Ramanujian's Deathbed Problem Cracked

Jake's Mom sends word of the serendipitous solution to a decades-old mathematical mystery. Researchers from the University of Wisconsin have unraveled a major number theory puzzle left at the death of one of the twentieth century's greatest mathematicians, Srinivasa Ramanujan. From the press release: "Mathematicians have finally laid to rest the legendary mystery surrounding an elusive group of numerical expressions known as the 'mock theta functions.' Number theorists have struggled to understand the functions ever since... Ramanujan first alluded to them in a letter written [to G. H. Hardy] on his deathbed, in 1920. Now, using mathematical techniques that emerged well after Ramanujan's death, two number theorists at the University of Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them."

Comments

Good job!

[UbuntuDupe]

The summary didn't refer to Ramanujan as "the Indian math guy" this time! Great work! (Don't ask how I remember that one.)

Although, it could do with one less "i" ...

Re:Good job!

[Slooze]

Heheh...no kidding. When I saw "Ramanujian" in the header, my first thought was, "An Armenian created a math problem?!"

Re:Good job!

[The Clockwork Troll]

This does seem like good work, but realistically we won't know how important it is until it appears as a deus ex machina device on NUMB3RS.

Spelling error

[kraemate]

Spell error in story title! Its Ramanujan, without the 'i'.

Re:Spelling error

[boingo82]

But...but....with the "i" it almost anagrams to "marijuana"!

Re:Spelling error

[Tilzs]

I think you imagined the "i"

Re:Spelling error

[gsn]

Ramanujan was already a complex guy.
Trying to Wick rotate him would be a pretty negative thing to do.

Curiously enough

[TubeSteak]

Now, using mathematical techniques that emerged well after Ramanujan's death, two number theorists at the University of Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them.

There's gotta be a Scientology joke in there somewhere

Bloody lack of details...

[UnknownSoldier]

Since the article STILL doesn't define what a mock theta func is, what is, and how can it be applied?

Guess the wiki still needs to be updated

There is (as yet) no generally accepted abstract definition of a mock theta function; Ramanujan's own definition of the term is notoriously obscure.

--
  "I want to work in Theory -- everything works in Theory!" -- John Cash, id

Re:Bloody lack of details...

[jd]

A quick search shows that mock theta functions are a special case of Jacobi theta functions which are a form of Jacobi Elliptic Functions which are a type of elliptic function. Ok, this explains next to nothing.

Arxiv doesn't appear to carry the paper, and only two papers in it relate to mock theta functions at all. One of them is a transformation formula for second-order mock theta functions and the other talks about mock theta functions as quantum invariants, whatever that means. A glance at the paper suggests that mock theta functions relate to a key element in topology, but my maths isn't nearly good enough to tell you exactly what is being described.

Re:Lack of information

[arlo5724]

The mock theta functions are special functions that describe of host of phenomena, the most interesting of which is probably its relation to modular forms. There has been a great deal of controversy as to how these functions should actually be defined in the abstract sense and for the most part any serious attempts at figuring them out have involved using nothing more than the functions that Ramanujan himself wrote down in a notebook right before he died. It will probably be some time before this "solution" appears in a final, published form so don't get your hopes up unless you have connections to number theorists close to the activity. If you are at a university you can look up scads of articles on the topic from JStor, or just browse the bounded periodicals in the library.

This is cool and all, but the real kicker will be if Peter Sarnak from Princeton proves the Riemann Hypothesis (rumor has it he is on the way to doing so).

Ramanujan

[theurge14]

From what I've read about Ramanujan, what I still can't understand is how a guy from a poor background with little to no formal schooling is able to just sit around and write in a notebook and come up with the equations he did. I just have to wonder what it was in nature that made him so more adapted to mathematics than the rest of us mere mortal humans. This guy was on a completely different level. Mozart comes to mind when I think of him.

Re:Ramanujan

[teetam]

He was poor and from a poor country, but he did go to school and learn math there. He just happened to be fascinated by it and continued to work on it, neglecting everything else. He obviously also had a knack for math. That has nothing to do with poor or rich.

Math, being theoritical, does not require a lot of external resources (like laboratories etc.)

Re:Ramanujan

[ezzthetic]

He won prizes at school for his maths prowess, and went to university on a scholarship. He lost the scholarhip due to his obsessive inability to do other aspects of the curiculum that were not maths related, or which were offensive to his Brahman beliefs. There was never any doubt that he was mathematically gifted, and his mother promoted him intensively. There seems to be a myth that he was an illiterate peasant who happened to stumble on a maths book came from, but I don't know where it came from.

Re:Ramanujan

[OldManAndTheC]

It's sad to think that geniuses may languish among the world's millions of underprivileged children who lack access to education. When you think of the potential impact of a single person of the caliber of Mozart, Ramanujan, etc., our civilization could be missing out on some truly wonderful things.

