Contemporary Geometry: does it exist?
January 18, 2010 9:45 PM   Subscribe

Contemporary Geometry: does it exist?

Who or what is at the leading edge of geometry (theoretical, practical)? What are the newest applications of geometry, from fifteen years ago to today? Please explain it in layman's terms if you can. thanks.
posted by Jason and Laszlo to Education (14 answers total) 5 users marked this as a favorite
Contemporary mathematicians deal with geometry as an aspect of group theory, which is a broad field and home to a tremendous amount of ongoing research.
posted by mr_roboto at 10:18 PM on January 18, 2010

IANAGeomancer, but here are a couple of things:
- proof of the four-colour theorem is new, but that could be called a graph-theory result not geometry
- elliptic curves are starting to be used a fair amount in cryptography
- geometric algebra is becoming the In Thing in certain applications (fundamental physics, computer graphics) as it can replace a lot of Euclidian+Quaternion reasoning quite neatly.

Sorry, that's all I've got.
posted by polyglot at 10:22 PM on January 18, 2010

The four color theorem was proven over thirty years ago, and yeah, that's graph theory, not geometry.

Wiles' proof of Fermat's last theorem (mid 1990s) was, to a large degree, algebraic geometry.
posted by Flunkie at 10:32 PM on January 18, 2010

I don't know if you consider topology to be geometry. It sort of is. Anyway, topology has turned out to be really important for things like automated layout of circuit board, and automated routing of signals in ICs.
posted by Chocolate Pickle at 11:04 PM on January 18, 2010

Analysis, Geometry, and Algebra are usually considered the classical broadly construed fields of mathematics proper. Combinatorics is the more recent fourth broadly construed field of mathematics itself.

All other mathematical disciplines generally fit under some combination of these four headings. Of course, there are other broad fields like number theory or logic whose classification depends upon the technicalities.

Indeed, high power mathematics usually falls under two or three of these headings. For example, symplectic topology and differential geometry are considered part of geometry proper, but they definitely depend upon algebraic underpinnings. Algebraic geometry requires an even more substantial algebraic basis.

Similarly, it's extremely common that the most interesting results inside one field heavily employ techniques from other fields. For example, the Poincaré conjecture to be about the most important theorem of the last five years, while the result is an almost exclusively geometric, the proof is largely analytic. Btw, I've most often seen the classical point & lines notions of geometry arise as techniques used for classifying algebraic objects, such as finite simple groups, not other geometric ones.

p.s. Statistics isn't exactly considered part of mathematics, more a mixture of probability (analysis) with not purely mathematical considerations. Computability and algorithms are mathematical subdisciplines of combinatorics that are practiced inside computer science, another non-mathematical discipline.
posted by jeffburdges at 11:34 PM on January 18, 2010

I think Wiki is your friend (here and particularly here), though it doesn't quite get down to the last 15 years in any detail.

As CP mentioned above, if you'll consider topology to be one of the main streams of thought that flows out of classical geometry, then the solution to the Poincaré conjecture is certainly one of the most interesting and major recent developments (previously and previouslier on mefi).

As long as we're talking about the legendary mathematical achievements of the past 15 years or so, we might as well throw in the solution to Fermat's last theorem--because the whole approach to that certainly had some very strong geometric elements.

What you'll notice, though--even just reading through the summaries of the solutions to the Poincaré Conjecture or Fermat's last theorem, and even if (as most of us don't) you don't understand much of anything the articles are summarizing--is that a lot of modern mathematical thought is very, very eclectic.

No one just sits in their office sort of concentrating on "geometry" with their compass and straightedge. If you have a geometrical type problem you're like to try applying the tools of algebra (ie, groups & rings etc), analysis, topology, perhaps even a computational approach to certain aspects, etc etc etc.

