Wednesday, September 30, 2009

in which, not being well-read, i am sidetracked by reading.

.. 68 references and counting,

which means i've reading reviews, checking up on facts,
and occasionally hunting down and browsing through articles.

there's a lot i never learned.

i'm not a very well-read mathematician .. not by my own standards, at least. sometimes i wish i were born earlier, when the analysιs of metric spaces was in its true nascence or before.

this field follows a long history, from many traditions.

when i was a graduate student, it had already begun to take shape. subsequently one can now train a student in purely metric ways and avoid classical formulations altogether .. not that one should.

for example, in heιnonen's lectures on analysιs on metric spaces, the first discussion of capacιty (chapter 7) doesn't really do justice to the history. by this i mean that the "mod=cap" theorem feels tautological. the definitions make it so.

in contrast, at the time it required nontrivial efforts by gehrιng and zιemer to prove the "mod=cap" theorem in the classical setting. the point is that one had to construct a function with the right gradιent, not just an upper gradιent ..

.. never mind the rich geometry one can do with capacities. in some sense, they are the analytιc cousins of hausd0rff measures; you can even study rectifiabilιty of sets in terms of capacitιes.

i've been feeling the same way about many other topics, which puts me at an impasse:
  1. there's no time for me to re-learn the past. there's two more years of postdoc left, and one more year before i fill out the job applications.

  2. without some awareness of the past, and how some constructions came to be, i have that much less intuition with how to proceed, much less pose well-motivated problems.

    if you learn only axiomatic definitions, then how do you escape pseudo-problems arising from the continual manipulation of axioms?

oddly enough, i had meant to write this not to as an uneasy invective, but as something positive. what i really wanted to say was this:

the more i read about them,
the more i am impressed with euclιdean spaces.

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