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.

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.

$1B EU QM computing initiative + QM theory/ applied highlights Q1 2016

➡ 💡 ❗ ⭐ 😮 😀 😎 ❤ hi all, more signs that physics is in a golden age esp in the area of quantum computing and quantum mechanics. EU just announced a $1B initiative to fund quantum computing. this combines with two other existing/ similar projects to analyze the brain and advance graphene technology/ understanding.[a]

the Manifesto[a1] is a sign that we are in dramatic times. you will not much hear scientists ever use the word Manifesto in their entire lifetimes. (marx was a rare exception.) another word that you will rarely hear scientists use is “revolution” but they are calling quantum computing the 2nd quantum revolution. as the chinese say, may you live in interesting times.

the announcement is utterly buried in a bureacratic document[a3] under the section “What is the financial impact of the Cloud Initiative? Where will the money come from and how will it be invested?” but Nature blared it anyway.[a2] apparently those bureacrats are a little gun shy after criticism of the $1B brain initiative as too ambitious or infeasible etc.

these are other big 2015 Q1 developments/ highlights. as usual its hard to keep up with lightning fast advances in this field. even the canadian prime minister is talking about QM these days.[b1][b2] DWave seems to be further confounding its critics with real, measurable performance.[b9][b4] somewhat similar to the EU announcement, chinas Alibaba announced a major QM research initative.[b8] they are intending to start out with encryption applications and hundreds of engineers, working toward QM computing technology.

SAT induction revisited wrt Bertrands postulate, collatz vs prime similarity

hi all. was thinking really hard this week and then came up with some cool ideas about prime theorem proving and more advanced stuff. it all led back to an idea thats now over 2yrs old on this blog, “SAT induction”.[a]

there is a “hall of mirrors” type aspect of working on undecidable problems. there are many equivalent ways to formulate the same undecidable problem. its like a sort of core that crosscuts many fields, some unexpected. (lately there is some excitement that it shows up in theoretical physics etc).

however, think that there is a way out of the “going in circles” and SAT induction seems to be it.

consider what a proof of infinite primes would look like using SAT instances. the immediate idea is to use Bertrands postulate, also proved by Erdos in a surprising elementary proof. in SAT one might encode instances of the problem in the form “if there is a prime with n bits, then there is a prime with n+1 bits”. (alternative idea, very similar: “there exists a prime with n bits”.)

then one could lay out these finite instances of the infinite problem, say F_n for each n. this would lead to an array of SAT instances. (over 2 decades ago, built SAT circuits that factor numbers in binary. these circuits are similar.)

now, encode the SAT instances as graphs. and look for graph similarities between G_n and G_n+1. and then one is attempting to build graph G_n in terms of graph G_n-1. this is where the machine learning problem comes in. but for some problems, such as Bertrands postulate encoded, presumably there is not an extremely difficult solution that could be found by machine learning. one finds a systematic relation between G_n and G_n+1 and then one has to show that this also leads to the property that if F_n is satisfiable then so is F_n+1 (based on the graph relation). this reminds me of the concept of “satisfiable/ unsatisfiable core” that is being explored in SAT research, suspect it would be relevant.[c]

this is an almost radically new form of automated theorem proving that is not explored almost anywhere in the literature. and, it doesnt entirely sound like implausible/ inconceivable science fiction. there are decades of results in automated theorem proving and massive effort in the area, many different software algorithms/ packages developed and research programs (see eg wrt this problem [b]). but none come anywhere close to this.

undecidability: the ULTIMATE challenge

(hi all, my favorite “haunts” in cyberspace lately seem not so busy or as engaging as in past, not sure why that is, so out of sheer lack of alternatives am going through my big link piles/ archives and finally writing up some of my big topics saved over years.)

this is a very big/ deep/ mysterious topic that have been researching and musing on for decades, almost since a teenager, starting with Hofstadters award winning/ near-legendary musings.[a5][a6] the topic also unifies many different threads/ post on this blog. (the delightful comic comes from a google image search result on the term “undecidability”… (doncha just luv AI?! which maybe even could show signs of a (great!?!) sense of humor at times?)

its one of those crosscutting areas where it can seem like very many people are inquiring into it, and almost nobody is, all at the same time. undecidability seems to be the extraordinary mathematical equivalent/ model of terra incognita aka middle age maps that stated at the edges—literally thought to be the same as the edges of the earth or even existence—“here there be dragons”.

aka (mixed metaphor overload alert…) the lynchpin. the 3rd rail (of math/ computer science). haunted house/ mansion. the bermuda triangle. UFO. (boogeyman? monster? under the bed?) or how about some great physics analogies? dark matter/ energy? black hole(s)? etc!

one might say this is a major phenomenon of the massive iceberg, where 90% (or more!) is unseen, or the elephant, in the aphorism about the blind men and the elephant. its a very big elephant in this case! that may even be the understatement of the year! which reminds me, as an old anecdote went in my high school calculus book (Finney?), talking about the self-referential derivative of e^x as e^x, quoting an old joke/ legend/ folklore about what the indian guru said when asked about what held the earth, and said “an elephant”, and then asked “whats below that”, replied, “its elephants all the way down!”

over the many years, have discovered there is a strong crosscutting connection between these many seemingly disparate areas of computer science in math. they are basically all united by the phenomenon of turing completeness, or the flip (dark) side, (the sheer difficulty of) undecidability. maybe in some ways have some better picture of this than others, but even my own picture is quite fragmented and cloudy at times. ie just another blind mans ramblings. (“in the kingdom of the blind, the one-eyed man is king”?)