Shinichi Mochizuki | |
---|---|
Born | [1] | March 29, 1969
Nationality | Japanese |
Alma mater | Princeton University |
Known for | Anabelian geometry Inter-universal Teichmüller theory |
Awards | JSPS Prize, Japan Academy Medal[1] |
Scientific career | |
Fields | Mathematics |
Institutions | Kyoto University |
Doctoral advisor | Gerd Faltings |
Shinichi Mochizuki (望月 新一, Mochizuki Shin'ichi, born March 29, 1969) is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory,[2][3][4][5] which has attracted attention from non-mathematicians due to claims it provides a resolution of the abc conjecture.[6]
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abc conjecture proof PUBLISHED|Latest news you definitely have missed|Proof by Shinichi Mochizuki
Transcription
A proof has been announced of a unsolved, so far, conjecture called the abc Conjecture. If this proof is right, then it's going to be news on the scale of Fermat's last theorem was in the '90s, which was this big unsolved problem. And was this huge event. So it's really exciting. We don't know if this proof is right yet, so a Japanese mathematician called Mochizuki has released these papers, and altogether, it's 500 pages long. He's been working on this for a very long time. He's come up with his own theory of maths-- a whole new body of maths-- and he's called it interuniversal geometry. I know nothing about that. Very few people do. Even the experts don't know much about this at the moment. So it's going to take a very long time to make sure if the proof is right, because they're going to have to learn this whole new theory of mathematics. So the abc Conjecture involves the most simple formula you can think of. It's this-- a plus b equals c. It doesn't get much easier than that. And that's where it gets its name from. The rules are, these are whole numbers, and they don't share any factors. So that means if I can divide a by 2, then I'm not allowed to divide b by 2. Or if I could divide a by 3, then b is not allowed to be divisible by three. They're not allowed to share any factors like that. All right, let's try an example that works. 1,024 plus 81 equals 1,105. Right, now let's just check they don't share any factors. In fact, I've picked these on purpose. This one is 2 to the power of 10, and this one is 3 to the power of 4. So they don't share any factors there. Oh, and this one, I'll do the same, is 5 times 13 times 17. Now, this is what I want you to notice. On the left hand side, you've got lots of prime numbers. We've got all 10 of them over here, and another four. Loads of them over here. On the right hand side, you only have three, and this is what you tend to see most of the time. This is what's normal. If you get lots of primes on the left, you only get a few on the right hand side. So this is what the conjecture is about. I want to show you one where it doesn't work. 3 plus 125. That's equal to 128. Let's just check they don't share any factors then-- well, that's 3, this is 5 cubed, and this, 128, is 2 the power of 7. Now, this one is not like the first one I showed you. You've only got a few primes on the left, but we've got loads more on a right. So you've got more on the right than you do on the left. That's unusual. That's weird. So rather than not working, like I said earlier, it's the unusual example. These don't happen so often. So the technical way to say this is this. Times those primes together. So I'm going to do that. So 2 times 3 times 5 times 13 times 17, and that equals to 13,260. It's a big number, and it's bigger than the right hand side, which was this. That's what normally happens. OK, so if you do this, you get a bigger number. I'm going to show you this one that I said was unusual. If we do the same thing-- 3 times 5 times 2, that's equal to 30, and that's smaller than 128. So that's the difference. So this is unusual. This is much smaller than the right hand side. This number, when you multiply the primes together, is called the radical of ABC. It's called the radical because it is [INAUDIBLE]. The abc Conjecture is the radical-- which I told you how to work out, that's this-- the radical of abc is bigger than the right hand side. I said that was c. That's what you get normally. In fact, the conjecture is more than that. It talks about the powers of that, too. But there are exceptions, and these are the exceptions. When k equals 1-- that's the power is 1-- there are infinitely many exceptions just like the one I've just shown you there. Infinitely many, even though I said these were the rare ones, the unusual ones, there are infinitely many of them. But if you take a power bigger than 1, even if it's only a little bit bigger, even if it's like a power of 1.00001-- tiny, tiny little bit bigger-- if it's bigger than 1, then you get finitely many exceptions. And this is a little bit surprising, because, yes, if it's just a little bit bigger than 1, you get finitely many exceptions. You could count them off. You could write them down. You could say, here are all the exceptions for this power. And that's unusual. That's unexpected. Now, this is the conjecture. It's very abstract. It's very pure. This was made in the '80s, this conjecture. But if this can be proven, what it's going to do is it's going to prove a whole bunch of other stuff at a stroke. And that's why it's big news. Originally, they thought that Fermat's last theorem, which I talked about being solved in the '90s, they thought this was the way to solve it. Because there is a way that, if you can solve this, you can solve a version of Fermat's last theorem. It didn't turn out that way, because Fermat's last theorem was solved first. I heard of it, I think, before it went around the nerdy blogs, and I thought, well, we could talk about it on Numberphile. But then, it's still not been checked, so maybe we shouldn't talk about it on Numberphile. But then when all the blocks started going mad about it, I thought someone might ask us. I mean, that's how you probe extra dimensions. That's how you probe the very small.
