Question

I am wondering what are the prerequisites for being a modern geometer? It seems that the amount you have to know is just huge: differential geometry, differential topology, algebraic topology, algebraic geometry, symplectic geometry, ... and then there is all kinds of overlap. So my question is the following: what should you know (from the topics I mentioned and the topics that I forgot) to call yourself a geometer (lets say by the time you get your Ph.D.)? And where are the best books/articles covering all that?

Answer 1

The question asked effectively is "What does one have to know ... to call oneself a geometer?" and as several comments note, that question is tough to answer, nonspecific, and the answers are scary ... and yet the question swiftly picked up two "favorite" votes too.

If the question were rephrased as "What ideas are geometers pursuing?" then the MathOverflow community might be able to supply answers that are more specific, useful, and inspiring to students beginning their research.

Notable mathematicians have written many fine essays on this topic. Commonly these essays are more-or-less centered around a guiding idea that was articulated by Mac Lane in his Mathematics, Form and Function (1986) as follows (and I'm posting this as an "answer" solely to be able to format this quotation properly):

Analysis is full of ingenious changes of coordinates, clever substitutions, and astute manipulations. In some of these cases, one can find a conceptual background. When so, the ideas so revealed help us understand what's what. We submit that this aim of understanding is a vital aspect of mathematics. [...]

Effective or tricky formal manipulations are introduced by Mathematicians who doubtless have a guiding idea---but it is easier to state the manipulations than to formulate the idea in words.

Just as the same idea can be realized in different forms, so can the same formal success be understood by a variety of ideas. A perspicacious exposition of a piece of Mathematics would let the ideas shine through the display of manipulations.

Nowadays the notion of "geometry" has become so broadly generalized, as to be effectively identical to Mac Lane's notion of "a piece of mathematics whose ideas shine through the display of manipulations."

For systems engineers nowadays (me in particular) geometry is largely about the dynamical flow of complex systems ... and of course we want our engineering understanding to "shine through the display of manipulations" .. but it would be a grave mistake to imagine that geometric understanding of dynamical systems is all that geometry is about ... because geometry has evolved to become a much broader notion than that.

Therefore, a reasonable piece of advice to young researchers nowadays—in math, science, engineering, and even medicine, it doesn't much matter which—is not to ask oneself "What articles and books should I read?" without first asking oneself the organizing question "What articles and books will I someday want to write? What will be the core ideas? How will I explain these core ideas clearly?"

As soon as you can write down those ideas in even a hazy and uncertain form, then your research career will have begun ... as you learn, your ideas will slowly take concrete form ... and almost certainly you will be led to the study of geometry in its many modern forms.

Perhaps this is why, 2400 years after Plato's Academy first affirmed it, the members emblem of the American Mathematical Society until recently (and maybe still?) said in greek: "Let none but geometers enter here".