numerative ombinatorics

A pplications Enumerative Combinatorics and Applications ECA 1:1 (2021) Interview #S3I3 ecajournal.haifa.ac.il

Interview with Doron Zeilberger Toufik Mansour

Doron Zeilberger received a B.Sc. in mathematics from the Uni- versity of London in 1972, and a Ph.D. in mathematics from the Weizmann Institute of Science in 1976, under the supervision of Harry Dym. Professor Zeilberger has important contributions to the fields of hypergeometric summation and q-Series and was the first to prove the alternating sign matrix conjecture. He is considered a champion of using computers and algorithms to do mathematics quickly and efficiently, and his results have been used extensively in modern computer algebra software. Profes- sor Zeilberger’s distinctions include the Lester R. Ford Award in 1990, Leroy P. Steele Prize for Seminal Contributions to Re- search in 1998 for the development of WZ theory with Herbert Wilf, and the Euler Medal in 2004. In 2016 he received, together with Manuel Kauers and Christoph Koutschan, the David P. Robbins Prize of the American Mathematical Society. Professor Zeilberger was a member of the inaugural 2013 class of fellows of the American Mathematical Society.

Mansour: Professor Zeilberger, first of all goals of your research? we would like to thank you for accepting this Zeilberger: Since, very soon, pure human- interview. Would you tell us broadly what generated mathematics will be done better and combinatorics is? faster by computers, I dedicate my research to Zeilberger: Combinatorics is everything. All teaching computers to do mathematics, that in our worlds, the physical, mathematical, and my case is mostly combinatorics and number even spiritual, are inherently finite and discrete, theory. In fifty years, computers will not need and so-called infinities, be their actual or po- us, but until then, it is fun to act as “coaches”. tential, as well as the ‘continuum’, are ‘optical Mansour: What were your early experiences illusions’. with mathematics? Mansour: What do you think about the de- Zeilberger: See my interview with Ron Aha- velopment of the relations between combina- roni from 20151. torics and the rest of mathematics? Mansour: Were there specific problems that Zeilberger: Some of it is definitely good, but made you first interested in combinatorics? for sociological, psychological and political rea- Zeilberger: I started out in analysis, and sons, derived from the fact that (so far) math- tried to solve a problem, suggested by my ad- ematics is done (mostly) by humans, the ob- visor, Harry Dym, by doing a discrete analog. session with ‘relations to other parts of math- I never solved the original problem, but I fell ematics’ has gone too far, and to a large part in love with the discrete. To be honest I never was driven from the need for respectability, and liked the continuous, especially the way it was feeling part of the ruling-‘mainstream’. treated in the usual real analysis courses based Mansour: What have been some of the main on the ‘theological’ (and meaningless) notions The authors: Released under the CC BY-ND license (International 4.0), Published: November 3, 2020 Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected] 1http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/netgar.html 2http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html Interview with Doron Zeilberger of ‘real’ numbers (what an oxymoron!)2. Mansour: What would guide you in your re- Mansour: We would like to ask you about search? A general theoretical question or a your formative years. What was the first rea- specific problem? son you become interested in mathematics and Zeilberger: Whenever I see a problem that specifically combinatorics? Did that happen ‘can be taught to a computer’ I love it. under the influence of your family, or some Mansour: When you are working on a prob- other people? lem, do you feel that something is true even Zeilberger: Again see 3. before you have the proof? Mansour: What was the reason you chose the Zeilberger: As I said many times, the notion Weizmann Institute for your Ph.D. and your of proof in mathematics is misguided and it advisor, Harry Dym? should be called fully formal proofs. There are Zeilberger: I read a beautiful article that many statements that are obviously true, for Harry wrote in Advances in Mathematics, on example, Goldbach conjecture, the irrational- de-Branges spaces. Even though it was offi- ity of Euler’s constant, the Riemann Hypoth- cially analysis, it had beautiful algebraic struc- esis, and P 6= NP , that are obviously true, ture. At the end I winded up not working on but we human beings do not yet have a formal it, but that was what lead me to Harry Dym. proof. Hopefully computers will help us soon. Mansour: What was the problem you worked Mansour: You criticize being too obsessed on in your thesis? with rigorous mathematical proofs. Why? Zeilberger: As I said above, I tried to tackle Zeilberger: As I said above, and in more de- a continuous problem by first doing a discrete tail in the following essay 5, the notion of ‘rig- analog hoping to ‘take the limit’ at the end, orous proof’ has nothing to do with wanting but then abandoned the original problem and to know the truth about the mathematical uni- worked on discrete analytic functions. See my verse, it is just an (often fun, I admit) intellec- Ph.D. thesis4. tual game, and also a competitive sport, and Mansour: How was the mathematics at the also a (fanatical) religious dogma. Weizmann Institute at the time? Mansour: What three results do you consider Zeilberger: Very stimulating. My teachers the most influential in combinatorics during Harry Dym and Yakar Kannai were really in- the last thirty years? spiring. Zeilberger: These are too numerous to list, Mansour: Would you say a little bit about but I like those results that in hindsight turned your most influential results and why they have out to have one-page proofs, that previously been influential? only had partial results with long and boring Zeilberger: To use the cliche the whole is proofs. larger than the sum of its parts, I hope that - The Adam Marcus-Gabor Tardos’s proof6 my most significant contribution is in the real- of the Stanley-Wilf conjecture7. ization, that the future of mathematics is in the - The Ellenberg-Gijswijt’s proof (based on a direction of close collaboration with our sillicon previous proof of Croot-Lev-Pach of an analo- friends, until they won’t need us anymore. gous result) about three-term arithmetical pro- Mansour: Would you tell us about your friend gressions8. and collaborator Herbert Wilf and your joint - The Hao Huang’s ingenious proof of the works, specifically on the development of the sensitivity conjecture9. Wilf-Zeilberger theory? These are true gems, and they also show the Zeilberger: He was a great mensch and a true inherent triviality of human-generated mathe- giant. He was also ahead of his time, using matics. All these conjectures that great human computers and devising algorithms to handle minds could not do for many years, turned out combinatorial objects. to have one-page proofs. 3http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/netgar.html 4See http://sites.math.rutgers.edu/~zeilberg/DZthesis.pdf 5https://sites.math.rutgers.edu/ zeil berg/mamarim/mamarimhtml/hersh90.html 6 https://www.sciencedirect.com/science/article/pii/S0097316504000512?via%3Dihub 7 https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/martar.html 8 https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/f3n.html 9https://annals.math.princeton.edu/2019/190-3/p06

