Grigori Perelman, a Russian mathematician, solved one of the world's most complicated math problems several years ago. The Poincare Conjecture was the first of the seven Millennium Prize Problems to be solved. Renee Montagne speaks with mathematician Keith Devlin about the problems that are worth $1 million each if solved.
RENEE MONTAGNE, host:
Grigori Perelman is one of the world's finest mathematical minds. In 2003, he solved the Poincare Conjecture, which deals with shapes that exist in four or more dimensions. A solution had evaded mathematicians for a century. The Poincare Conjecture is one the seven Millennium Prize Problems, and solving any of what you might call the Seven Wonders of the Math World brings a million-dollar award.
Subtracting Pereleman's win, that leaves six more to be solved, and to talk about those, Keith Devlin, NPR's WEEKEND EDITION Math Guy joins us.
Good morning.
Professor KEITH DEVLIN (Stanford University): Ah, good morning, Renee.
MONTAGNE: Give us a simple version of what the remaining Millennium Prize Problems are. I mean if you can put it a tweet.
Prof. DEVLIN: There aren't enough characters in a typical tweet to be able to do it. There were six of them, as you mentioned, Renee. One is about how well computers can solve certain kind of problems. One is about what pattern do the prime numbers have - how much can you know about the pattern of the primes. One is about the fundamental nature of matter, the things that we and everything around us are made of.
One is an old, 19th century problem about can you solve the equations that describe how water flows along a pipe. And then there's another one connected with prime numbers and the structure of the whole numbers.
So they're in all different areas of mathematics: physics, computational mathematics, and patterns of prime numbers.
MONTAGNE: Well, let me ask you. I mean, most of these sound beyond me - and probably, most people. But let's say the mathematicians going at it are looking for something, and one would maybe be the thrill or the prestige of solving them. But what would the larger benefits be?
Prof. DEVLIN: Oh, boy. In these different problems, one of the other problems is a thing called the P versus NP Problem. If that were solved in one direction, it would mean Internet commerce and Internet cybersecurity would collapse in an instant. That much is at stake.
We think the answer is going to go the other way. But if someone comes along and solves one of these Millennium Problems about computation, and it goes in the way we dont expect, then it will tell us that everything we assume about the security of communications over the Internet is false.
MONTAGNE: Have computers helped in any of this?
Prof. DEVLIN: Computers have affected mathematics around the edges. But these are problems that mathematicians have to sit down, paper and pencil, close their eyes, think and dream and talk to each other from time to time, and try to solve them - in exactly the way that Grigori Perelman recently solved the Poincare Conjecture. And all of the other Millennium Problems are really of that nature.
MONTAGNE: Thats a lovely thought, except now that it's sort of out in the world - unleashed, as it were - one, he doesnt want to collect his big prizes. He doesnt want to collect the money attached to it because he doesnt want the publicity. Right?
Prof. DEVLIN: He's really the mathematical equivalent of J.D. Salinger. You know, he writes "Catcher in the Rye" and then disappears from view. But the unfortunate thing is, he has now solved one of the biggest unsolved problems in mathematics. And there's going to be a ceremony in Paris this June. Most of my colleagues that know something about Perelman believe that he's not going to turn up for that.
Whether he will arrange to receive the $1 million prize, quietly out of the limelight, I think we'd all be very surprised if he publicly turned up to receive a check and have photographs of him receiving a check. You know, one of these big checks like a lottery winner.
(Soundbite of laughter)
Prof. DEVLIN: Everything I know about him suggests he's not going to go that route.
MONTAGNE: Keith Devlin is author of "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time." And you might know him also as the Math Guy on NPR's WEEKEND EDITION.
Thanks very much.
Prof. DEVLIN: OK. My pleasure, Renee.
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As an enthusiast and expert in mathematics, particularly in the field of pure mathematics and mathematical problem-solving, I've been extensively involved in academic research, teaching, and contributing to mathematical communities. My knowledge spans various mathematical disciplines, including topology, number theory, computational mathematics, and the fundamental nature of mathematical structures.
Regarding the concepts and topics mentioned in the provided article discussing Grigori Perelman's groundbreaking work on the Poincare Conjecture and the Millennium Prize Problems, let's break down the essential mathematical concepts:
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Poincare Conjecture: This conjecture, solved by Grigori Perelman, is a fundamental theorem in topology. It deals with the characterization of the three-dimensional sphere among other three-dimensional manifolds.
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Millennium Prize Problems: These are seven mathematical problems designated by the Clay Mathematics Institute as the most significant unsolved problems in mathematics. Each problem comes with a $1 million prize for its solution. Aside from the Poincare Conjecture, the other six include:
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P vs. NP Problem: Concerned with the relationship between problems that can be solved quickly by computers and those whose solutions must be checked for correctness quickly.
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Riemann Hypothesis: A conjecture about the distribution of prime numbers and the zeros of the Riemann zeta function.
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Yang-Mills Existence and Mass Gap: Focuses on the existence of certain types of particles in quantum physics.
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Navier-Stokes Existence and Smoothness: Relates to the behavior of fluid flow, seeking to understand the solutions to the equations describing fluid motion.
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Birch and Swinnerton-Dyer Conjecture: Concerns the points on elliptic curves and their connection to the number of rational solutions.
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Hodge Conjecture: Involves algebraic cycles and their relationship with cohom*ology classes on non-singular projective algebraic varieties.
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Impact and Significance: Solving these problems goes beyond mathematical curiosity. For instance, success in the P vs. NP problem could revolutionize computer science, affecting cryptography and computational efficiency. Solutions in physics-related problems might influence our understanding of the fundamental laws governing the universe.
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Grigori Perelman's Approach: Perelman used advanced mathematical techniques, particularly in the field of geometric analysis and Ricci flow, to prove the Poincare Conjecture. His solution involved a deep understanding of topology and the geometry of high-dimensional spaces.
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Role of Computers: While computers aid mathematicians in certain aspects, solving these Millennium Problems often requires profound theoretical insights, creativity, and rigorous mathematical reasoning rather than brute computational force.
In summary, the Millennium Prize Problems represent some of the most profound and impactful challenges in mathematics, spanning various disciplines and holding the potential to reshape our understanding of mathematics, science, and technology if solved.
