Posted by: Alexandre Borovik | August 19, 2008

Fixed part, moving part

An e-mail from Peter Dalakov:

I am sending this in connection to your call for stories, though it doesn’t really fit your criteria: I am sending a couple of young age stories which are not quite mathematical, and a few mathematical, but adult…

I am male, my school and university (masters) education has been in my native tongue, my PhD study has not. I have just finished my PhD and will be a postdoc.

When I was about 5(?), I had a problem understanding the concept of Earth’s rotation: I expected that if the Earth really rotates, then when I wake up at night I should see the room “flipped”.

( Other questions that bugged me at the time (or earlier) were “Does it really get dark when I close my eyes?” and “How is it possible that there is no air in cosmos if air=nothing?”. )

I had a short moment of bewilderment (around 14 or 15 or 16?) when I learned that the graphs of $f(x-a)$ and $f(nx)$ when compared to those of $f(x)$ behave “in the opposite way” to one’s expectation. I got it very quickly and easy, but I remember being surprised.

And this flows into similar adult experiences:

Does the “transition matrix” transform the basis or the coordinates? (Actually, many books hide the appearance of the inverse of the transpose by suitably defining the transition matrix) Given a matrix of a linear map, am I writing the map between the vector spaces or between their duals? Do the transition functions of a vector bundle transform the frame or the sections? Am I looking at the sheaf $\mathcal{O}(D)$ or $\mathcal{O}(-D)$ ($D$ – divisor on a variety), and do its sections have a pole or zero at $D$?

This is all one and the same question and one learns to recognise it, but it is amazing how persistent it is.

What I am going to say now probably won’t make sense. It seems that often there is a ‘fixed part’ and a ‘moving part’ in a problem, and I have the feeling that my mind often gets confused by looking at the ‘moving part’ and forgetting that it ‘moves’ with respect to the ‘fixed part’.

This refers both to mathematical and worldly experiences. In math this is often a question about inclusions vs natural inclusions (equalities). In daily life one example of a similar problem is learning the streets of a city. I have the feeling that often my brain stores some images separately, say, the direction in which I traverse a street for the first time gets stored with higher priority. Or there may be several images of the same place (from different viewpoints) which don’t exactly glue. As if the brain prefers to work with the moduli stack and has a problem when passing to the moduli space.

Please, send me more stories like that!