math highlights 2018¾

hi all havent written on misc math topics for quite awhile. was reminded of the area by the recent fields medal award to Birkar and the wild theft imbroglio.[a13-15]

these are a collection of fascinating/ fun links.

highlights for me: wiles abel prize.[a11] but do think its a real pity it took ~2½ decades to award! ouch! alas, it shows how conservative prize committees are. it seemingly wasnt awarded until they were convinced his heroic effort had more applications in math and could be built on by others. to me solving a 3½ century open math problem eluding the worlds greatest mathematicians of ages deserves a large prize in a short amt of time, but this also reflects some of the larger math community conservativism: eg Collatz is open ¾ century but is not taken that seriously by mathematicians at large. gauss wrote evasively about FLT centuries ago and one might say the semi-stigma never went away…

my favorite area is auto thm proving and theres misc developments in the area, but alas nothing earthshaking as hoped and not even much of a discernable increase in momentum.[e] but, have learned to be very patient in this area and think fortunes will change. esp notable was the Yedidia-Aaronson paper on constructing a small Turing Machine that verifies ZFC very similar/ related to some of my own research into TM compilers.[e7][e8]

the abc conjecture resolution complexity seems to have gone on several years now with no end in sight and is a bit scandalous, now rather singular in the history of math.[b] to me it shows partly how complex modern math has become where new deep results may span hundreds of pages. as noted numerous times in this blog over the years there are a lot of analogies between math and computer code and that seems to be tightening in the 21st century. the ongoing abc work is sort of the “mother of all code reviews”. have also said here “math is like algorithms that run in human brains”.

deep links between the riemann conjecture and physics are very intriguing, feel like someone really should write a book on this or at least a huge survey.[c] there are other connections that show up.[d] have blogged on these yrs ago but it deserves another highlight.

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norbert blum P vs NP attack goes CYBER-SCIENCE-VIRAL!

⭐ 💡 ❗ 😀 😎 ❤ hi all! want to write a lot on this, have all kinds of impressions and things to say on one of my favorite topics in the world for 2½ decades now and which this blog is partly dedicated to, but am starting out by just collecting some links vacuumed up mostly today for others to check out, the top locations for commentary. unf there is not really a key/ central place so far where all the action is happening. will be adding to this post as time goes on. its very hard to figure out when to time a post wrt very fast moving developments like this, but my trigger just hit mostly right after RJLipton[a10] blogged about it. oh yeah, that comment by gowers that “I think it may reach the level where the experts feel that it needs to be checked” is highly triggering for me too (referring ofc to himself in 3rd person, maybe being more than a bit )( coy about this— doncha just love hardcore mathematicians! which reminds me of the joke about shoelaces…) … oh and then theres also fortnow who literally wrote the book on it! (oh yeah wrt that theres this by RJL too!) 😛

it looks like the Norbert Blum paper[a14][a15] basically went “pop/ scientific viral” today or so probably after mentions on Reddit[a8][a9] and Hacker News.[a11] of the experts Trevisan[a1] was the first to wager a reaction along with some curious onlookers like Baez[a2] and verging-on-jester-or-gadfly-or-even-troll Motl[a5] who along with some mild conspiracy theory about CS research(ers) (lol!) says, wrt the #1 top problem in TCS…

Except that unlike the case of string theory, there exists absolutely no rational evidence that there exists something stunning waiting to be discovered.

@#$% 😡 👿 ❗ whoa, fighting words and/ or near-sacrilege there! excuse me, what is “stunning” is that the problem with $1M prize over 1½ decade has eluded/ stymied/ thwarted the worlds greatest scientific minds for almost ½ century now! and the proof either way, in whatever shape or form, will be assuredly stunning to anyone who isnt jaded or a nihilist! hey physics bozo! think of a great experiment that guaranteed leads to an extraordinary answer either way! arent there any examples like that in your field anyway? or is maybe someone looking a little envious over there? or out of the loop or maybe dont even know how to flirt at all?

the TCS stackexchange post by a n00b newbie (who didnt know enough to not ask! hows that for “zen beginners mind”?) is now up to 136-4 votes and 28K hits (thursday eve 8/17).[a3] (joy that the lame heavyhanded/ constricting/ dictatorial policy of the self-appointed cyber spoilsports and partypoopers also long criticized here in this blog, is overruled by striking mass opposition!) that post slipped my radar for a few days! found the paper on the 1st day (monday this wk) but thought it might stay “unviral” (you know, like those killer viruses locked in the… thawing… arctic, right?) based esp on the last Hauptmann attempt.[a12][a13]

discovered the big commotion today via reddit myself and spent excited hours poking through sites! but my hits on the Hauptmann blog post had been creeping/ climbing up massively all week, merely from google searches (many from germany), and figured it was due to questions on the Blum attempt. turns out theyre both profs at the same university! (Bonn/ Germany) pardon me but wtf?!?… some kind of story there eh? are any reporters listening?

cyberspace is such a joy sometimes, in my quasi-bipolar days, mostly-sometimes seemingly very flat for days, weeks, months, even years at a time, this is a massive manic spike for me… the adrenalin is flowing! didnt sleep as well this week! there is so much cybersynchronicity to this, must admit, its a very rare event, am myself (and defensively, some others are too) having a bit of a mini-cyber-orgasm so far… 💡 😮 🙄 ❗