Re:Ramanujan

[MrBoombasticfantasti]

Still, that means that 2/3 of his discoveries are new and original!

Might it be that education structures the mind to follow the known paths? Perhaps by not knowing the 'usual' solutions, you can come up with a more elegant and deep solution?

Re:Ramanujan

[rxmd]

Quoted from Hardy "So the real tragedy of Ramanujan was not his early death at the age of 32, but that in his most formative years, he did not receive proper training, and so a significant part of his work was rediscovery..."

At the same time, Hardy acknowledged that "on the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain." (From Hardy's article in "The American Mathematical Monthly" 44.3 (1937), p. 137-155.)

Ken Ono's seminar

[alpha_foobar]

It appears that Ken is holding a seminar at UW on March 29 2007 (http://math.uwyo.edu/DEPTCOLLOQ.asp#Mar%2029). We will probably have to wait until then for any details.

Re:Real World Uses?

[arlo5724]

To answer this very loosely, parts of these functions are bounded by geodesics with cusps at the corners, and this means that any geodesic structure of this type (certain types of chemical structures and a slew of phenomena in relativistic physics) can be partly described by those pieces of these functions and that it is possible that these functions represent a certain type of generalization for these structures, allowing scientists to better describe some existing structures with similar modular forms and even some that exist only in thought.

Obligatory

[Bananatree3]

"I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

the man who knew infinity

[phreakv6]

not totally offtopic but i would like to recommend this amazing book (the man who knew infinity) to anyone interested in reading his biography. its one of the best biographies i've ever read.

Re:Lack of information

[Dunbal]

You mean there was even MORE math after "Integration by Parts"? Sheesh you guys need to get a life :P

Disappointing

[grimdawg]

As a young mathematician-in-training (just finished my undergrad degree), it disappoints me to see the kind of coverage the maths community gets.

It takes a near-century-old problem to be solved to pop a maths story on slashdot - and TFA holds no details. To get on any kind of mainstream news, the Poincare conjecture needs to be solved, and then we get "Perelman proved a rabbit was a sphere".

Mathematics at universities worldwide is being dumbed down for the pursuit of the cashed-up Engineering student. Mathematicians get no kind of acclaim for their work - even compared to other 'unglamourous' pursuits. People these days don't seem to appreciate the debt they owe to mathematics.

What's it going to take for mathematicians to get some mainstream coverage? A sex scandal?

Re:Disappointing

[Nicky G]

Sex scandal? Uh, yeah... don't hold your breath.

Re:Disappointing

[jd]

It would have to be imaginary, or complex. But that's a bit of a tangent from the point. The TFA is obtuse, cos() it doesn't exp()lain anything much. It would seem that the Slashdot crowd are caught on it Hooke, line and sinker, though. Of course, any maths problem is as easy as Pi, if you use sufficiently advanced techniques. However, if the problem cannot be differentiated meaningfully, can it be integral?

Re:How to solve a mathematical mystery

[Dunbal]

How will I intellectually masturbate?

      Simple. Redefine the universe's parameters such that intellectual masturbation is no longer necessary, and place yourself in the appropriate set. You're a mathematician. You can do ANYTHING. Duh!

Ease of understanding & teaching.

[Anonymous Coward]

Ease of understanding & teaching.

I really think the reason why a lot of people are bewildered with math (& thus ignore it) is that they were never really able to approach it properly. Mathematics has a tendency in university to not explain itself properly. Things that I found rather simple in the end were just never explained clearly, were incomplete explanations, assumed you knew & understood concepts from other, unrelated courses, or were given "pseudo-explinations" that kind-of explained something but not properly, giving potential incorrect understandings that could be disastrous later (think high school math).

The entire cutter mentality that math classes can tend to be in university don't help much either (what is probably the #1 reason why people drop their hard science/engineering/comp sci courses?? Probably MATH!)

Once I figured whatever a concept really meant in math, I realized reading the textbook after the fact (sometimes several courses later) they use terms and concepts that aren't explained at all or they use really obtuse english sentences while simply defined symbolic language could easily show the concept. Actually most of it I found rather simple & clear in the end once I got to understand it but found that the textbook just explained it, badly or with huge gaps in their explinations.

Re:Ease of understanding & teaching.

[muecksteiner]

You have a very good point about math generally not being taught as well as it could be.

Not in the sense that the curricula should be dumbed down in any way - this would not work out well in the long run.

But there definitely is a streak of the beloved "if it was hard to code, it should be hard to understand" mentality to be found in mathematics.

Introductory math courses at universities usually do not have concepts of such bewildering complexity on the curriculum, that they should be considered to be as "hard" as they turn out to be for everyone.

