By popular demand the first text is the entire speech of my 3 minutes thesis presentation, which didn't at all fullfil the purpose of explaining what my thesis was about but made quite an impression on some people in my research department.
Imagine that as a child somebody had taught you a game. A game which alike Chess has very simple rules as well as a huge operational space, but which unlike Chess is not played on a board. It's an abstract mental game. As you play you quickly discover that this game has got a very particular feature: whomever plays it discovers languages, methods and tools that allow no less than the understanding and mastery of Reality: from understanding the hyperspace shape of the universe to being able to make my phone still work on crappy signals.
The game is called mathematics, of course, and, as I like explaning to people in the street, has little to do with the horror misleadingly called "mathematics" that they encountered in high school.
In the mind of mathematicians, mathematics start with simple objects and the simple relationships between them. But quickly we discover that collections of similar objects can have very regular shapes, patterns and structures associated to them. Sometimes we call those aggregations "mathematical spaces". Of course, as we come across them we give them names --so that we can refer to them later-- and then we discover that those spaces relate to each other in non trivials ways; and that leads to new objects and new parts of the game...
In fact, from your early university education up to end of your life as a mathematician, the most beautiful outcome of our species mental activity, slowly unfolds and reveals itself inside your mind.
So then what is the problem ? Well, the problem is that we still know very little about the mathematical universe. As part of my PhD, for instance, I focus on a not very well known --but relatively painful-- collection of objects called Hankel Operators, and in fact an even smaller clique called Compact Hankel Operators. The aspect of them that drives my work is the study of what is referred to as their Spectral Theory.
I would love to explain how those objects are built, but that would take time; a lot of time. Instead, I will tell you something funny. I have no idea whether there are real life applications of my work; and to be honest with you, I just don't care. This said, if past history is any indication of the future, then I know that probably long after my death, somebody, somewhere, maybe somewhere else in the Galaxy, will design a system whose understanding will be partially relying on that one pure maths thesis written at Kings College London in the early 21st century. A thesis called "Spectral Theory of Compact Hankel Operators".
Thanks for listening. I hope you enjoyed it :-)
The second text is an email I wrote today as a follow up of a lunch discussion with some collegues. (Subject line was "Universes")
Quick clarification while being in the tube :-)
The human mind has discovered three different universes so far: (1) the physical universe (the thing in which we are, and have been since about 13 billion years ago) and probably a larger structure called the multiverse, (2) the mathematical universe and (3) the computational universe.
The physical universe is easily identifiable, of course. We attempt its understanding through the discovery of the Laws of Nature. Quantum mechanics, General relativity, String Theory are all relatively recent and very elegant models that match our observations so far. The structure called the Standard Model, for instance, which describes elementary particles of matter (as well as their interactions) is the most studied, experimentally confirmed, structure ever formulated by Man. And while being on the subject of the physical universe I have to mention stone age attempts that are still with us under names such as religion or spirituality.
The mathematical universe is where I spend my time. We have started to explore it about 6,000 years ago, when tax collectors needed the ability to rebuild lands after the seasonal flooding of the Nile. At the time they used straight edges and compasses, but what was happening is that their minds were starting to manipulate abstract concepts such as time, space and numbers that many thousand years later is encapsulated in the amazing mathematical knowledge we have accumulated and carry on doing so. Our entire world relies on mathematics.
The computational universe is the youngest (not even 100 years old). We have spent so many centuries under the effects of the Greek civilisation beliefs in the purity of mathematics that it took us an amazingly long time before starting to understand the notion of Abstract Process, which is not a native notion of the physical universe nor of the mathematical universe. Basic questions of the computational universe are, for instance, What is information and how to manipulate it ? Are there natural transformations that describe its behaviour ? What is artificial intelligence ? as well as (my personal favourite) What is the actual name of the abstract process that Natural Evolution (Darwinian Selection) is the implementation of ?
My point at lunch time was that we are indeed just at the beginning of our exploration of the computational universe. People focus on computers and computer networks, but in fact those toys are for the computational universe what telescopes are for astronomy. Programming and development will at some point stop being like alchemy and will become a proper science. Then the next two or three hundred years should be interesting :-)