The reason I have to keep relearning this stuff is that I do not use/make use/practice this language on a daily basis in my work. My bread-and-butter mathematical activity is in formal symbolic exact solutions to polynomials and transcendental functions and differential equations using the various formulations of the Lagrange Inversion Formula (LIF). The problem is that I end up working very much alone and isolated, which makes it very difficult for me to maintain connected with other mathematicians doing this kind of work. Life and work are like an endless struggle of juggling water with your hands, desperately trying to prevent what knowledge one has accumulated in the past from dropping on the floor and evaporating.
One of the things that makes subjects like this so hard and confusing to newbies, and certainly was to me, was that, to newbies, everything looks like you're not really saying much or doing anything. e.g. an inclusion map from a variety, V, into the outer space, C^n, sounds completely trivial, and therefore, unnecessary. Why introduce an inclusion map? But, what novices to this subject forget is to keep in mind at every step in terms of which variables or which functions a given variety is known or defined. i.e. {(1,3,6,x4,x5)| x4,x5 are in C} -> {(x1,x2,x3,x4,x5)| x1=1,x2=3,x3=6,x4,x5 in C} is an inclusion map of a variety into C^5 where we explicitly know the values of the points on the variety But, if the variety V were given as the closure of the ideal generated by f(x1,x2,x3,x4,x5)=(x1^7*x3^87-8*x2^55+1991*x3)*x4*x5 g(x1,x2,x3,x4,x5)=(x1^4-7*x1^2*x2^55+1991*x3^2)*x4*x5 h(x1,x2,x3,x4,x5)=(x1^16-51*x2^55+1991*x2*x3^5)*x4*x5 then it might not be so obvious what the inclusion map is.
The reason I have to keep relearning this stuff is that I do not use/make use/practice this language on a daily basis in my work. My bread-and-butter mathematical activity is in formal symbolic exact solutions to polynomials and transcendental functions and differential equations using the various formulations of the Lagrange Inversion Formula (LIF).
The problem is that I end up working very much alone and isolated, which makes it very difficult for me to maintain
connected with other mathematicians doing this kind of work. Life and work are like an endless struggle of juggling water with your hands, desperately trying to prevent what knowledge one has accumulated in the past from dropping on the floor and evaporating.
One of the things that makes subjects like this so hard and confusing to newbies, and certainly was to me, was that, to newbies, everything looks like you're not really saying much or doing anything. e.g. an inclusion map from a variety, V, into the outer space, C^n, sounds completely trivial, and therefore, unnecessary. Why introduce an inclusion map?
But, what novices to this subject forget is to keep in mind at every step in terms of which variables or which functions a given variety is known or defined.
i.e. {(1,3,6,x4,x5)| x4,x5 are in C} -> {(x1,x2,x3,x4,x5)| x1=1,x2=3,x3=6,x4,x5 in C} is an inclusion map of a variety into C^5
where we explicitly know the values of the points on the variety
But, if the variety V were given as the closure of the ideal generated by
f(x1,x2,x3,x4,x5)=(x1^7*x3^87-8*x2^55+1991*x3)*x4*x5
g(x1,x2,x3,x4,x5)=(x1^4-7*x1^2*x2^55+1991*x3^2)*x4*x5
h(x1,x2,x3,x4,x5)=(x1^16-51*x2^55+1991*x2*x3^5)*x4*x5
then it might not be so obvious what the inclusion map is.