The Poincare Conjecture and Father’s Day

Gabriellebirchak/ July 2, 2018/ Contemporary History, Modern History

My love for math began when I was about sev­en. From ele­men­tary school through high school, on some morn­ings I would find a math puz­zle next to my cere­al bowl. My dad, before he would go to work, would write down a puz­zle and set it next to my break­fast set­ting. When I worked with him at Sund­strand Avi­a­tion after I grad­u­at­ed high school, some­times I would even find a puz­zle in my lunch bag. My dad real­ly loved me. I know this because he called me Stinky Pants. But, I LOVED my dad! Those math puz­zles sealed the deal, and etched his name in my heart as the world’s great­est dad!

To this day, I am still addict­ed to math puz­zles. They are my rab­bit holes that relieve that itchy sweater in my head.*

If you are like me (math nerd), you are prob­a­bly very famil­iar with Clay Math­e­mat­ics Institute’s Mil­len­ni­um prize prob­lems. The Insti­tute pro­posed sev­en prob­lems. Any per­son (or peo­ple) who can solve them will receive a one mil­lion dol­lar prize! To me, this sounds like heav­en! It is a rab­bit hole with the prize at the end of it.

There were sev­en prizes, and now there are six. The prob­lems include:
· P vs. NP
· Hodge Con­jec­ture
· Rie­mann Hypoth­e­sis
· Yang-Mills exis­tence and mass gap
· Navier Stokes exis­tence and smooth­ness prob­lem (my own per­son­al favorite)
· Birch and Swin­ner­ton-Dyer conjecture

I will pro­vide a brief fol­low up blog to this on these prob­lems, includ­ing Navier Stokes, which is my favorite. Navier Stokes includes tur­bu­lence, vor­tices, and three-dimensions…oh and a smooth­ness prob­lem that either nev­er occurs or presents pock­ets of ener­gy per unit mass! It is chaos…kind of like rais­ing teenagers.

There were sev­en prob­lems, but on March 18, 2010, the Clay Math­e­mat­ics Insti­tute announced Grig­ori Perel­man as the recip­i­ent of the award for his work on the Poin­caré conjecture.

So, what exact­ly is the Poin­care Con­jec­ture? It is a con­jec­ture based in topol­o­gy that states that for every sim­ply con­nect­ed, closed 3‑manifold is home­o­mor­phic to the 3‑sphere. For non-math read­ers: it states that a sphere can exist in mul­ti­ple dimen­sions. A 1‑sphere is a cir­cle. A 2‑sphere is a two-dimen­sion­al sur­face that sits in three-dimen­sion­al space, and a 3‑sphere is a three-dimen­sion­al sur­face that sits in four-dimen­sion­al space.

We can see a sphere in two dimen­sions, even though it looks like a cir­cle. Fur­ther­more, this sphere could also exist in four dimen­sions, even though it wouldn’t look exact­ly like a sphere. It revolves around the con­cept that we live in a world that we see as three dimen­sion­al, yet, hypo­thet­i­cal­ly, if we were to enter a four, five, or six-dimen­sion­al world would we still be able to rec­og­nize a sphere uti­liz­ing a math­e­mat­i­cal per­spec­tive. Or as come­di­an and math­e­mati­cian Simon Pam­p­e­na says, “It means that the sim­plest closed object in any dimen­sion is the sphere!”

So, stand­ing on the shoul­ders of Colum­bia University’s Richard Hamilton’s the­o­ries that were poised to solve it, Perel­man solved the problem.

Perel­man had already done sig­nif­i­cant work up to this point on this con­jec­ture. As a Russ­ian math­e­mati­cian, he made con­tri­bu­tions to Rie­mann­ian geom­e­try and geo­met­ric topol­o­gy that were out­stand­ing. In 1996, the Euro­pean Math­e­mat­i­cal Soci­ety offered Perel­man an award for his valu­able work and find­ings on the soul con­jec­ture. Perel­man reject­ed the award! In 2006, he was offered the Fields Medal for his analy­sis on the geo­met­ric struc­ture of the Ric­ci flow. He reject­ed this award as well. His response? “I’m not inter­est­ed in mon­ey or fame; I don’t want to be on dis­play like an ani­mal in a zoo.”

