Warning: Very long post. Maybe boring for those not into either films or mathematics.
Disclaimer: The word 'isomorphism' here is used in a very loose, non-mathematical sense. Probably a better word would be 'similarity/analogy' but that wouldn't sound cool enough.
Another description of the post is in terms of functions that map domain spaces (film directors) to extremely 'dissimilar' target spaces (mathematicians).
Nanga Fakir has always wondered at isomorphisms lurking at a structural level in a wide variety of apparently unrelated disciplines of human endeavor. This post is about such isomorphisms in the (apparently unrelated) fields of Mathematics and Films - in particular, the similarities in the manner of work of some mathematicians and some directors. It is unfortunate that NF doesn't know even half as much as he would want to know about either discipline and so the possibility of errors is significant. (Hence it is a blog post and not an article in the New Yorker.)
1) Kim ki Duk & John Milnor : The former is a Korean film director and the latter is an American Fields Medalist.
That which brings them both together is the quality of minimalism in their works.
Kim ki Duk has been known to make films with hardly any dialogue at all and with each passing work, relying more and more on non-verbal cues, quiet, subtle gestures and dry, unusual humor. Milnor on the other hand is not only a legendary mathematician who's won all possible math prizes left, right and center, but has also won the Steele Prize for mathematical exposition. The books and notes written by him are thoroughly minimalistic - in fact sometimes notoriously so. (The standard lemma-theorem-proof-corollary style is followed in a sparse, somewhat austere style and the Proof sections of his books are peppered with remarks like "Not hard to prove", "Easily shown to be true" etc. - much to the frustration of your average math graduate student.)
Perhaps it was Hemingway who said that perfection is achieved not when there is nothing left to add, but when there is nothing left to take away. Both of them seem (consciously or otherwise) to adhere to this view, for in both their works, taking anything away from what is there, necessitates the collapse of the entire edifice - the mark of a true minimalist. (NF sometimes fancies himself as a minimalist in the sense that he tries hard to put the minimal amount of effort required to barely get along. He wonders if he's a true minimalist though. Any lesser effort and we'll know the answer to that question.)
2) Andrei Tarkovsky and Bela Tarr & Alexander Grothendieck : There is no catchword here (like 'minimalism') that can be invoked to compare the two directors with Grothendieck (seen in picture here). In fact what is probably more at work here is the obsession with purity, perfectionism and mind-boggling care for details which is the hallmark of all these artistes.
This is easily seen to be so in the case of the two auteurs in question - both of them are obsessed with insanely long cuts (some of them as long as eleven minutes), excruciatingly slow camera movements, artsy and extremely highbrow literature, shots of breathtaking beauty and concerns with questions that are of deep philosophical value. The camera lingers on each shot, hovering on the subjects as if unable to tear itself away, trying to absorb the fleeting moment and record it forever. The concerns for purity and technical wizardry over audience-friendliness (and here the audience are considered to be arthouse regulars and not those who line up to buy tickets for the latest Transformers flick), the demand for patience and faith in the vision of the director (which is rewarded at the end by a deeply ambiguous and artsy climax which may or may not make any sense) and the paramount consideration of cinematic beauty over everything else make them to be champions of cinema as a pure, high art medium.
In order to see for yourself, look at the following videos:
a) A sequence from Tarkovsky's Stalker.
[probably one of the best sequences in world cinema]
b) A 'short' scene from Satantango - the seven hour plus long movie (yes, you read that right!) from Bela Tarr.
Like his counterparts in cinema, Grothendieck (Fields Medalist, greatest mathematician of the 20th century, according to some) was obsessed with purity and methods of great abstractness over more obvious routes to problem solving. In fact, his was such an abstract take on matters in Pure Mathematics that it is inaccessible to a lot of working mathematicians themselves, let alone the common man. His style of Mathematics - often dubbed 'Fortress Mathematics' (built around his cult of followers) was a major dampener for many-a-young wannabe mathematician who were overwhelmed by the technical virtuosity demanded of them to even begin to understand what was going on. Again, concerns for stellar, unimaginably stringent purity and an open disdain for anything even slightly contaminated (read less general/abstract - woe to applied mathematicians - the hacks who dare defile the sanctity of math!) make him the counterpart to Tarkovsky and Bela Tarr.
3) Quentin Tarantino & Benoit Mandlebrot and Paul Erdos : Cinema for Tarantino is not a meditation on deep matters (cross Tarkovsky, Tarr). He's definitely not into minimalism (cross Kim ki Duk). What's he into then?
Filmmaking for Tarantino is an expression of sheer joy. It's also about being all-over-the-place like a jumping, hyperactive kid suffering from attention deficit disorder high on methamphetamine. It is about exuberance, aesthetic violence, non-linearity, dialogue (his characters never seem to shut up) and a breakneck, blitzy pace. He doesn't care about conventions or boundaries or classifications. He breaks rules, bends them, amalgamates genres, rips off others' work and gives them a new spin and constantly reinvents and pushes the boundaries of filmmaking all the fucking time.
Somewhat similar was the approach to mathematics taken by Mandlebrot and Erdos. Mandlebrot was repelled by the Fortress Mathematics so in vogue when he took up studies of Math. His was a more intuition based, a more qualitative approach (often called 'Open Mathematics' by some). He was genre defying and refused categorization of any kind. His work in pure math soon spilled onto areas like Information Theory, Fluid Mechanics, Economics and Physics (one of his papers was on the stability of cotton prices over the years). Nassim Taleb considers him his guru and chief mentor.
Erdos is a legend in himself. His versatile work in Graph Theory, Number Theory, Analysis, Combinatorics and several other disciplines was the work of someone obsessed with and totally in love with what he was doing. [Perhaps it was he who said "When I am depressed I do math to become happy. When I am happy, I do math to stay happy".] Again, the hyperactive-all-over-the-place-kid effect can be seen here. His encouragement of budding young mathematicians, geeky sense of humor, astoundingly large collaborative body of work that changed the way Mathematics was done and championing of Open Mathematics makes him the ideal counterpart of Tarantino in Mathematics - perhaps even more so than Mandlebrot.
Perhaps more such maps will be discovered by readers of this blog, maybe not in the same two fields, but perhaps in others where such isomorphism have not yet been discovered.