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Mansour: What are the top three open ques- one of the foremost experts in combinatorics, tions in your list? we would like to ask some more specific math- Zeilberger: Let me answer a different ques- ematical questions. You have met several tion. The three steps in my research agenda combinatorial objects throughout your career. are: Which one is your favourite? 1. Make the mainstream obsolete. Zeilberger: Permutations, and their cousins, 2. Make the offbeat mainstream. alternating sign matrices. 3. Go to 1. Mansour: How do you define “Experimental Mansour: What kind of mathematics would Mathematics”? you like to see in the next ten-to-twenty years Zeilberger: Experimental Mathematics used as the continuation of your work? to be an oxymoron, but in the future it would Zeilberger: Computer-generated, or at least be the opposite, a redundancy. All mathematics computer-assisted proof of (at least the part that is worth doing) would (i) the Riemann Hypothesis10 be ipso facto, experimental. (ii) P 6= NP Mansour: Would you tell us about your (iii) the Collatz conjecture11 thought process for the proof of the alternating Mansour: What would you say about some of sign matrix conjecture? How did you become the major directions in combinatorics for the interested in this conjecture? How long did it next two decades? take you to figure out a proof? Did you have Zeilberger: I am not qualified to answer a “eureka moment”? regarding normal human-generated combina- Zeilberger: I heard it from Dave Robbins torics. My personal agenda is to teach your (1942-2003), one of my great heroes, at a com- computer to do your math for you. binatorics meeting at Oberwolfach, in 1982. It Unfortunately, if you want an academic job went through several iterations, where the ref- in a traditional math department, you need eree, that turned out to be Dave Robbins him- to do the same-old, currently mainstream re- self, kept founding gaps. Luckily with the help search. of almost 90 checkers, and a detailed Maple Mansour: You have “Doron Zeilberger’s Col- package, all the gaps were filled. A few months lection of Quotes” on your web page. What is later, Greg Kuperberg observed that it is an al- your favourite quote? most immediate corollary of a result in math- Zeilberger: It is hard, but how about “My ematical physics due to Izergin and Korepin, occupation is an open question. I was once and the proof turned out to be ‘almost’ a triv- an assistant professor of mathematics. Since iality, but this did not detract from the original then, I have spent time living in the woods of much longer proof that used the methodology Montana.” –Theodore J. Kaczynski, New York of constant terms, and incidentally proved a Times Jan. 23, 1998, p. A18 much more general result (that Gog=Magog Mansour: Would you tell us about your in- also for trapezoids, not only for triangles), for terests besides mathematics? which the ad-hoc Kuperberg-Izergin-Korepin Zeilberger: Solving two-move chess prob- approach is not (as far as I know) applicable. lems. Since I was good at math I was ex- Mansour: The study of permutation patterns pected to be a good chess player. The truth has seen great advances in the last thirty years. is that I was (and still am) terrible. Because Any comments on the research in this direc- of this inferiority complex, I started doing two- tion? movers thirty years ago. Even in this endeavor Zeilberger: Yes indeed, it is a booming field, I am still bad, but much better than I was and you, Toufik, are a major player! thirty years ago. I soon will try to study three- Mansour: Why is it very difficult to count movers. It also makes me appreciate my stu- some combinatorial objects? Do you think dents who are not naturally good at math, but that someone will be able to compute the through hard work manage to do well. Stanley-Wilf limit corresponding to the pat- Mansour: Before we close this interview with tern 1324 in the next decade? 10See http://www.claymath.org/millennium-problems/riemann-hypothesis 11See https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/

