Motivation and Details:
Mathematics and mathematical physics have been my passion since long ago. I started reading Einstein's book of Relativity at the age of 13, then I thought much about it, and, again and again, I imagined other ways of presenting it. For this and other subjects, I mainly worked on my own, by thinking, in search for the best way to understand the subjects, to grasp their deep mathematical meaning and elegance.
I felt that the maths (and physics) courses I attended at school and in "classes prépas" (intensive undergraduate classes), were not as good as what could ideally be done that I dreamed about, how I believed (as my experience showed me) mathematics really are and should be presented.
Not to speak about the problem of the institutional context of my exprerience (that it was an intensive preparation to contests), I think that the contents of a number of courses could be made better than what they are now.
I would distinguish in this problem a few aspects which can be more or less present or intricated:
- Sometimes, the choice of what to present to the students is not the best: some difficult, lengthy and monotonous developments happen to be presented years before what is interesting and elegant. (For example, real analysis is often developped in lengthy details long before complex analysis is introduced, but complex analysis is more interesting).
- But also, one given subject can be not presented in the best (elegant) approach. This is less a matter of choice (that could be repaired by merely listing the subjects in the right order with the right references) than the fact that the good approach, either has not yet been written until now, or is unknown by most teachers.
- Sometimes also, key ideas are known but omitted, or not given on time, like the fact that the determinant of a matrix represents the volume of the parallelepiped. The work would be to choose better the properties of the objects to present earlier as a way to introduce them.
These problems are related in the fact that a subject that would ideally deserve to be introduced at the undergraduate level for its interest and simplicity (usefulness or elegance), is not and cannot be, as long as the only presentations of it that can be found on the market appear as complicated, which makes them only accessible to the graduate level or higher.
What interests me most is to develop such new approaches of fundamental subjects, to make them appear clearer and more accessible/interesting than how they now look in the litterature. I think these questions have important consequences on the future of mathematics, both for the general image of mathematics to the public, and for providing the students with a larger mathematical culture (to give them a better taste and experience of mathematical elegance, and a better chance to learn more subjects by saving time and efforts with a more efficient presentation of the basic subjects)
I started writing such presentations, but only in French up to now.
I am ready to make some English versions soon.
I wrote about Special relativity theory (without the usual expression of the Lorentz transformation formulas), the cosmological models, affine geometry, vector spaces and duality, complex numbers, operations on quantities, centers of gravity, projective geometry, quadratic spaces, the Riemanm sphere and hyperbolic plane and their generalizations (geometries of inversions or with a constant curvature), the black hole's horizon, and a geometrical description of forces in a plane.
Then I worked for my PhD on a more standard research subject: the Configuration
space integral for links and tangles. I choosed the subject myself: after
reading some articles in the domain of Vassiliev Invariants, I imagined
a construction that complemented other works dealing with the problem of
comparing this integral with the Kontsevich integral. I first imagined
it in terms of nonstandard analysis, but the reactions of other mathematicians
lead me to search for a reformulation of this construction in terms of
a compactification. I made it successfully, contributing to a better formalization
of the construction of this integral formulated by other mathematicians
and physicians on the subject, into a more rigourous approach. My intoduction
in French presented a "classical mechanics" version of the Chern-Simons
gauge theory, whose quantum field version (seemingly the only one considered
until now) was known as what gave the configuration space integral as a
Now the first part (construction of the integral for links) has been published. The referee had asked me to make some more developments to make the second part (extension to tangles as a parallel construction to the Kontsevich integral) more rigourous and readable. So it is a work that can be done in the future.
But in the last two years, my main research (or what I consider as such)
consisted in making a new start of my project : I wrote some chapters that
aim to be the beginning of a future book of mathematics and physics presenting
some fundamental theories as I see them, as they could be introduced in
an undergraduate level.
I already wrote about : some comments introducing to mathematics, physics and their fundations, more detailed explanations of the meaning and developments of Special relativity theory, the least action principle, an introduction to spinors, the Boltzmann law and the statistical definition of entropy, an introduction to quantum mechanics with the EPR paradox, a certain "dynamical" approach of the fundations of mathematics (naive set theory and model theory).
They are here : Relativity; in French: geometry-logics.
I taught mathematics for two months and a half in some Romanian high school classes, and then also presented my approach of the Special relativity theory a number of times in other Romanian high school classes (for an hour or a couple of hours each time) and also to undergraduate students. I did that often in French but sometimes also in English, which was not a significant obstacle for me. I had a positive feedback from the pupils and teachers who generally liked what I presented there. Already before, I had presented my PhD in English, then this two years trip in Romania and (at the end) in Poland made me train more my English, so even if it is not perfect I think it will not be a problem for me to teach mathematics in English.
In the future, I plan to write other developments, including: more about the relations between geometries and about their axiomatisation; linear algebra, tensor calculus, classical mechanics, general relativity and quantum mechanics; set theory, universal algebra (as a unified version of algebra including categorical properties, and the completude theorems of propositional calculus and model theory).
But I would be even more interested in presenting directly these subjects to students. Not only it would provide me a concrete motivation for continuing my work, but also the interaction with the students would be a great experience that could help improving this work. I think such courses could also bring some new interest and open fundamental questions that could be good research subjects for students who may contribute, even maybe at the undergraduate level.
Of course, I will also be ready to adapt to the directions and recommandations
of the University/College that would accept me, to respect the coherence
of their works, programs and schedules.