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May1613   Kanatahodets: I agree that for Hou it would be better to concentrate on chess; she has comparative advantage in this matter. she will never be as good as in chess in any of profession. Lasker was a decent mathematician, but not of Poincare or Hilbert dimension. moreover, he knew that he will never reach such heights in math as he did in chess. 

May1613   Kanatahodets: Karyakin also tried to be someone except chess but he failed. McShane changed his focus from chess just because he cannot live on chess only. he is also smart and he knows that he cannot reach magnus' level. he can be one of the best, say like Karyakin,Radjabov, Nakamura. But effort is high and the outcome is uncertain. that's why he changed his profession to one where it is easier to make a living. 

May1613   Kanatahodets: Nicholas cited Lasker, it is true. Emmanuel could be of the level of great Noether, but still this is not the Hilbert's level. It is true that competition and level of math is much higher than of chess. i would compare the first one with NBA and the second one with NHL. more players in math, higher stakes. Someone said, if chess would be popular as math we would have 20 MC! I would add than 10 of them would be from China! I've checked recently the faculty at Princeton math  I haven't found a single Russian but many Chinese. This is very serious, it is no chess. 

May1613
  HeMateMe: What is Yifan studying, in school? What if she discovers biology? Will her chess suffer? 

May1613   Kanatahodets: regardin Lasker math. there's a notion of lasker ideal. as far as i remember it is an intersection of prime ideals. i think lasker proved that factorization of any PI ring by such an ideal is a direct sum of prime rings. Then any prime PI ring, at least finite dimensional, to my knowledge is easily describable. Say for finite dimensional prime ring is a simple ring and thus can be characterized as a sub ring of nxn matrix ring. 

May1613   Kanatahodets: it is a hardly difficult result, but at that time it was something taking into account their cumbersome notations. Even Hilbert theorem 90 can be proved now by any grad student. So lasker worked in abstract algebra. In this field Goettingen school started its spur with Emmi Noether and her students. So lasker couldn't compete with great Emmi and he realized that it is safer to come back to the pastures where he was unbeaten. 

May1613
  norami: < Kanatahodets> I've read that the chance that a number is prime is the inverse of it's natural log, but that hasn't been proven and proving it is the most important problem in mathematics. Any truth to that? 

May1613   Kanatahodets: <norami: < Kanatahodets> I've read that the chance that a number is prime is the inverse of it's natural log, but that hasn't been proven and proving it is the most important problem in mathematics. Any truth to that?> NO!!! It was proved in XIX century by Hadamard and VPoussen. Truly genius result! But Laster's theorem is related to prime ideals of noncommutative rings (he could consider commutative rings  I don't know). It is an analog of a prime number. For example for Z all ideals are pZ where p is a prime number. When you factorize Z/pZ you get Z_p  the field which is very easy to analyze. 

May1613   Kanatahodets: Norami, you may have in mind the Riemann hypothesis. This is truly THE MOST important problem in all math. BTW, prime number theorem and the Riemann hypo are related but for the theorem you need a much weaker version of RH. RH is the behemoth of all problems. 

May1613
  norami: I thought the prime number theorem was that as x approaches infinity the percentage of integers less than x that are prime approaches the inverse of the natural log of x. But that's not the same thing as RH which says prime numbers form a random sequence over the integers under the probability model of the inverse of the natural log. In plainer but less precise English, the probability a number is prime is the inverse of the natural log. At least, that's the way to bet. 

May1613   Kanatahodets: < norami: the probability a number is prime is the inverse of the natural log. At least, that's the way to bet.> That is true and it was proven long time ago! 

May1613   Kanatahodets: < norami: ...RH which says prime numbers form a random sequence over the integers under the probability model of the inverse of the natural log.> This doesn't make any sense. Sorry. 

May1613
  norami: Whatever RH is, I have one more question. WHY is it the most important problem in mathematics? 

May1613   Catholic Bishop: <I haven't found a single Russian but many Chinese. This is very serious, it is no chess.> Russians and Eastern Europeans usually do very well at the International Maths Olympiad. Chinese and other East Asians also do pretty well. The only slightly curious exception is India, ranking consistently lower than little countries like Taiwan and Hong Kong at these competitions. 

May1613   Kanatahodets: <Catholic Bishop: Russians and Eastern Europeans usually do very well at the International Maths Olympiad. Chinese and other East Asians also do pretty well.> This is very old data; China has 1015 teams of equal strength and it dominates IMO. Still I don't take IMO seriously. Who cares about fast problem solving? 

May1613   Kanatahodets: < norami: Whatever RH is, I have one more question. WHY is it the most important problem in mathematics?> If I remember well if RH is true it will give us a lot in terms of knowledge of distribution of prime numbers. Much more important than Fermat theorem! 

May1613   nok: <Lasker was a decent mathematician, but not of Poincare or Hilbert dimension.> Uh, you can't hold that against him. 

May1613
  perfidious: <nok: <Lasker was a decent mathematician, but not of Poincare or Hilbert dimension.> Uh, you can't hold that against him.> Same with the fact that he was not a giant in bridge, while a most capable player, on the order of Terence Reese. It speaks volumes of Lasker's brilliance that he was able to perform to the degree that he did in these disciplines, plus chess. 

May1613
  HeMateMe: Lasker was also a master of checkers, and he wrote one or two instruction books on the game of backgammon, which, though not nearly as deep as chess, does have a skill component. 

May1913
  HeMateMe: Hou finishes at 3/4/4, negative score. Strange. She still seems to be finding her range. She's the best, when she's on. 

May1913   dehanne: Hou might be losing interest in chess. 

May1913   Alien Math: Hou still shows interest in chess her blog notes 

Jun2213   Thanh Phan: <HeMateMe: What is Yifan studying, in school? What if she discovers biology? Will her chess suffer?> Apparently at Peking University Institute of International Relations. 

Jun2813
  notyetagm: Wow, how did Khotenashvili go from winning the last GP event to finishing dead last in this one? FIDE Women's GP Geneva, 1st => Women Grand Prix Geneva (2013)/Bela Khotenashvili FIDE Women's GP Dilijian, 12th => FIDE Women's Grand Prix Dilijan (2013)/Bela Khotenashvili 

Jun2813   haydn20: < norami: Whatever RH is, I have one more question. WHY is it the most important problem in mathematics? > Technically, The PNT is the claim that the primecounting function pi(x) is asymptotic to x divided by the natural log of x. That is, pi(x)/(x/ln(x)) > 1 as x > infinity. We can restate this as pi(x)/x is approximately equal to 1/ln(x), i.e., the density of the primes among the natural numbers is about 1/ln(x). This means, for example, that the probability that a number less than 1000 is prime is about 1/ln(1000) = 0.145. [The actual density is 0.148.] Since 1/ln(x) > 0 as x > infinity, we can loosely say that the probability that a given number is prime is 0. There are various expressions for the error in this approximation. If we had a proof of the Riemann Hypothesis, we
could substantially improve the error estimate. In addition, since the zeta function of the RH pops up all over the place, God only knows what effects the new math that seems necessary to prove the RH will have. 



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