Mirzakhani math superwoman 1977-2017

😦 😥 ❤ a few years ago Mirzakhani was credited in this blog with her striking bolt-from-the-blue achievement, and now its with major regret to write this “postscript” in this case also a “postmortem”. (“bolt from the blue” is a turn of phrase used by Heisenberg to describe the Einstein EPR1935 paper and Bohrs response to it, which historical echo seems somewhat fitting/ apt here. coincidentally, it just now occurs to me, Turings amazing paper introducing or even “solving” the undecidability problem was released one year later in 1936. two problems that have been particular extreme focii of attn for me over the years, and is there maybe some deeper resonance that could be observed beyond superficial connections?)

ironic or a bit eerie that Mirzakhani died at age 40, exactly the cutoff age for the fields medal, a topnotch prize with a rare age limit (over the years, not without some element of professional/ expert questioning verging on controversy). on the other hand esp in “winner take all capitalism”, the golden rule is he who has the gold, makes the rules…

have long thought the fields medal prize value ought to be much higher. its status is far higher than the numerical reward in dollars. ($15K, ~1.5% of a Nobel despite being called the “Nobel of mathematics” or ~0.5% of the Millenium prize won by her cohort Tao [a8]—who btw apparently still has said absolutely nothing about his award(s) in all his copious blogging and advice pages and elsewhere!)… but maybe the comparatively micro-money-prize is fitting for mathematicians who tend to profess to be unmotivated by material rewards? or maybe better to avoid such “calculations”? 😮 🙄

(brief tangent, inserting prize trivia or nontrivia depending on your pov, am reminded some of the TCS $5k Godel and Knuth prizes. extremely prestigious, rather tiny by Trumpian standards. another interesting case study is the Turing prize which was apparently unfunded for ~¼ century, was $¼ million ($250K) for ~½ decade, and went to $1M the same year Mirzakhani won, 2014. yeah as iconoclast Kary Mullis observed, prize money awards are one of the most irrational things around… near miss with Nobel himself, long story…)

am not familiar with Mirzakhanis work in particular but the identification of it as “science fiction mathematics” is personally meaningful/ delightful (have myself admittedly consumed vast quantities in written/ visual/ hollywood form of the literary genre, no intention of pulling/ holding back on that in future). as regular readers know, there is a lot of flavor of dynamical systems analysis in these pages lately wrt Collatz conjecture, and there can be regarded as some rough/ abstract connection.

Royen proof of gaussian correlation inequality

hi all theres been a recent shock of awareness of the Royen proof of gaussian correlation inequality, pop-sci publicized by Wolchover for Quanta, a big milestone… this is a nearly ½-century-open problem![a] Quanta funded by Simons institute is one of the top outlets for scientific/ mathematical writing around today. a real community resource/ treasure!

the Royen proof is not exactly my area so cant write a lot on it but do note that its a key case study in dynamics of scientific peer review, and seems like it has some parallels to the ongoing mochizuki proof analysis.[b] it took over ~1½ year for “community” to begin to grasp the correctness of this proof and Wolchover has a nice historical timeline for how others began to notice/ accept it, a mapping of the spread of awareness. it did not help that Royen was somewhat isolated and did not seem to personally contact any cohorts for peer review. he published openly but it got lost in the noise. it shows how community acceptance is sometimes far from a black/ white binary decision, esp for “big problems”.

is there any way to improve peer review? its definitely a bit of an achilles heel of the scientific process. my feeling is that there is no way to improve it very much except maybe to try to increase transparency somehow. its very similar to the problem of “fake news”. how do you measure quality in content? we live in the vast Information Age but as has long been noted, theres a big difference between Information and Wisdom, and in a way peer review is the major mechanism that is designed to separate/ discriminate the two.

math celebration 2016: pythagorean triples empirical attack conquest

actual U Texas Stampede supercomputer where proof was run [a2]

➡ 💡 ❗ ⭐ 😎 😀 hi all. RJLipton recently covered the amazing/ brilliant breakthrough of solving the pythagorean triples problem by empirical work namely a reduction to SAT and analysis by a supercomputer by Heule, Kullmann, Marek and its a nice pivotal trigger/ tipping point for my own writeup along with related stuff.[a] (have been waiting for opportune time to write this up since may.) proofs like these are a complex or “complicated relationship” for mathematicians (aka facebook-speak), a love-hate affair. (did the extraordinary/ breakthrough/ revolutionary/ paradigm-shifting 4-color computer proof ever win any awards? and how much despair/ handwringing and further effort has there been over it over the decades?)

the breakthrough is celebrated but mathematicians would like to see shorter proofs that are human-comprehensible, so there are mixed/ ambivalent feelings about it within the community. have written on this topic quite at length in this blog even since its beginnings, and this latest breakthrough is delightfully affirmationally crosscutting across many of this blogs categories, and think this is the tip of the iceberg of 21st century mathematics in a way not yet fully recognized. its a dramatic, vivid realization/ materialization of an idea suggested a few years ago here called “SAT induction.” think that these types of proofs will lead to new theory that is indeed human comprehensible but some of the isolated theorems will be claimed first by computer analysis before the more thorough theory catches up to integrate them.