the millenium problems (what's going on with math and why you, who are friends of an aspiring mathematician, should care =P)

just finished the latest book on my personal reading list (the millenium problems by keith devlin). this calls for a book report of sorts :-) enjoy, and feel informed, evn if it is just vaguely, at my summary of a summary of what hardcore math people are interested in these days =P

question: what are the millenium problems?

answer: back in 1900, david hilbert, mathematician extraordinaire, made a list of the top problems he'd like to see solved in the next century... all but one were solved. in hilbert's spirit, the clay mathematics institute made a list of seven "millenium problems" which they presented in 2000 as the new hilbert's list of big important hard problems to solve. the remaining hilbert problem is one of the seven, along with 6 other significant questions. As a measure of recognition of how difficult these problems are CMI (clay math institute) has put a $1million bounty on a complete solution to each of these problems...

with how increasingly abstract and specialized research mathematics is, devlin's book is an overview of each of the problems for mathematicians who are not specialists in these areas as well as for ambitious non-mathematicians... his descriptions of the first 5 problems are accessible... of the last two, he even admits "these are really hard to put in simple terms... good luck with these last 30 pages, and don't feel bad if you give up"...

all that said, here's the list of the millenium problems, very simply what they're about (lara's even more simplified summary of devlin's summary), and why you should care (besides the fact that you know me (an aspiring mathematician) and probably others of my math friends and this is what we all care about :-P)

here goes:

(1) the Riemann Hypothesis:

in simple math terms: all zeros of the Riemann zeta function (other than the negative even integers) lie on the line Re(z) = 1/2 in the complex plane

in relatively non-math words: there's a particular equation number theory people are interested in, because if it does what they think it does, we know a lot about how the prime numbers are distributed

why do you care? if this equation can be solved and the prime numbers do what mathematicians think they do, this affects the way internet security works (which is based on the idea that it's really hard to factor/predict insanely large prime numbers)

(2) Yang-Mills Theory and the Mass Gap Hypothesis:

in simple math terms: for any compact, simple, guage group, the quantum yang-mills equations in four-dimensional euclidean space have a solution that predicts a mass gap

in relatively non-math words: physics is looking for a grand unified theory of everything that accounts for the mechanics (which we witness day to day), quantum theory (what happens on a tiny scale inside atoms/etc.), and relativity (what happens on a giant cosmic scale)... physicists operate on many assumptions that according to all scientific evidence are indeed true... however, the math to support what modern physics is working in doesn't exist yet... mission: fill in the missing mathematics!

why do you care? solving the yang-mills equations would give the mathematical backing for some things physics knows and uses on a regular basis... being able to give the mathematics behind the mass gap hypothesis, would be significant for physicists as well though, explaning why the phenomenon that m=E/c^2 (related to E=mc^2) seems to imply you can get mass from pure energy... however there's a nonzero minimum energy level needed to produce such mass... solving the mass gap hypothesis would account for why this is so, which is currently a mystery.

(3) P vs. NP:

in simple math terms: can all problems that can be solved in nondeterministic polynomial (NP) time really be solved in polynomial (P) time?

in relatively non-math words: if there is a "P" time algorithm to solve a problem, no matter how much data you give it, a computer should be able to solve it in a non-ridiculous amount of time. however, many problems that businesses need to solve or approximate from day to day (even scheduling workers/tasks optimally) are "NP" problems, meaning even just adding slightly more data/workers to sort makes the computer solving time go up ridiculously. the idea of this problem is to determine if such "ridiculous to compute" problems really do have "P" time algorithm solutions that we just haven't found yet

why do you care? if P=NP we know that algorithms exist to solve things more efficiently, and this is a good thing :-)... if not, then it's ridiculous to keep working on finding polynomial algorithms for some tasks because they don't exist (and for the record, the money's on the latter being true)... in a word, it's all about time management

(4) the Navier-Stokes Equations:

in simple math terms: the navier-stokes equations given initial conditions can be solved for all times t 0<=t<=T where T is very small. mathematicians want to know if a general solution exists and if so what is it?

in relatively non-math words: the navier-stokes equations describe how liquid moves in 3 dimensions. however, no one can pinpoint how to solve them even though we know they work... in that respect, kinda like the issue with yang-mills above =P

why do you care? if solved, navier-stokes would give immense insight into the dynamics fluid flow over surfaces -- a solution could result in better design of planes or boats... or in better understanding how blood moves inside us leading to new life-saving devices we don't know enough to make yet!

(5) the Poincare conjecture:

in simple math terms: can a 3-manifold have the loop shrinking property and NOT be equivalent to a 3-sphere?

in relatively non-math words: in studying topology, the above conjecture has been proven for 2 dimensions and for 4 and higher dimensions... 3 is the missing gap no one seems to fill in

why do you care? if, as suspected, it turns out to be true, we finally fill in a long missing piece of the puzzle of topology... if for some weird reason it turns out to be false (although all the money's on it being true), then there are some really quirky things going on in the universe

(6) the Birch and Swinnerton-Dyer conjecture:

in simple math terms: the existence of many solutions to a particular equation mod p (for many primes p) guarantees the original equation (not mod-anything) has infinitely many rational solutions

in relatively non-math words: in algebra you learned how to "solve" equations for x and y (e.g. x^2 + 5x -6 = 0... x = ?)... this is basically a question about solving more complicated equations where we just want to know answers where x and y are rational (can be written as fractions)

why do you care? this problem has to do with elliptic curves... in the spirit of being as vague as possible, answering more questions about elliptic curves would have repurcussions in stuff anywhere from number theory to geometry to cryptography to the mathematics of data transmission... vague description, but useful :-P

(7) the Hodge conjecture

in simple math terms: every harmonic differential form (of a certain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles

in relatively non-math words: you learned at some point (i hope) that math equations can describe particular objects you can "see" (x^2 + y^2 = 25 for example describes a circle of radius 5)... an algebraic variety is the "object" described by a system of equations -- maybe you can visualize it maybe you can't... this conjecture guesses that a particular type of algebraic variety (call them H-objects) can all be built out of really simple building blocks, so to speak

why do you care? this is after all the most complicated to explain of the 7... because it seems bizarre anyone could have intuition like this on something so abstract... i can't tell you what good application it would have, but if this turns out to be true it would say a lot about how amazing human intuition can really be :-)

if you're still reading, i'm impressed... don't you feel at least somewhat informed/slightly smart now? to see the "real" versions of the 7 millenium problems, rather than my abstraction of an abstraction, go here: http://www.claymath.org.

later dudes =P