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Zeilberger: Because life is hard. The few lem with magnetic field, and the 3D version, combinatorial objects that we can count ex- thereby getting us to shake hands (once the actly are the trivial ones (in the eyes of God). pandemic is over) with the King of Sweden. So even counting alternating sign matrices is Mansour: Is there a specific problem you trivial, compared to counting 1324-avoiding have been working on for many years? What permutations. I doubt that the exact num- progress have you made? ber of 1324-avoiding permutations with 10000 Zeilberger: The Riemann Hypothesis, the terms, the number of 10000 × 10000 Latin Collatz conjecture, P 6= NP . No luck, so far. squares, or the number of self-avoiding walks Mansour: Are you working on an interesting of length 10000 will ever be known. Also the problem with your famous “co-author” Shalosh Stanley-Wilf limit would never be known to an B. Ekhad these days? accuracy of more than 20 decimal digits. Zeilberger: Yes, we try to count (with Mansour: You recently have an interesting Manuel Kauers and his computer) Standard paper titled as “A simple rederivation of On- Young Tableaux with forbidden runs, that, sager’s solution of the 2D Ising model using ex- contrary to the opinions of the editors of perimental mathematics.” Do you expect some the journal Algebraic Combinatorics, are both other interesting results in this direction? deep and original. Zeilberger: Yes. I hope (when I have time) Mansour: Professor Doron Zeilberger, I that my coauthor, Manuel Kauers and I can would like to thank you for this very interesting generalize the method (with the help of our interview on behalf of the journal Enumerative beloved computers) to the still open prob- Combinatorics and Applications.

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