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Interview with Gil Kalai Toufik Mansour numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 2:2 (2022) Interview #S3I7 ecajournal.haifa.ac.il Interview with Gil Kalai Toufik Mansour Gil Kalai received his Ph.D. at the Einstein Institute of the Hebrew University in 1983, under the supervi- sion of Micha A. Perles. After a postdoctoral position at the Massachusetts Institute of Technology (MIT), he joined the Hebrew University in 1985, (Professor Emeritus since 2018) and he holds the Henry and Manya Noskwith Chair. Since 2018, Kalai is a Pro- fessor of Computer Science at the Efi Arazi School of Computer Science in IDC, Herzliya. Since 2004, he has Photo by Muli Safra also been an Adjunct Professor at the Departments of Mathematics and Computer Science, Yale University. He has held visiting positions at MIT, Cornell, the Institute of Advanced Studies in Prince- ton, the Royal Institute of Technology in Stockholm, and in the research centers of IBM and Microsoft. Professor Gil Kalai has made significant contributions to the fields of discrete math- ematics and theoretical computer science. Two breakthrough contributions include his work in understanding randomized simplex algorithms and disproving Borsuk's famous conjecture of 1933. In recognition of his contributions, professor Kalai has received several awards in- cluding the P´olya Prize in 1992, the Erd}osPrize of the Israel Mathematical Society in 1993, the Fulkerson Prize in 1994, and the Rothschild Prize in mathematics in 2012. He has given lectures and talks at many conferences, including a plenary talk at the International Congress of Mathematicians in 2018. From 1995 to 2001, he was the Editor-in-Chief of the Israel Journal of Mathematics. He is a member of the Israel Academy of Sciences and the Humanities, the European Academy, and an honorary member of the Hungarian Academy of Science. Mansour: Professor Kalai, first of all, we choice. would like to thank you for accepting this in- Mansour: What do you think about the de- terview. Would you tell us broadly what com- velopment of the relations between combina- binatorics is? torics and the rest of mathematics? Kalai: Hmm, you have started with a dif- Kalai: I personally think that many flag- ficult question. Combinatorics is a very rich ships' contributions in combinatorics are the and inclusive subject that deals mainly, albeit ones developed within the field itself: Tutte's not solely, with discrete objects. Can we say, theory for enumeration of planar maps1, perhaps, that combinatorics to mathematics Sz´emeredi'stheorem,2 the graph minors the- is a bit like mathematics to science? Anyway, orem,3;4 etc. So I do not think that connec- whatever it is, it was my choice for my pro- tions with other areas are needed to justify fessional life, and I am very happy about this combinatorics. But still, there are beauti- The authors: Released under the CC BY-ND license (International 4.0), Published: July 27, 2021 Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected] 1W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74(1) (1968), 64{74. 2E. Szemer´edi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299{345. 3N. Robertson and P.D. Seymour, Graph minors. XX. Wagner's conjecture, J. Combin. Theory Ser. B 92 (2004),325{357. 4L. Lov´asz, Graph minor theory, Bull. Amer. Math. Soc. 43 (2005), 75{86. Interview with Gil Kalai ful connections with the rest of mathemat- in math pretty early on, and I was also inter- ics; sometimes a result of a method in com- ested at a young age in other areas of science binatorics resonates with other areas, and at (largely because of excellent popular books I other times, methods from other areas are the read.) My mother gave me her calculus book only known ways to prove combinatorial re- from high school (she did not like mathemat- sults. These connections are beautiful as well. ics that much, but realized that I liked it), and Richard Stanley5, who was my postdoctoral I remember that trying to read it, I could un- advisor, found amazing applications in com- derstand various things (like functions) but I binatorics of commutative algebra, algebraic got stuck on the expression f(x+∆)−f(x). I geometry, and other areas. knew that x is a variable that represents num- As the field of combinatorics widens, it is bers but I did not understand how numbers also important to find relations between dif- could be added to triangles. ferent areas of combinatorics itself. I remem- Another source of information was a pop- ber that at MIT, I was quite happy to notice ular science journal called \The Young Tech- some