What Are the Newest Fields of Research in Mathematics?

[MathematicalPhysicist]

which is the newest field in maths?
when i mean "new" i refer to minimun 25-50 years of research in the field back from when it was founded?

perhaps it's category theory or are there newer?

 

[selfAdjoint]

Category theory was around when I was in grad school, 40 years ago.

 

[fourier jr]

There may be categories in a course I'm planning on doing in the fall. From what I've heard about them, they're a TOTAL waste of time; I'm not really looking forward to that. But then, we might not do categories at all & do more modules, fields & noncommutative rings. That would be much better.

As for recent math, I would say that dynamical systems has really taken off in the last few decades. It's not brand new since Poincare was doing something like that a hundred years ago, but the field has really caught the attention of people in the last couple decades.

 

[matt grime]

if you don't want to do category theorythen you should think about stopping maths. Or does the unification of the language of topology, algebra, theoretical computing, geometry, and mathematical physics not mean much to you? every mathematician in the algebraic area must know what they are, otherwise he cannot call himself educated.

 

[fourier jr]

But categories don't say anything new, it's just a different way of looking at everything. Nothing new is proved.

 

[selfAdjoint]

That's like saying autos don't do anything new, they are just a combination of engines, which were already known, with wheels, which were invented ages ago.

The people who don't use categories are the ones who can't see the point of them. So many neat concepts in math turn out to be functors, the statements of so many theorems simplify so greatly, and one's arsenal of proof techniques expands so mightily, and then somebody who doesn't know says, "It's all trivial". It's a way of expressing a whole lot of math.

The same thing was said about Lagrangean and Hamiltonian dynamics.

 

[matt grime]

What, in your opnion does consitute something new, Fourier jr? Playing at your game, nothing new has been done in mathematics since we invented the natural numbers. One can even define mathematics using categories rather than sets. If people didn't have categories then we might still be waiting for Atiyah's topological quantum field theory. No mathematician may call himself that if he works in algebra and doesn't know about categories. Even just a smattering. I wouldn't expect you to know what a homotopy colimit is, but if you don't know what a functor is then you're undereducated. Homology groups are functors, strings are objects, branes are morphisms...

here is a proof of something that only uses categories, and is about non categories really:

let L and R be a pair of adjoint endofunctors such that the counit of the adjunction is surjective on all objects in some a module category (of a finite dimensional group algebra, say) then there exists a relative homology theory in which the homologically trivial objects are summands of objects of the form LX.

the proof only uses abstract ideas about categories

 

[fourier jr]

It's not what I said, it was written in a textbook written by Lang, I think, a respected author. He wrote that it's "abstract nonsense" and my prof for that course told us a bit about categories & what the different sides say about them, He said the critics say that nothing new has been done in group theory for example, with categories; you just give things different names. That's what I meant when I said nothing new has been done with them. Sounds pretty redundant to me... I guess it would be useful to know what they are just so you'd be able to hold a conversation with someone, but they're certainly not necessary to do math, not even algebra.

 

[matt grime]

abstract nonsense is a commonly used joke term, which means the proof does not contain mathematics that is directly related to the proposition's content and is predictable.

Your professor is wrong about groups and categories. But then I do research in group theory using abstract category theory. And if you look above at the sample proof I offered you'd see that I exactly proved something about groups using category theoretic general nonsense.

I notice you side step the issue of what you consider to be genuine new mathematics.

If you wish to be taken seriously as a student of mathematics you need to shake the idea that category theory is silly. I hope you don't think the same about geometry (proper algebraic geometry such as quasi;coherent sheaves over P(n))

 

[e(ho0n3]

I think the point of category theory is to give you a different perpective on things. It's like with algebra and geomtry: certain algebraic proofs are more intuitive if you view them from a geometrical standpoint. Whether a particular field of mathematics is useful to you depends and what you want to use it for. I mean, why study statistics when all I want to do is algebra.

 

[meteor]

There's a new branch called tropical geometry, and I think that is really brand new

 

[jcsd]

jcsd analysis is pretty new, it attempts to divide numbers into homosexual and hetrosexual.

 

[Janitor]

I remember that around 1980 some writers were all hot and bothered over something called 'catastrophe theory.' I've not heard much about it in the last ten years, so maybe it was just a fad that fizzled out.

 

[MathematicalPhysicist]

jcsd analysis is pretty new, it attempts to divide numbers into homosexual and hetrosexual.

i prefer the "theorem" that says there isn't even one uninteresting number.

 

[matt grime]

coarse geometry is new, getting back to the topic in hand.
mathematics is seeing a rejuvenation of interest from the physical sciences ; chemistry and biology are benefiting from and in some cases driving new work in mathematics. in our department today the colloquium is on molecular motors. others have included bioinformatics as well as the more mathematical subjects involving prime numbers.

 

[arivero]

Non Commutative Geometry, the art of studying the ring of functions over a manifold instead of the manifold itself, was suggested around 1980 and developed during the next 15-20 years. Now it is a bit sleep.

 

[sol2]

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. - Hermann Weyl

I borrowed this from John Baez's site on Topos Theory

Something had to arise out of Smolins attempt at consolidation in Three Roads? For a laymen like myself, the above quote is important. Some will know my feelings on "origination." On how such maths will materialize?

This is an important question for me. What kind of Quantum geometry can arise from Quantum Gravity?

 

[matt grime]

I would suggest that noncommutative geometry is far from asleep and is going to be very fashionable for a few years yet. Many aspects of geometry algebra and topology are being assaulted with research in this vein, but then it would require you to know lots of category theory in the algebraic sense so some may think it worthless; the analytic version doesn't.

 

[arivero]

I would suggest that noncommutative geometry is far from asleep and is going to be very fashionable for a few years yet

Hmm, yep, indeed the whole spring http://www.ihp.jussieu.fr/ceb-progscient.html of the institut Henri Poincare was dedicated to it, this year.

I am more worried about the absence of advances in applications to theoretical -fundamental- physics. I am afraid that the Connes-Lott scheme keeps frozen in the "reality" formulation, no clear hints of neutrino mix (except some work from Gracia Bondia et alia), and no deeper analysis of the Higgs sector. And I expected more from that hint of Brouder about the relationship between renormalization and numerical Runge-Kutta methods.

 

[MathematicalPhysicist]

another question has there been any research being done in arithematics (if something new can be found) besides the formalisation of peano axioms in natural number which are basic to arithematics?

 

[matt grime]

another question has there been any research being done in arithematics (if something new can be found) besides the formalisation of peano axioms in natural number which are basic to arithematics?

plenty. there's a result showing there are theorems true in N that are not deducible from the peano axioms, conway has shown how to divide by three without appeal to the axiom of choice. try looking up some of the logic journals and symbolic logic stuff