Hardest Mathematics field?

Forum for the GRE subject test in mathematics.
toopham
Posts: 20
Joined: Sat Apr 10, 2010 9:36 pm

Hardest Mathematics field?

Postby toopham » Sat Apr 24, 2010 9:39 pm

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?

topgun70009
Posts: 3
Joined: Fri Apr 23, 2010 2:05 am

Re: Hardest Mathematics field?

Postby topgun70009 » Sun Apr 25, 2010 8:39 pm

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.

origin415
Posts: 61
Joined: Fri Oct 23, 2009 11:42 pm

Re: Hardest Mathematics field?

Postby origin415 » Mon Apr 26, 2010 8:12 am

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.

CasinoMadrid
Posts: 5
Joined: Thu Apr 29, 2010 9:42 pm

Re: Hardest Mathematics field?

Postby CasinoMadrid » Thu Apr 29, 2010 11:03 pm

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.

lambdafunctor
Posts: 5
Joined: Mon May 17, 2010 3:24 am

Re: Hardest Mathematics field?

Postby lambdafunctor » Mon May 17, 2010 11:03 pm

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.

Manifold
Posts: 1
Joined: Mon Jul 11, 2011 2:30 am

Re: Hardest Mathematics field?

Postby Manifold » Mon Jul 11, 2011 2:34 am

I think it would be... Nonlinear Dynamical Systems!! This a field with a mix of Partial Differential Equation, Real Analysis, Topology, Bifurcation Theory and Numerical Analysis.

dragoonlp
Posts: 54
Joined: Thu Mar 25, 2010 4:38 pm

Re: Hardest Mathematics field?

Postby dragoonlp » Mon Jul 11, 2011 2:49 pm

Yea, as an undergra student, I just took a very, very low level algebraic geometry in my school. It is hard, definitely!!
But here is my idea: well, I admit that algebraic geometry is difficult, but I do think the PDE is hard too. Maybe not hardest one, but, how can you compare PDE and algebraic geometry who is more harder? by what sense?

just an idea~~ :D

YKM
Posts: 45
Joined: Mon Aug 08, 2011 10:25 am

Re: Hardest Mathematics field?

Postby YKM » Tue Aug 16, 2011 4:14 am

Based on what I read from the history of Math, most ppl voted that number theory is indeed the hardest field. There is an old saying, "if a problem can't be solved in 100 yrs, then it is probaly a problem from number theory.... "

cauchy2012
Posts: 22
Joined: Sat Aug 13, 2011 9:04 am

Re: Hardest Mathematics field?

Postby cauchy2012 » Tue Aug 16, 2011 5:55 am

Come on! I think noncommutative geometry is the hardest field.

noidea
Posts: 2
Joined: Fri Aug 12, 2011 7:27 am

Re: Hardest Mathematics field?

Postby noidea » Tue Aug 16, 2011 7:32 am

I think, Analytic Number Theory.

aeginotu
Posts: 9
Joined: Mon Mar 21, 2011 9:09 pm

Re: Hardest Mathematics field?

Postby aeginotu » Tue Aug 16, 2011 11:23 pm

I don't think there's a hardest field. There are fields where one has to know a lot of prerequisites to get into it, and among those the most 'difficult' I think are algebraic geometry and noncommutative geometry. I'm not really sure which has more prerequisites.

Fyrstenburg
Posts: 1
Joined: Fri Feb 17, 2012 4:30 am

Re: Hardest Mathematics field?

Postby Fyrstenburg » Fri Feb 17, 2012 4:47 am

Topics which synthesize analytic and algebraic techniques are typically the most difficult, because most people have trouble with one or the other. Some such areas, all of which also have an enormous number of prerequisites, are algebraic geometry, algebraic number theory, ergodic theory and arithmetic combinatorics. However, analytic number theory is not a comparatively difficult field; it is simply that the problems which still exist in number theory that are anticipated to be solvable analytically are rather ridiculous. Neither is proper algebraic topology, although an increasing number of problems in algebraic topology have been solved by considering them in the context of geometric topology, which is most definitely a very difficult field. But in nearly any field, there are sub-fields which can be enormously difficult to work in. Some examples in fields that were not mentioned above might be model theory (in logic), Lawvere theory (in category theory), mechanism design theory and latin squares (in combinatorics), disk algebra (in complex analysis), and L-Theory (in K-Theory). In noncommutative algebra and algebraic number theory, there are a number of structures and techniques which are very specialized and extremely difficult: nonassociative rings, quasigroups, cyclotomic fields, Iwasawa Theory, etc.