That is prob­a­bly why when he was offered the Clay Mil­len­ni­um Prize in 2010, he (drum roll please!) reject­ed the award. Perel­man stat­ed that he believed that his con­tri­bu­tion to solv­ing the con­jec­ture was real­ly no greater than Hamil­ton, whose the­o­ries Perel­man uti­lized to solve Poincare’s con­jec­ture. Perel­man fol­lowed up his rea­son­ing by stat­ing that he dis­agreed with the orga­nized math­e­mat­i­cal com­mu­ni­ty, stat­ing that it was not fair that the Clay Insti­tute did not also offer the prize to Hamil­ton. “I don’t like their deci­sions, I con­sid­er them unjust,” Perel­man told Inter­fax. Then, it got cat­ty. Fields medal­ist Shing-Tung Yau deem­pha­sized Perelman’s work in the proof to high­light Cao and Zhu’s work on the con­jec­ture. In an inter­view with The New York­er, Perel­man respond­ed, “I can’t say I’m out­raged. Oth­er peo­ple do worse. Of course, there are many math­e­mati­cians who are more or less hon­est. But almost all of them are con­formists. They are more or less hon­est, but they tol­er­ate those who are not hon­est. … It is not peo­ple who break eth­i­cal stan­dards who are regard­ed as aliens. It is peo­ple like me who are isolated.”

Russ­ian geome­ter Mikhail Gro­mov summed it up per­fect­ly when he said, “The ide­al sci­en­tist does sci­ence and cares about noth­ing else.”

So, there it is. Perel­man is a man of prin­ci­ples and con­vic­tions. OK, so he is a bit eccen­tric. Per­son­al­ly, I love eccen­tric. That’s why I live in Los Ange­les — it’s the cap­i­tal of eccen­tric. He sounds like some­one I’d love to hang out with. He lived in his mother’s base­ment (smart finan­cial move), and once told a jour­nal­ist to leave him alone because he was pick­ing mush­rooms. But, he could have received a mil­lion dol­lars! He refused it. That was not a smart finan­cial move (I won­der what his mom thought about that!). But, ulti­mate­ly Perel­man solved the prob­lem because he loved math.

I get that. Well, not on THAT kind of lev­el. But I get it. My dad taught me that the val­ue of work­ing out a math prob­lem should be for the sheer joy of solv­ing it and dis­cussing the prob­lem at the end of the day. That is what math is about. Not the fame, not the recog­ni­tion, just the joy of math.**

Every morn­ing I think about all the math puz­zles my dad left for me. Those math puz­zles were my Dad’s way of telling me that he loved me beyond words…beyond num­bers. Much like the Navier-Stokes exis­tence and smooth­ness prob­lem, life has hand­ed me tur­bu­lence. Some­thing tells me that the equa­tions to solve some of those tur­bu­lences and pock­ets of chaot­ic ener­gy include lit­tle math puz­zles left beside cere­al bowls for my two boys to solve. Though the itchy sweater in my head may nev­er come off, there is an unde­ni­able swell of admi­ra­tion and love in my heart for the great­est Dad in the world. He too was a man of prin­ci­ple and con­vic­tions. But he also loved being the cen­ter of atten­tion. He had a laugh that could fill a room. He had a brain that nev­er stopped learn­ing. And, he had an altru­is­tic heart that nev­er stopped giving.

It’s been years since my dad passed. My dad was my #1 super­hero, my first dance part­ner, and my first math teacher. I love my dad. And, I loved those math puz­zles! They changed my life. Those math prob­lems didn’t just teach me math. They gave me insight into the human spir­it and the pow­er of parental love. Because of those scrib­blings, I dis­cov­ered that father­hood, par­ent­ing and self­less­ly liv­ing one’s life for some­one else is the most ful­fill­ing way to live a life.

Dads, please keep up the great work! Keep shar­ing your pas­sions with your kids. Keep remind­ing them of all the great things they can learn. Keep inspir­ing, keep encour­ag­ing, keep being present, and keep being a Dad! Good dads make this world an incred­i­ble place! And there is no doubt, we need more incred­i­ble in this world. And, thank you for all you do! As a daugh­ter who has noth­ing but rev­er­ence for awe­some dads, Hap­py Father’s Day! 

- Gabrielle

*See The Itchy brain sim­u­la­tion, Big Bang The­o­ry, sea­son 7, episode 8.
** Still…a mil­lion dol­lars! Who pass­es up money?!