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.
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.
⭐ ⭐ ⭐ 💡 ❗ 😮 ❤ 😀 😎
SO EXCITING/ “JAZZED”!
hi all it was only about 2 mos ago a big ambitious update on quantum mechanics shifts was posted on the “fluid paradigm shift”. if you ask me this is all at least as epochal as the invention/ discovery of quantum mechanics itself about a century ago. and (unbeknownst to me at the time) theres been a striking new development.
around here updates to QM are measured in years and not months. have been blogging now a solid ½ decade on the topic and thats just the recorded history, my inquiries go back decades. but as mentioned in that last blog “my neurons are really buzzing”. something is in the air, the zeitgeist is electric right now. lightning is striking. all my spider senses are tingling. normally would not write very quickly on all this but heres a “flash update” based on the sheer significance of the announcement/ finding.
all the research is paying off. a breakthru has arrived. there is now SOLID EXPERIMENTAL EVIDENCE FOR SUBQUANTUM EFFECTS.[a] thats right they have now been MEASURED and published in a reputable physics journal (physical review letters… uh hey preemptively addressing skeptics obsessing about peer review etc, didnt einstein + bohr + other founders have papers in there?).
that breakthru credit goes to the VINANTE-ADLER team. they have been working on this line of inquiry maybe about 2 decades judging by their papers.
yep, its definitely a marathon at this point, maybe now mostly comprised of month-long sprints (applying “agile” philosophy to mathematics/ research in the weird hybrid style typical of this idiosyncratic approach… thats not a bug its a feature™… or maybe in some sense that the historic Thai cave rescue was also marathon…). heres another new idea building on prior ones. earlier analysis looked at the “sibling trajectory” and found a strong correspondence with the trajectory. a fairly basic idea is to compare how much of the sibling trajectory matches wrt the climb length of the glide. in a sense the sibling trajectory “covers” some percent of the climb length.
using the latest generated glides (
mix26 scheme) this finds that in ~40% of the cases the sibling trajectory covers the full climb. in other words in 40% of the cases the computation to find the highest point of the glide is exactly repeated in a “lower” glide/ trajectory. in the remaining “uncovered” cases theres still 1.0-0.3 = ~70% coverage at worst and more typically/ on avg in the ~95% range. the graph is the ‘icm’ variable in impulse format which is matching length of trajectory and sibling divided by ‘cm’ glide climb minus 1, here 0 means matching the exact length. as seen in the graph the non-fully matching siblings still cover most of the climbs.
hi all, its been a very busy year for QM computing and it seems as if an inflection point has been reached. google/ IBM are announcing designs with more than 50 qubits.[b] intel is getting into the game. there is a lot of recent innovation going on with qubits in silicon, a trend spotted here previously.[a2] simulation is a very big topic, there are two types: simulating QC calculations on a conventional computer, and using QC computers to simulate physics problems such as atomic or molecular interactions.[a4] as far as “combine trendy buzzwords” its sometimes AI + ML + QC.[a3] this seems to me to be “jumping the gun” because our QCs are not even that powerful yet, but its reasonable to explore more abstractly.
another hot topic is “quantum supremacy” the idea that QC computing can be demonstrated in some sense to be “faster” than conventional computing on “some/ any problem”.[c] that problem is now being defined a bit circularly by google et al as “quantum calculations” but nevertheless results are that supremacy by that metric is real at around 50 qubits in the sense that these calculations are out-of-reach of conventional machines. (the scientists among us know in complexity theory nothing is described in terms of constant threshholds, in fact they are rejected as meaningless, but theres always been a lot of handwaving in this field!)
another huge event was the launching of stackexchange quantum computing site which seems to be healthy so far and its sponsorship by the new unusual startup strangeworks.[f] another very big deal/ gamechanger/ milestone is that the US congress is discussing some QC related funding/ development legislation/ bills.[g]
almost 2 decades ago some of the 1st popsci books came out on quantum computing. a few of us read those. its energizing/ enthralling/ inspiring/ exciting that a “relatively short time” later today its now a worldwide reality being pursued by top corporations/ even leading govt research programs. the wild/ starry-eyed early promise has not yet materialized; a general purpose machine stills seems off in the distance, and its not yet clear a QM computer will have the same revolutionary economic impact as the integrated circuit/ microprocessor, but the technology is advancing steadily now and looks like it will have a permanent niche somewhere. as the old expr goes its gone from glimmer in the eye to something real and thats really something to celebrate.
hi all. on vac this week & doing some new stuff (happy BTD US). there is a semifamous thousands-year old quote by sun-tzu maybe not yet contained in this blog (its been going thru my mind for quite awhile now, but wasnt able to find it in the blog via google). it is a quite favorite quote of business consultants which might tell you something about modern “leave no prisoners” business attitudes/ culture in our at-times militaristic/ hypercapitalistic modern age. (dramatic alpha-male stuff, but to put it more bluntly, one with a conscience/ empathy/ independent mind might wonder about the “fine print,” ie how many “enemy…” men did sun-tzu kill personally or oversee killing as a general? …or even humans which includes women/ children? oh but ofc its utterly metaphorical right?) 😮 😳
Strategy without tactics is the slowest route to victory. Tactics without strategy is the noise before defeat. –Sun Tzu.
collatz has been described as the very impenetrable/ unconquerable adversary. strategy/ tactics both play a key role and have commented at length on both. they are like a yin-yang combination. victory will likely not come without some kind of balance between the two.
have been more/ very tactical for quite awhile but have been musing on some overarching strategy/ perspective/ pov lately, thinking it all over at current point. this involves more abstraction.
couldnt find this basic idea pointed out in old blogs. was it? the key question is to prove f(x) < g(x) for all x. here f(x) is collatz stopping distance or some similar metric and g(x) is “any recursive function” (either time/ space bounded). now apparently f(x) in many related forms has extreme entropy, the “needle in haystack” property, and also “fat/ long tails” distribution, and fractal. earlier blogs have outlined the idea that it appears that victory seems to lie on the path of decreasing or minimizing entropy somehow. g(x) can be regarded as an orderly function from analytic mathematics and f(x) is “far from it” in the sense of being extremely disorderly.