the word “search” is something to ponder wrt this effort in math “re-search“. there are different optimization algorithms as a theme but in general theyre different types of searches for trajectories. in a way its all about searching for some kind of invariant property of trajectories, both with the computer/ algorithm and more generally in the research (“program”). have come up with many ingenious search methods and that effort continues.
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.
⭐ ⭐ ⭐ 💡 ❗ 😮 ❤ 😀 😎
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.