Forum index » Hardest Mathematics field?

This forum seems to not be very active. I think we should have more interesting topic that we can discuss.
This is my attempt.

I have a friend at MIT grad school who took algebraic geometry and told me it was the hardest class he took. He told me about a grad student at MIT who tried to do his phD thesis on algebraic geometry. After years of trying he switch to combinatorics. Then after he graduate he now work in the industry and not in academia.

I just came back from a math conference about combinatorics. The last speaker talked about local and global rigidity of framework graphs. I did not understand most of the talk but I believe it has to do with algebraic geometry. Usually after a talk there is usually at least 1 question asked but after this one everyone was quiet.

I am under the impression that algebraic geometry is the hardest mathematic field.
What do you guys think?

My vote definitely goes to algebraic geometry as the hardest subject in mathematics. One of my professors spent 3 whole years at UC Berkeley just working on his thesis in algebraic geometry.

I've heard many a grad student complain that algebraic geometry is the hardest subject to learn, at the very least. Theres a lot of different topics you need to have mastered before you can begin doing real work, and mastering one of those topics relies on you mastering all the others.

When I asked about how long it typically takes to get your PhD at Washington, they said it depended on your field: a field like combinatorics, where there isn't much you need to know before you can start doing real work would probably only take 4 years, however algebraic geometry would probably take 6.

I don't know about hardest to do, though.

I have a friend taking an intro class in algebraic gemoetry right now. Sounds like it's a total pain...and makes me not want to take it.

Keep in mind, though, that the purported difficulty of algebraic geometry is due to the comprehensiveness of the field. It unites ideas from pure geometry, pure algebra, topology, combinatorics, algebraic topology, differential geometry, real and complex analysis, and (especially recently) higher structures (i.e., all algebraic spaces are collectively classified as belonging in the localic ringed topos of etale sites, and one can construct sheaves with such topoi which encode geometric information about these spaces; e.g. an algebraic stack).

I work in algebraic geometry, and it is for its richness and beauty that I study it. It can seem daunting at times, but one cannot go without a passing knowledge of some algebraic geometry if one wishes to be taken seriously in the modern research environment. I mean, I started off being intrigued by category theory and algebraic topology, and that led me to algebraic geometry, which in turn has led me to understand and be interested in topics ranging from representation theory to hopf algebras to string theory.

So don't sell the discipline short; if you want to learn it, I suggest reading the following texts, then checking out Hartshorne's "Algebraic Geometry;"

+Calculus on Manifolds: exterior forms, tensors, a bit of complex analysis
+Introduction to Topology (Gamelin): Develops the necessary topology in a very algebraic way
+Algebra: Chapter 0 (Aluffi): Abstract algebra with some categories and homological algebra
+Categories for the Working Mathematician (Mac Lane): Categories!
+An Invitation to Algebraic Geometry (Smith): Beginning AG--Grassmannians, Schemes, Affine Projective varieties--all at a very non-intimidating pace

...now try Hartshorne. Trust me, you won't regret it; the subject of AG is the vantage point from which the beauty from the rest of mathematics may be observed.