Biography
Early life
Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki.[7] When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974–76).[8] Mochizuki attended Phillips Exeter Academy and graduated in 1985.[9]
Mochizuki entered Princeton University as an undergraduate student at the age of 16 and graduated as salutatorian with an A.B. in mathematics in 1988.[9] He completed his senior thesis, titled "Curves and their deformations," under the supervision of Gerd Faltings.[10]
He remained at Princeton for graduate studies and received his Ph.D. in mathematics in 1992 after completing his doctoral dissertation, titled "The geometry of the compactification of the Hurwitz scheme," also under the supervision of Faltings.[11]
After his graduate studies, Mochizuki spent two years at Harvard University and then in 1994 moved back to Japan to join the Research Institute for Mathematical Sciences (RIMS) at Kyoto University in 1992, and was promoted to professor in 2002.[1][12]
Career
Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998.[13] In 2000–2008, he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve.
On August 30, 2012, Mochizuki released four preprints, whose total size was about 500 pages, that developed inter-universal Teichmüller theory and applied it in an attempt to prove several very famous problems in Diophantine geometry.[14] These include the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. In September 2018, Mochizuki posted a report on his work by Peter Scholze and Jakob Stix asserting that the third preprint contains an irreparable flaw; he also posted several documents containing his rebuttal of their criticism.[15] The majority of number theorists have found Mochizuki's preprints very difficult to follow and have not accepted the conjectures as settled, although there are a few prominent exceptions, including Go Yamashita, Ivan Fesenko, and Yuichiro Hoshi, who vouch for the work and have written expositions of the theory.[16][17]
On April 3, 2020, two Japanese mathematicians, Masaki Kashiwara and Akio Tamagawa, announced that Mochizuki's claimed proof of the abc conjecture would be published in Publications of the Research Institute for Mathematical Sciences, a journal of which Mochizuki is chief editor.[18] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".[18] The special issue containing Mochizuki's articles was published on March 5, 2021.[2][3][4][5]
Publications
- Mochizuki, Shinichi (1997), "A Version of the Grothendieck Conjecture for p-adic Local Fields" (PDF), International Journal of Mathematics, Singapore: World Scientific Pub. Co., 8 (3): 499–506, CiteSeerX 10.1.1.161.7778, doi:10.1142/S0129167X97000251, ISSN 0129-167X
- Mochizuki, Shinichi (1998), "The intrinsic Hodge theory of p-adic hyperbolic curves, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica: 187–196, ISSN 1431-0635, MR 1648069
- Mochizuki, Shinichi (1999), Foundations of p-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1190-0, MR 1700772
Inter-universal Teichmüller theory
- Mochizuki, Shinichi (2011), "Inter-universal Teichmüller Theory: A Progress Report" (PDF), Development of Galois–Teichmüller Theory and Anabelian Geometry, The 3rd Mathematical Society of Japan, Seasonal Institute.
- Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF).
- Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF).
- Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF).
- Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF).
References
- ^ a b c d Mochizuki, Shinichi. "Curriculum Vitae" (PDF). Retrieved 14 September 2012.
- ^ a b Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory I: Construction of Hodge Theaters" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 3–207. doi:10.4171/PRIMS/57-1-1. S2CID 233829305.
- ^ a b Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 209–401. doi:10.4171/PRIMS/57-1-2. S2CID 233794971.
- ^ a b Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 403–626. doi:10.4171/PRIMS/57-1-3. S2CID 233777314.
- ^ a b Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 627–723. doi:10.4171/PRIMS/57-1-4. S2CID 3135393.
- ^ Crowell 2017.
- ^ Leah P. (Edelman) Rauch Philly.com on Mar. 6, 2005
- ^ MOCHIZUKI, Kiichi Dr. National Association of Japan-America Societies, Inc.
- ^ a b "Seniors address commencement crowd". Princeton Weekly Bulletin. Vol. 77. 20 June 1988. p. 4. Archived from the original on 3 April 2013.
{{cite news}}
: CS1 maint: bot: original URL status unknown (link) - ^ Mochizuki, Shinichi (1988). Curves and their deformations. Princeton, NJ: Department of Mathematics.
- ^ Mochizuki, Shinichi (1992). The geometry of the compactification of the Hurwitz scheme.
- ^ Castelvecchi 2015.
- ^ "International Congress of Mathematicians 1998". Archived from the original on 2015-12-19.
- ^ Inter-universal Teichmüller theory IV: log-volume computations and set-theoretic foundations, Shinichi Mochizuki, August 2012
- ^ Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
- ^ Fesenko, Ivan (2016), "Fukugen", Inference: International Review of Science, 2 (3), doi:10.37282/991819.16.25
- ^ Roberts, David Michael (2019), "A crisis of identification", Inference: International Review of Science, 4 (3), doi:10.37282/991819.19.2, S2CID 232514600
- ^ a b Castelvecchi, Davide (April 3, 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. Retrieved April 4, 2020.
Sources
- Castelvecchi, Davide (7 October 2015), "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof", Nature, 526 (7572): 178–181, Bibcode:2015Natur.526..178C, doi:10.1038/526178a, PMID 26450038
- Crowell, Rachel (19 September 2017). "On a summary of Shinichi Mochizuki's proof for the abc conjecture". American Mathematical Society.
- Ishikura, Tetsuya (16 December 2017). "Mathematician in Kyoto cracks formidable brainteaser". The Asahi Shimbun.
External links
- Shinichi Mochizuki at the Mathematics Genealogy Project
- Personal website
- Papers of Shinichi Mochizuki
- A brief introduction to inter-universal geometry
- On inter-universal Teichmüller theory of Shinichi Mochizuki, colloquium talk by Ivan Fesenko
- Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki by Ivan Fesenko
- Introduction to inter-universal Teichmüller theory (in Japanese), a survey by Yuichiro Hoshi
- RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory, March 2015, Kyoto*
- CMI workshop on IUT theory of Shinichi Mochizuki, December 2015, Oxford*