Originally Posted by DrBenway
I'd be curious to know your area of research, if you could put it in terms that a layman could understand. Number theory? Or something I haven't ever heard of?
I work in an area called low-dimensional topology (Low-dimensional topology - Wikipedia, the free encyclopedia
) which is essentially studying the "shape" of 2, 3 and 4-dimensions spaces that are locally modelled on Euclidean space (R^2, R^3, R^4 respectively). Topology is similar to geometry except one does not pay attention to measurements or angles (you can stretch and bend your space and it is still equivalent to the original). You could think about networks or circuits where the important thing is how nodes are connected, not how far apart two nodes are or at what angle they meet. One of the essential tools in low-dimensional topology is a knot theory (which is also an rich and highly exciting area of research on its own). Currently I am doing a lot of work in knot concordance which is a subfield of knot theory (Knot theory - Wikipedia, the free encyclopedia
) that examines the way that knot can bound surfaces in 4-dimensions. This subfield is especially useful in the understanding of 4-dimensional spaces. It is a very exciting area of research that uses tools from many other areas of mathematics: abstract algebra, number theory, functional analysis, etc. Also, one of the big problems in low-dim topology has recently been solved (Poincare Conjecture
). The interesting thing is that it was solved by a geometer, Grigori Perleman. There are some interesting stories that go along with this if you want to look it up. Unfortunately, we still do not have a purely topological proof of the Poincare Conjecture.
Hope that is helpful.