If we are talking about at least moderately broad fields, I would say that algebraic geometry, algebraic number theory, ergodic theory and arithmetic combinatorics are the most difficult fields to work in. Algebraic number theory is a bit of an odd man out, though; the material is certainly difficult, but the difficulty with algebraic number theory really lies in the fact that you need to be a true master in a huge number of areas, any one of which is a field in and of itself. For instance, a great algebraic number theorist will be an expert in algebraic geometry, but the converse is not necessarily true. If we are allowing sub-fields, then I would vote for geometric class field theory, crystalline cohomology, Iwasawa theory, geometry of numbers, descent theory, capacity theory on algebraic curves, arakelov geometry, latin squares and Milnor K-Theory. I know very little about mathematical physics, so I can't comment on anything from that field. Some problems in combinatorial game theory and geometric group theory are also intractable, but I would not say that those are exceptionally difficult fields (merely that exceptionally difficult problems can arise in them).

A rule of thumb: if you're wanting to avoid difficult subjects, then avoid anything with "cohomology," "arithmetic" or "geometric" in its title.


A final note: the Inverse Galois Problem and the Erdos Conjecture on Arithmetic Progressions are two of the most difficult problems in mathematics, and neither are strictly from any of the topics listed above. Every field has extremely difficult problems in it. Somebody who is an expert in algebraic geometry, for instance, would likely have less success on a big problem in the field than somebody who is moderately proficient in algebraic geometry, but is also proficient in a number of related fields. Big problems in common topics are not going to be solved by using techniques specific to that area; if they could be, then they already would have been. It is more useful to know some about many fields, so you can see a problem in a potentially new way than people who have already looked at it with a more narrow lens.

akpandey1
Posts: 5
Joined: Thu Dec 29, 2011 2:03 am

Re: Hardest Mathematics field?

Postby akpandey1 » Fri Feb 17, 2012 12:43 pm

I can't say it is hardest but Harmonic Analysis is one beautiful piece of mathematics. You take a group and then you try to calculate all it's representations and then through all irreducible representations you try to develop the whole Fourier analysis theory on this group. I like the subject as your goals are quite clear and if at some point you are stuck then you can easily switch to fields like Representation theory and Operator Algebras.

dasgut
Posts: 250
Joined: Thu Jan 21, 2010 12:20 am

Re: Hardest Mathematics field?

Postby dasgut » Fri Feb 17, 2012 12:54 pm

akpandey1 wrote:I can't say it is hardest but Harmonic Analysis is one beautiful piece of mathematics. You take a group and then you try to calculate all it's representations and then through all irreducible representations you try to develop the whole Fourier analysis theory on this group. I like the subject as your goals are quite clear and if at some point you are stuck then you can easily switch to fields like Representation theory and Operator Algebras.


NO NO! Math finance is the best! I mean what other field gives you the chance to possibly name a result "The Mo' Money, Mo' Problems Theorem?"

omgmath
Posts: 19
Joined: Sun Nov 06, 2011 11:45 am

Re: Hardest Mathematics field?

Postby omgmath » Fri Feb 17, 2012 1:26 pm

akpandey1 wrote:I can't say it is hardest but Harmonic Analysis is one beautiful piece of mathematics. You take a group and then you try to calculate all it's representations and then through all irreducible representations you try to develop the whole Fourier analysis theory on this group. I like the subject as your goals are quite clear and if at some point you are stuck then you can easily switch to fields like Representation theory and Operator Algebras.


Very interesting! Are there good surveys that introduce these ideas? Thanks

akpandey1
Posts: 5
Joined: Thu Dec 29, 2011 2:03 am

Re: Hardest Mathematics field?

Postby akpandey1 » Sat Feb 18, 2012 10:36 am

omgmath wrote:
akpandey1 wrote:I can't say it is hardest but Harmonic Analysis is one beautiful piece of mathematics. You take a group and then you try to calculate all it's representations and then through all irreducible representations you try to develop the whole Fourier analysis theory on this group. I like the subject as your goals are quite clear and if at some point you are stuck then you can easily switch to fields like Representation theory and Operator Algebras.


Very interesting! Are there good surveys that introduce these ideas? Thanks


Surely the wikipedia article will help:
http://en.wikipedia.org/wiki/Harmonic_analysis
You can check out the references mentioned below. If you want to know about applications of harmonic analysis in physics, you can look at the book "Harmonic Analysis in Phase Space" by Folland. Another good book is "Analysis on Lie groups" by Jacques Faraut. Hope it will help.

mmcbride
Posts: 1
Joined: Sun Aug 11, 2013 11:07 pm

Re: Hardest Mathematics field?

Postby mmcbride » Sun Aug 11, 2013 11:11 pm

Real Analysis. I'm sorry but out of all of the courses I have completed, this was by far the biggest challenge. Whether you agree or not, I will say with complete confidence that many of us struggled to get an 'A' in Analysis. At least if you had my professor :)




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