unexpected connections between a prob- nician"9 that my uncle had. I remember that lem of Stanley, motivated in algebraic com- I read there about a young boy that had binatorics, and a method of Kleitman6 from completed two patents three months after his extremal combinatorics. eighth birthday. Since I was seven years old Mansour: What have been some of the main at the time, I was reassured that I had plenty goals of your research? of time, and need not worry about it. Then Kalai: I started with combinatorial aspects I forgot all about it until it was too late: at of convex geometry, Helly-type theorems, and the age of nine I stumbled upon it again, and the theory of convex polytopes, and up till today, I have still not written any patent. now, I keep thinking also about related prob- I had a similar experience with the analy- lems. In these fields, the g-conjecture7 for sis of Boolean functions. I thought that it spheres was my personal holy grail, but I would be nice to extend the theory to the could not solve it. I also wanted (and in- non-commutative representation theory, but I deed still want) to understand flag numbers of also thought that I should not start with such polytopes and related cellular objects. Later a non-commutative project before the age of on, I was influenced by Nati Linial to work 40. But then at the critical time, I totally with him and Jeff Kahn8 on a certain prob- forgot about it. lem on Boolean functions and, since then, Mansour: Were there specific problems that the analysis of Boolean functions has become made you first interested in combinatorics? a central part of my research together with Kalai: I became interested in problems I had applications to probability and computer sci- encountered in class. So in my first university ence. year, I took a seminar on tree enumeration us- Mansour: We would like to ask you about ing J. W. Moon monograph10 and I became your formative years. What were your early intrigued with enumerating trees and related experiences with mathematics? Did that hap- combinatorial identities. In my second year, I pen under the influence of your family or some took a convexity class and became interested other people? in Helly-type theorems11. (Talking about J. Kalai: My father showed me the formula for W. Moon, I was trying to find his picture on- (a + b)2 when I was very young and it im- line for my blog, but Google only gave me pressed me. I think I knew that I was good pictures of the moon. Any help would be wel- 5R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics, Volume 41, Birkhauser, Boston, 1996. 6M. Saks, Kleitman and combinatorics, Discrete Math. 257 (2002), 225{247. 7K. Adiprasito, Combinatorial Lefschetz theorems beyond positivity, arXiv:1812.10454. 8J. Kahn, G. Kalai, and N. Linial, The influence of variables on Boolean functions, 68{80, in Proc. 29th Annual Symposium on Foundations of Computer Science, 1988. 9https://he.wikipedia.org/wiki/%D7%94%D7%98%D7%9B%D7%A0%D7%90%D7%99_%D7%94%D7%A6%D7%A2%D7%99%D7%A8. 10J. W. Moon, Counting labelled trees, Canadian Mathematical Congress, Montreal, 1970. Available at https://www.math. ucla.edu/~pak/hidden/papers/Moon-counting_labelled_trees.pdf. 11L. Danzer, B. Grtinbaum, and V. Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence, R. I., 1963, 100{181. ECA 2:2 (2022) Interview #S3I7 2 Interview with Gil Kalai come.) we met another expert in representations of Mansour: What was the reason you chose Sn that could help with the problem. the Hebrew University for your Ph.D. and Mansour: What was the problem you your advisor, Micha Perles? worked on in your thesis? Kalai: I was at the Hebrew University al- Kalai: I worked on the following conjecture ready as B.Sc. student and as an M.Sc. stu- of Katchalski and Perles12;13: If you have n dent and actually Micha Perles gave the two convex sets in Rd and no d + k + 1 of them courses I mentioned in the previous ques- has a point in common then the number of tion. Micha Perles was extremely generous (d + 1)-tuples that has a point in common and brilliant, and he was also very modest n n−k is at most d+1 − d+1 : After I solved the and I liked it; I even tried to adopt some of problem, I went on to solve a more general Micha's modesty in spite of my earlier reverse conjecture by Eckhoff14 on face numbers of disposition. nerves of families of convex sets in Rd. Mansour: How was the mathematics in Mansour: What would guide you in your Jerusalem at that time? research, a general theoretical question or a Kalai: I liked the atmosphere in Jerusalem specific problem? then and I still do now.
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