Who wants to be a millionaire?

Gabriellebirchak/ July 20, 2018/ Contemporary History, Modern History

Image from Who Wants to Be a Millionaire

Some­times win­ning $1,000,000 is sim­ple. You can scratch a tick­et, or beat the sta­tis­ti­cal odds of a lot­tery draw­ing, or earn a spot on Who Wants to be a Mil­lion­aire? Easy, right?! A not so easy way to win $1,000,000 is to solve one of the six remain­ing Mil­len­ni­um Prob­lems. It used to be sev­en prob­lems until Math­e­mati­cian Grig­ori Perel­man solved the Poin­care con­jec­ture, and then, in Perel­man style, denounced his award and refused the mon­ey. The back­sto­ry on the rejec­tion of his award is that he believed that his con­tri­bu­tion to solv­ing the con­jec­ture was real­ly no greater than Colum­bia University’s Math­e­mati­cian Richard Hamil­ton, whose the­o­ries Perel­man uti­lized to solve Poincare’s con­jec­ture. Okay, so he’s a man with con­vic­tions. Let’s give him that.

So now, who is ready to give up on Scratch­ers and solve the remain­ing problems?

Homer in the Third Dimen­sion, Sea­son 7, Episode 6: Where P=NP

First, we have P ver­sus NP, intro­duced by Amer­i­can-Cana­di­an Com­put­er Sci­en­tist and Math­e­mati­cian, Pro­fes­sor Stephen Cook of the Uni­ver­si­ty of Toron­to. This par­tic­u­lar prob­lem is a com­put­er sci­ence prob­lem that asks whether every prob­lem that can be solved and ver­i­fied quick­ly can also be quick­ly solved and ver­i­fied again.

In short, P ver­sus NP is like com­par­ing a set of prob­lems that are easy to solve and ver­i­fy ℗ against prob­lems that are easy to ver­i­fy but can­not be quick­ly solved (NP). We know that an algo­rithm exists for prob­lems in the P set, but we don’t quite know if an algo­rithm exists for the prob­lems in the NP set.

Pro­fes­sor Pierre René Vis­count Deligne of the Insti­tute for Advanced Study intro­duced Hodge con­jec­ture. Briefly, the Hodge con­jec­ture is a topo­log­i­cal prob­lem and it states that com­pli­cat­ed alge­bra­ic vari­eties can be reduced to alge­bra­ic cycles that are com­bi­na­tions of sim­pler forms. It incor­po­rates many con­cepts in math­e­mat­ics that include topol­o­gy, geom­e­try, cal­cu­lus, and alge­bra. David Met­zler of Albu­querque Acad­e­my has a real­ly great series of six videos that helps explain Hodge con­jec­ture start­ing here:

Rie­mann Hypoth­e­sis, based in num­ber the­o­ry and intro­duced by Pro­fes­sor Emer­i­tus Enri­co Bombieri of the Insti­tute for Advanced Study, asks for the proof or dis­proof that all non-triv­ial zeros of the Rie­mann zeta func­tion lie on the crit­i­cal line of ½ with the excep­tion of the triv­ial zeros along the neg­a­tive axis. I love this one! One of my favorite videos on Rie­mann zeta func­tion is an expla­na­tion by Pro­fes­sor Edward Frenkel of the Uni­ver­si­ty of Cal­i­for­nia Berke­ley. It’s a fan­tas­tic video pro­duced by Num­ber­phile and Brady Haran.

Dis­cov­ery of the Omega Minus Particle

The Yang-Mills exis­tence and mass gap was pre­sent­ed by Math­e­mat­i­cal Physi­cists Arthur Jaffe of Har­vard Uni­ver­si­ty and Edward Wit­ten of the Insti­tute for Advanced Study. Jaffe and Wit­ten cre­at­ed a the­o­ry that address­es strong and weak nuclear forces. The the­o­ry is based on dif­fer­en­tial geom­e­try and quan­tum mechan­ics. In the Yang-Mills the­o­ry, there is a mass gap, also known as a non-zero Mass. It is lit­er­al­ly a gap between strong and weak nuclear forces! This gap rep­re­sents the dif­fer­ence between the low­est ener­gy state and the high­est ener­gy state. Even though phys­i­cal exper­i­ments show that there is a mass gap, and even though com­put­er-based math­e­mat­i­cal mod­els show that there is a mass gap, we are still try­ing to explain, math­e­mat­i­cal­ly, the Mass Gap in the quan­tum appli­ca­tions of their for­mu­las. In my opin­ion, this is one of the most excit­ing Mil­len­ni­um prob­lems pre­sent­ed. Some the­o­rists say it is much like the ear­ly days of cal­cu­lus or even quan­tum physics when tan­gi­ble results were being dis­cov­ered even though the process of ful­ly reduc­ing log­i­cal aber­ra­tions through the­o­ret­i­cal­ly rig­or­ous math­e­mat­i­cal expla­na­tions hadn’t been estab­lished yet. The Yang-Mills exis­tence and the mass gap is like for­ward­ing to the end of the movie, see­ing the most amaz­ing end­ing, yet miss­ing the cli­mac­tic adven­ture at the end of Act 2! The excite­ment is yet to come!

