Well I am happy to say that there has been some learning going on at my desk. During the past few days I have been going through the sections three sections of the second chapter of Nakahara's book. That would be basics of maps, vector spaces, equivalence relations and topological spaces.
In theory I should know must of it since I took classes on Topology, Linear and Abstract Algebra, and Set Theory. The truth is that I have forgotten must of it, which is a shame. But reading it has been easy. At least I have been able to work out the exercises. Then I turned my attention to chapter five on manifolds. I choose to "ignore" the chapters on homology and homotopy groups. I technically read the first part on homology groups (on simplexes and simplicial complexes) but that was on an airplane heading to Puerto Rico and later back to NYC.
The first task is to define what a manifold actually is. Once this is done you go on and define what a vector is on a manifold. With vectors one can define the dual space of linear functions and then you can construct higher tensors.
I got kinda stuck while discussing differential forms. It was kind of sad, since I know that this exercises is easy. Basically one has to prove that given an r-form and a q-form the exterior product between them will be the same as that with reverse order multiply by a minus one to the power of the products of r and p. And also one has to show that for r an odd number, the exterior product of an r-form with it self vanishes. Both of this identities follow from the definitions of a differential form and the exterior product, it just include a counting part. Sadly I suck at counting things.
That was my least favorite part in statistical mechanics: counting micro-states or configurations. It is very sad that I have been always so stubborn and skeptic about counting and number theory in general. I am adding it to the list of things I should learn right before I die.
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