However, they still are the bane of undergrads everywhere, and sometimes I wonder if the obtuseness of these courses is not just an in-joke perpetrated by the mathematicians.

If you are not smart enough to "get it" in the arcane way the stuff is being presented, you woul not hack it further down the road anyway - at least not in pure math, and they are not inclined to have pity on anyone who could not have gone down that road in the first place.

Or so the reasoning might go, when mathematicians are amongst themselves... :-)

Note that the remarks in this posting mostly apply to the teaching of the kind of "working math" that an engineer might use, which (to put it mildly) can still be pretty involved in terms of complexity, but always has a goal-oriented quality to it that pure math does not necessarily share. This residual "grounding in reality" usually makes the teaching of even advanced concepts much easier - a potential bonus that (at least in my opinion) is not used nearly as often as it could be.

A.

Re:Ease of understanding & teaching.

[Flyboy Connor]

I can relate to that. I studied math at a famous university for a couple of years before I dropped out. Here are some of the things I remember:

We started with over 100 students in the first year. By the third year, the number had dropped to less than 10 students. Half of those dropped out later. The professors were proud of this fact.

Each lecture took three hours, with one fifteen minute break. You were only allowed to ask questions in the last 15 minutes of the lecture.

Professors only took the trouble to learn students' names when they entered third-year courses.

I once wrote a research paper for one of the professors for a first-year course. In the very last paragraph of the paper I wrote a little joke. The paper was marked "A", then the "A" was crossed out, "C-" written below it, with an arrow pointing to the joke.

Math students had access to the faculty mainframe (this was in the early 1980s), but did not get instruction on how to use it, as opposed to physics students. The reasoning was that math students either should not need computers for their work, or should be smart enough to figure it all out by studying the manuals.

Professors often supplied example excercises. Students were encouraged to make these excercises and supply their answers to the professor. However, these answers were NEVER corrected, so that after a while students simply did not bother anymore.

Professors were notorious for not preparing lectures, and working out examples as they were going along, often failing to prove what they wanted to prove. One particularly telling incident was when a professor was working out a complex proof, starting at the top left of one of the two four-piece blackboards in the hall, and chalking down, very fast, formula after formula. I was trying to follow his proof, but, of course, was always several lines behind. But I thought I did understand it, and was approaching to where he was. When he was at the bottom-right of the second blackboard, he paused, and kept staring at the last line he had written, muttering to himself. While I was approaching this last line (making lots of notes, because OF COURSE these proofs weren't in the textbooks or anything), he started scanning back. After doing this for about five minutes, he suddenly walked over to the first board again, changed a plus into a minus in one of the first lines, then made lots of changes in the rest of what he had written, and finally wrote "Q.E.D." at the bottom-right. Then he closed the blackboards and sent us on our way.

Through this experience I thought I simply was not good enough at math. But when I switched to computer science, where math courses were taught by computer scientists, I passed with flying colours, usually as the best of the class. Not because the courses were easier, but because they were taught better.

you left math too early

[^Z]

There are smooth operators that act on imaginary numbers right by the corner. Then it gets really kinky. Consider improper integrals, strip functions, etc.

Indian mathematicians

[d0n quix0te]

India has had a long standing history in mathematics much of which predates that in the Islamo-christian tradition.

Formal mathematical schooling among Brahmins was particularly important among people in Tamil Nadu and Kerala, two of the sea-faring communities in India. Ramanujan belonged to the Iyengar tradition of mathematics (although many people related Iyengars to Yoga...) from Tamil Nadu.

Among other contributions of Indian mathematics include

Pre-ACE

The decimal system and the number zero
Inductive reasoning and the inductive method
Fractions
Equations
Mathematical tables
Binomial theorem
Pythogorean theorem
Area calculations
Conic sections
Irrational numbers
Boolean Logic
Null Sets
Transformations and recursions
Number theory
Trignometry
Formal language and grammar theory

Post ACE (pre renaissance)

Cubic and Quartic Equations
Pi as an infinite series
Geometric and Harmonic series
Series theory
Permutations and combinations
Cardinal numbers
Transfinite numbers
Set theory
Fibonnacci series
Derivative
Rolles theorem
Differentiation
Limits
Differential and integral calculus (predating Leibnitz and Newton by 200 years) ......
For a laundry list see

http://en.wikipedia.org/wiki/Indian_mathematics

Some of these brahmanic schools were far more advanced than European schools. Ramanujan had good schooling from a tradition steeped in mathematics. He was Europe's first direct exposure (as opposed to published books that were translated) to Indian mathematics hence the cult status.

Imagine a Narayana Pandit or a Chitrabhanu from the Kerala schools in Europe in 1500 AD spouting Calculus and Reimann's theorem (two well known theorems in India at that time)... they too would have been declared as geniuses.

-S