Year 12 Research assignment, Geometry. I'm currently deciding on a question to lead my research for a large final school year assignment. So far the question i have is: "Which geometric theorem has most influenced 21st century geometry?
First i would like some advice on the question itself, How does it sound, is it possible to make it better? 
Otherwise what are some theorems you can recommend me? What other research should i approach?
What does 21st century geometry look like? How is it used in the real world?
Thanks in advance!
 A: I have two suggestions for you. The first one is to study the developement of group theory in relation to geometry, which was developed in the late 19th and 20th century. I think it's one of the most fascinating parts of "modern" geometry, and you could get a much deeper understanding of it than something fancy from the 21st century, which I think is more important in a research project.
My second suggestion is to study the sphere packing problem, which is to find the optimal way to pack spheres in a given dimension. The case for two dimensions is pretty intuitive, and can be proved rigurously with some basic group theory. The case for three dimensions was solved in the 20th century, and it's one of the first proofs that used extensively computer calculations. The case for dimensions 8 and 24 was solved last year, so that gives you some 21st century stuff to look at, although it gets closer to topology than geometry. I think a project on this would not only be interesting because of the problem itself, but also because it shows how mathematics developed in the last century, by looking at the different methods used for the proofs.
Hope that was useful!
A: You should 


*

*limit yourself to the 20th century,  

*not restrict yourself to theorems (but consider "domains" instead),

*not look for "the most this or that" ("the best in the world" trend is - maybe - good for commercial purposes, but very elusive in scientific areas). Consider instead "an important domain". 
@user404944 has given you a list of themes. I could add all the themes where geometry uses graphs, combinatorics, etc. ; one example among many: the coloring theorems ("four colors theorem" for example, with recent progresses using new software tools, such as Coq). 
A big issue is that you have to master the subject. If it is too abstract, or too wide, it may be very difficult  for you to reach a good synthesis. 
For the choice of a subject, I advise you to consult synthetic presentations such as the wonderful recent book "Princeton Companion to Mathematics" by T. Gowers (http://press.princeton.edu/titles/8350.html). 
