the joy of (cyber) mathematics

1956Metrophi all. this is an informal blog written for the new community blog.[13] hope it works out!

have been using stackexchange computer science sites fairly heavily for over 2 yrs now. its been a highly educational experience to say the least. over several years stackexchange has evolved into an elite social network and meeting place for experts in mathematics[20] and computer science research (mathoverflow/cstheory sites) [6,7,8]. the user lists read like a “whos who” of major researchers in these fields. its a real opportunity to “rub shoulders” at times with some of the greats in cyberspace. “MO” (mathoverflow) recently surpassed the extraordinary 50K question count. is notable in how fast its taken off in a brief time and its scale and is one of the fastest-growing stackexchanges in recent years. one might say the site(s) run like sleek/powerful/well oiled machine(s). 😎

also, Ive been interested in famous open problems for many years. stackexchange is not always the ideal platform for info on them however there are scattered references across many sites. stackexchange has a truly awesome user interface and feature set and does have many gems of questions and answers that reward patient searches. it is unrivalled in many ways as a scientific research platform tracking many core/cutting-edge developments in math and computer science fields.

on good days stackexchange sites can be said to be evolving into leading online scientific societies in a new emerging age of collaborative cyberscience, identified recently in a great book by Nielsen called Reinventing discovery: the new age of networked science,[21] (but thats a big subject for a whole other blog!) and at the minimum all stackexchange sites are invaluable online educational resources for learners at all stages and of course esp for students. over the years have collected many stackexchange questions touching on personal areas of interest and blogged on various topics with copious se question references sprinkled in.

have been blogging here on TMachine blog for about 1½ yr.[12] its basically dedicated to theoretical and applied computer science. however, if advanced computer science is at times like a mazerati, lamborghini, or ferrari (eg in fueling multibilliondollar modern day corporations/algorithms like google/pagerank etc [19]), then mathematics is the elite turbo-charged V12 engine running on nitromethane that powers its central insights! 😎 (uh do any of those cars actually have a V12? oh nevermind…) 😛

so at the core many deep TCS theorems and research directions are based on mathematical properties. mathematics and advanced computer science (especially complexity theory) are highly intermingled subjects and that fusion is increasing all the time. have been inspired to write on some significant mathematical topics on the blog at this intersection.

a neat example is the famous P vs NP problem.[1] this was posed in 1971 by computer scientist Cook. it has remained open now for over 4 decades, so is like the Everest or even Olympic Mons of open problems (Everest was eventually scaled!). in later years mathematicians have become more interested in this problem for various reasons (although its difficult to summarize that shift, its possibly based on its difficulty but also complexity theory’s deep connections to extremal set theory). it has also been connected to some deep mathematical theory in recent times via a very highly advanced bridge called “geometric complexity theory”.[9] this problem has had a $1M “Millenium” prize attached for over 1 decade connected with the Clay Mathematics Institute.[10,11] there are many great questions related to P vs NP scattered on stackexchange and two different CS se sites even have “p-vs-np” tags. way cool!

not so long ago the 100th anniversary of Erdos birthday was celebrated.[2] this had a large amount of related material on the internet. Erdos was a brilliant and legendary mathematician and the most collaborative mathematician and probably scientist who ever lived based on his scientific paper coauthoring, numbering over a thousand, a modern day Euler (almost no other mathematicians in history can match or compare to that “prolificity” as far as number of papers and even Euler was not so collaborative!). Erdos theory wrt “sunflowers” (a theorem about sets of sets or equivalently hypergraphs) shows up in some deep proofs in complexity theory.

Erdos’ work in combinatorics was not as highly prized by all mathematical factions (eg there is a large continuous topology faction in mathematics that doesnt mix easily with discrete Erdos-style combinatorics) but his deep insights are proving to stand the test of time and even find very significant applications in TCS and new previously-unseen ones seem to be conceivable also.

another key topic that has interested me over many years is automated theorem proving.[3] this can also be seen as a dynamic, emerging, even at-times thriving bridge/gateway between computer science and mathematics. there is quite a bit of interest on this topic spread across various se sites.

there was even a recent exciting breakthrough in this area on the Erdos Discrepancy problem, notable in that it was covered in the mainstream media.[5] a major finding/advance was achieved based on formulating the problem as a propositional satisfiability problem (SAT) [17] and attacking it with state-of-the-art solvers. SAT is of course at the heart of CS theory of NP completeness, essentially the 1st problem proven to have the property.[18] automated theorem proving also ties in closely with empirical mathematics, an emerging area with increased attention and developments in the area.

have dabbled myself in experimental mathematics mixed with the notorious Collatz problem.[14,16] experts agree this is one of the hardest open problems in all of mathematics, open ~¾ century (even that may be an understatement, some think its maybe impossible to solve)! yet it has so many nice features to recommend it as far as overlapping with many cutting edge areas of math and CS research and is highly amenable to remarkable computer experiments, making it a worthy subject of even some undergraduate tinkering.[15]

another exciting moment happened with the Zhang twin prime breakthrough about a year ago.[4] this is an old open problem that is over two millenia in age, dating to Euclid and the Greeks, and shows how intensely challenging mathematics can be, but also the unbroken line/thread of development through the ages. number theory problems are some of the oldest and hardest ones in all of mathematics. Zhang was the first to show a finite/infinitely-occuring limit between twin primes.

its truly one of the extraordinary intrinsic features of math that a modern research mathematician could be working on exactly the same problem as an ancient sage. also after working in obscurity and outside academia (even at a subway sandwich shop!) Zhang is now regarded as something of an academic rockstar and has been invited to give talks at elite ivy league schools on his proof. the highly collaborative polymath-cyberspace team has very substantially improved his results.

anyway, all these referenced blogs below represent sizeable amounts of surfing and work in collecting the best links in cyberspace on these topics, many of them to stackexchange questions (preferrably high voted). hope you enjoy them and get a bit of the occasionally-exciting zeitgeist associated with the subject! (oh and by the way, the stackexchange chat rooms can be very fun & lively, at times, encourage you to try them out and might see you around in one!) 😀


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