What do you mean with modern math?
Well, basically everything on the style of "let´s take a paracompact subset which is also hausdorf, then..." that is, things exposed usgin point set topolgy and in general rigurous definitions.
In physics we studied (now its a bit different, and worst) analisis of one variable, vector cauculus and linear algebra in first course. Diferential equations, more linear algebra, some of tensor algebra and tensor analisis, and complex variables in second. Mathematical methods (special funtions, some integral transforms and in general everything in the style of the arfken or matwes walker book in third. Statistic and,if you where in theoretical physics gropup theory (as continuous groups, the teacher didn´t have any idea of what a topolgicla space, nor to say a manifold or diferential forms were, well, at least not before i teached him in the work necessary to can passs the asignature
)
I had to get all the modern math in the math licenciature, basically geometry, topology and functional analisys (i only got the asignatures necessary for physic, i am getting the others now).
Well, for exemple about categories theory my professors told me nothing more than it exist!
Feel happy about it. My fvourite branch of maths is topology. So I got the book which covered more topics in algebraic topolgy to complemente the asignature in fourth of maths. It is written by Spanier. And it begins introducing the notions of categories and functors betwen them and using it ocasionally. That´s all my knowledge about them.
And it was engought. It is not that i find then particularly difficoult.Simply they looked me unnecesary excess of formalism. When i readed Baez saying they were meant to be the future of LQG I thought theory had died even before reaching it´s infancy, in fact i left the subject behind for a time partly because it. I am glad to see it has not been the case. The work in black holes singularities has renowed my interest in the subjecto (and also seeing that categories seems to have voltilized from it
).
Of course these is a prejudice of my. But in topology I prefer the line of thinking and exposings of the books of Jonh Milnor, and categories don´t appear there xD. I see them usefull for organing ideas but i can´t realize how any basic understunding of something (nor to say quantum gravity) can merge from it (i am very happy to not have any reputation to keep and feel free to say these things withouth a deep knowledge of the matter
)
About mechanics i studied in second classical mechanics which included lagrangian mechanics and aplications. Hamiltonian mechanics. And the las theme was Hamilton Jacobi and action angle variables. Never agian used them. In theoretichan mechanics,fourth course, we saw constrained systems and the Dirac formalism. I sutied a bit of symplectic mechanics myself but never had ended to see the point in it (now thanks to these forum people i see the sense of all that annoying things i had readed just for,er, snobism?
. Ok, ok, supossedly i had in the far horizont the idea that it could serve me to understand the demostration of the kolmogrov´s theorem about the stability of the solar system, but i never went so deep.
So that´s why the first chapters of Rovellis books were hard for me. I just saw the exposure of the same things in all the available languajes of mechanics (two of which were unfamilar for me) withouth seeing a good reason for it.
A propossito Francesca io parlo Italiano e anche posso legerlo senza tropi problemi. Vero e che fago molti errori ma comme puoi ridere anche li fago in anglese, cosi se tu hai qualquna referenza interesanti in italiano per me non chie problema.
As i said a am studing now the 2003 review of Alejandro Perez in spin foams. Good to see that it could be possible to ask him my doubts. Althought if i expose then here maybe someone else has similar doubts and all of as gain something (that´s the reason for these forums, isn´t it?)
P.S. Murphy law strikes again. A lot of papers withouth any mention on categories and when i arrive at page11 of the Alejandro article I read that spin networks define a category and that spin foam theories are morphisms betwen them. Better i should have been quiet.
