❗ ⭐ ❤ 😎 😀 hi all. it was a very lively week last week it would seem "all cylinders were firing". the mini adventure crosscut other blogs, reddit/ social media, multiple stackexchange sites and chat rooms, mainstream media, top world mathematicians eg longtime favorite heroes eg Tao/ Gowers, and a world class award/ prize venue ie Heidelberg Laureate forum— (eg what (personal?) connection the skeptic naysayers ask?— yes! eg did get substantial hits from their blog referrals!). it is a relatively rare event for even several of those “different spheres” to intersect.
(looking on the bright side!) the Atiyah announcement ties in with some key interests/ themes of this blog covered in the past such as attacking top open problems, (cyber-) collaboration/ peer review, scientific psychology/ culture, tracking the award/ prize-winning leaders of the field, tracking current leading edge research/ controversy which sometimes also leads to real breakthroughs, empirical/ computational/ coding approaches to math theory research, physics/ math research connections/ overlap, number theory, etc., did have some real fun! it seemed to have paid off to jump in the fray with a big analysis that got many hits and a lot of comment feedback, the former uncommon and the latter a rarity for this blog. emotions ran high! the story was not the happy/ breakthru/ historic ending hoped for but it was nevertheless quite the story, and definitely still historic!
as stated in chat, do think instead of a “dud”, alas, a “Big Kahuna™ (Hooked)” somewhat on the level of Perelman or Wiles proof is on the intermediate horizon and will be seen in our lifetime. P vs NP, Riemann, Collatz, not sure which, they seem to come along about once per generation; anyway from this episode looks like all the troops are lined up and ready to pounce so to speak. (anticipating objections…)
some might laugh at putting Collatz on that list, but it really is that important to me at least, and suspect it could rise in wider community attention/ importance with an ingenious solution, esp if its a potentially broad new technique that could seemingly have applications to other problems, say algorithmic/ computational/ deep learning/ auto thm proving aspects— ie exactly what is long sought/ advocated/ promoted here, and have believed for decades something like that is lurking in the deep so to speak. for those students of history, there is no insurance that a problem thought significant will stay that way/ aka “stand the test of time” but likewise, sometimes seemingly inconsequential problems rise in importance. theres an element of fashion to science/ even math, but in a deeper sense… maybe more like uncovering hidden gold veins, possibly bulging deeper…
meantime have also been banging on some interesting angles with the collatz research, and there was some cross pollination as commenters reacted to my referencing Tao-Polymaths new “number crunching” Riemann attack. from long observation it seems some (minority of) mathematicans are “triggered” by CS-oriented approaches to math problems, and they have my full sympathies, but some are also not very conversant with or tend to pushback against long history of world class breakthrus in the area, covered in some other blogs here/ many writings elsewhere. everyone is entitled to their own opinions, but not their own facts™…
as promised/ foreshadowed/ alluded last month, did want to share/ unveil/ unleash my shiny new analogy/ “elevator pitch”, now established as a nearly monthly tradition around here. have been tracking physics research/ breakthrus in this blog since its inception. one lively area is fusion research. its very broad to try to look into, but basically there are new directions/ developments being pursued worldwide in this area and one hears of some advances. in particular “stellarators” are gaining some viability. some scattered refs in the blog, am collecting links for a new one.
in fusion research the fiendishy difficult problem is trying to control a levitating extremely-high-energy plasma using magnetic fields. in this way there is some similarity to atom trapping technology that advanced rather extraordinarily in our lifetimes, in the last few decades. however atom trapping, as scifi/ hitech as it is, seems to be relatively “easy” compared to fusion control.
in fusion research in contrast the plasma is highly unstable, much more than an atom is, the atom despite its internal energy fluctations is nearly stable in comparison. its looking like extremely sophisticated control technology is going to be required in stellarator fusion. it looks like it will require extremely fast computations to devise a control system that reacts and counteracts the very unstable/ dynamic “modulations” in the plasma. one is reminded of the classic Lorenz attractor problem which again is highly stable in comparison. lately there are new insights and maybe something like the worlds most sophisticated/ fast responding AI algorithms responding to complex measurements within milliseconds, and some very advanced theory may be invoked. another emerging beautiful combination/ fusion of physics + TCS… stay tuned more later…
anyway my new analogy with collatz is as follows: collatz trajectories are sort of wild, unstable, and even “fractal” like in the sense of a plasma. to solve the problem, one has to somehow “trap the plasma”. the plasma escapes different control methods, this is the sense in which some parameters measuring it become unbounded. others somehow bound it even as the plasma gets larger/ more energetic. the plasma is chaotic yet somehow there may be an (extraordinary, to say the least) orderly control method that converts disorder into order (down-regulates the entropy so to speak). these ideas were recently carried out in the sense of the nonmono run lengths of the right side of glides apparently looking bounded compared to left sides (despite “push-up” optimization applied on both, ie cf
bitwise8 experiment) and there are many other experiments with this theme eg graphs where some lines/ measurements trend up with increasing trajectory lengths as others flatline/ plateau.