The Navier-Stokes exis­tence and smooth­ness prob­lem is a love­ly and won­der­ful prob­lem for math­e­mati­cians like myself who enjoy Par­tial Dif­fer­en­tial Equa­tions. Math­e­mat­ics Pro­fes­sor Charles Fef­fer­man of Prince­ton Uni­ver­si­ty intro­duced this equation.

I per­son­al­ly became emo­tion­al­ly attached to the Navier-Stokes equa­tion as an under­grad, when a peer jok­ing­ly pro­posed that half of the class solve it, just to prove to the world of math­e­mat­ics that some­times under­grads know what they are doing. By the time we reached finals, we rather real­ized that we had no idea what we were doing.

Gif by Thier­ry Dugnolle

This gif depicts Leon­hard Euler’s (18th cen­tu­ry) equa­tion, which describes the flow of fric­tion­less and incom­press­ible flu­ids. It gets trick­i­er with Claude-Louis Navier’s appli­ca­tion of vis­cos­i­ty. Final­ly, Sir George Gabriel Stokes cre­at­ed solu­tions, but they were only applic­a­ble in two-dimen­sion­al flow. Now, throw in pres­sure, gra­di­ent tur­bu­lence, vis­cous flow, non-steady flows that depend on time, vor­tices, chaos, and three-dimen­sions, and you have a smooth­ness prob­lem that either nev­er occurs or presents pock­ets of ener­gy per unit mass.

In 2014, Math­e­mat­ics Pro­fes­sor Mukhtar­bay Otel­bayev of the Eurasian Nation­al Uni­ver­si­ty pre­sent­ed a pos­si­ble solu­tion uti­liz­ing an abstract approach. How­ev­er, his results are still being ana­lyzed. Math Pro­fes­sor Ter­ence Tao of the Uni­ver­si­ty of Cal­i­for­nia Los Ange­les has pre­sent­ed some won­der­ful analy­sis on Navier-Stokes and Otelbayev’s solu­tion that you can read here: https://terrytao.wordpress.com/tag/navier-stokes-equations/. Addi­tion­al­ly, in 2016 Otel­bayev pre­sent­ed more work on Navier-Stokes (https://aip.scitation.org/doi/abs/10.1063/1.4959619?journalCode=apc) that “clar­i­fied the cor­rect for­mu­la­tion of the bound­ary con­di­tions for the pres­sure in the environment.”

Final­ly, we have the Birch and Swin­ner­ton-Dyer con­jec­ture. Who doesn’t love donuts?!!!

Oxford University’s Regius Pro­fes­sor Sir Andrew John Wiles, who proved Fermat’s Last The­o­rem, pre­sent­ed the Birch and Swin­ner­ton-Dyer con­jec­ture. The con­jec­ture was born out of numer­i­cal inves­ti­ga­tions of ellip­tic curves con­duct­ed by Pro­fes­sors Bryan Birch and Peter Swin­ner­ton-Dyer at the Uni­ver­si­ty of Cam­bridge, using the EDSAC com­put­er. This beau­ti­ful con­jec­ture says that there must be a math­e­mat­i­cal­ly sim­ple way to deter­mine whether an ellip­tic curve, which is a cubic curve (to the order of 3) that is restrict­ed to a torus (a donut) has either finite or infi­nite ratio­nal solu­tions, based on whether the func­tion is not equal to zero or equal to zero. The details of this con­jec­ture are beau­ti­ful (or ele­gant as some would say), and for bud­ding math­e­mati­cians, it is delight­ful to dig into as you learn about Mordell’s The­o­rem, the rank of a curve and mul­ti­ple consequences.

Image by me

So, here are six Mil­len­ni­um Prob­lems to solve and win $1,000,000, thanks to the Clay Math­e­mat­ics Insti­tute! Remem­ber: they are called the Mil­len­ni­um Prob­lems, not Mil­len­ni­al Prob­lems. Mil­len­ni­al Prob­lems are much hard­er! Those include fig­ur­ing out how to pay down the expo­nen­tial­ly ris­ing inter­est on your stu­dent loans, or how to work five jobs and still get sleep. No, Mil­len­ni­um Prob­lems and its glo­ri­ous math­e­mat­ics are much easier!

For bud­ding math­e­mati­cians, you can read about the Mil­len­ni­um Prob­lems at the Clay Math­e­mat­ics Insti­tute web­site at http://www.claymath.org/millennium-problems

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