Much of the beauty of modern mathematics is in how much light can be shed on a problem by transforming it from one domain to another, often completely non-obvious one, and then applying the sort of machinery and modes of thought of a completely new field onto an old problem.
posted by flug at 11:42 PM on January 18, 2010

Anyways, all the broad disciplines of mathematics are actually "accelerating" in the sense that the rate of major new discoveries is increasing; this occurs despite the prerequisites for understanding these inventions continuing to grow.

Math bachelors and doctorates having strong applied sensibilities are very employable in industry and government, as well as the other sciences. For example, a differential geometer might study the configuration spaces that appear in manufacturing or consumer equipment.

In some sense, academic mathematicians are substantially less employable because there are few purely research positions without teaching duties, but almost all academic disciplines have an enormous problem with too many young PhDs choosing to remain in academia instead of leaving for industry.
posted by jeffburdges at 12:05 AM on January 19, 2010

Kepler's conjuncture about sphere packing was proved correct in 1998, they're still working on things in higher dimensions. Also related: the kissing number for spheres in four dimensions is 24, which was proved in 2003.

For applications, rigid origami/paper folding is used in a bunch of places where things need to be folded and unfolded, like aerospace (solar cells) and medicine (stents).
posted by anaelith at 2:38 AM on January 19, 2010

Enumerative Geometry is rather contemporary, interesting, and some of the main mathematics behind string theory. If you're looking for a decent book on the subject, I'd try Sheldon Katz's book on the subject.
posted by Lemurrhea at 3:59 AM on January 19, 2010

A whole bunch of mathematicians and other scientists work on topological statistics, attempting to analyze gigantic data clouds using more sophisticated geometric techniques that have previously been available. Very loosely: for about a hundred years statisticians have been fitting lines to data. Why not fit other shapes, when the data demands it? But what shapes? That's a fundamentally geometric question.
posted by escabeche at 4:28 AM on January 19, 2010 [1 favorite]

I am a geometer.

In addition to what the above posters have mentioned, there is an active field of research in discrete geometry; roughly, the geometry of stuff with corners and edges, e. g. Cubes, as opposed to the geometry of smooth things, such as spheres.

Areas of active research off the top of my head:

--polyhedra and polytopes with varying properties

--configurations of points and lines with various properties (I focus on classification of configurations with symmetry)


--arrangements of lines and pseudolines (or higher dimensional analogues)

--rigidity of linkages

--things like the art gallery problem fall into this category

(there's lots more, but it's the middle of the night so my brain is slow)

No one just sits in their office sort of concentrating on "geometry" with their compass and straightedge.

actually...I wrote an article a couple of years ago in which what turned out to be an incredibly useful theorem was proved purely with old fashioned Euclidean geometry.
posted by leahwrenn at 5:02 AM on January 19, 2010 [2 favorites]

Computational geometry is big in engineering. Example sub-areas:

* Parametric modeling: imagine a CAD model of a soda can. Now imagine an engineer has to make it half as tall. This has ramifications for the hole, the tab, the stamped perforation pattern on the top... A good CAD program has algorithms that can handle this automatically. This was a huge area of research in the 80s and 90s, but I think it's pretty much been figured out.

* Machining: imagine you're trying to develop a product that will be manufactured in a milling machine (basically a drill on a robotic arm). Certain shapes are impossible to produce: you can't make a hollow sphere, for example. Good CAD software can decompose a shape into parts that can be manufactured. This is currently a major research area, especially as interesting new tools like 3D printers come out.
posted by miyabo at 6:55 AM on January 19, 2010

i am also a geometer.

some very classical geometry is being repurposed in an attempt to discretize differential geometry. there are even applications to architecture.
posted by at 7:42 AM on January 19, 2010

For example, the Poincaré conjecture to be about the most important theorem of the last five years, while the result is an almost exclusively geometric, the proof is largely analytic.

and the question goes back 100 years to Poincare. More to the point, what Perelman allegedly proved was Thurston's geometrization conjecture about the classification of 3-manifolds, which was made in the 80's: I'm not sure that 15 years works very well for an arbitrary 'contemporary' time period.
posted by at 7:55 AM on January 19, 2010

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