavior of a function does not mean that the image set is nondefinible at all.
An easy conjecture is that disorder is a consequence of the axiom of choice. But can we establish a thorough theory of disorder, maybe even isolating differents degrees of disorder, of excluding it in some situations?

In my leaving exam dissertation, I tried to built a coherent mathematical interpretation of social changes, focusing on psychological responses rather than on (already widely investigated) economical aspects.
That attempt resulted in what I dare call an interesting alternative view of the problem. In fact, as far as I read or heard on the topic, it seeems like every social change must be economy-based, which is rather trivially not true in general.
Moreover, every psychological investigation seems to be doomed to be scientifically inconsistent, if not pseudo-scientific, which is a pity.
A mathematical theory solves both the problems: it provides an original sight and, even more important, is quantitative and then open to be judged objectively by the trial of facts.
Being this theory a "juvenile work", it requires deep revision and considerable widening. So, although applied Mathematics is not really my field, I will probably turn to it in future for revision.