numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 2:2 (2022) Interview #S3I6 ecajournal.haifa.ac.il Interview with Karim Adiprasito Toufik Mansour Karim Adiprasito completed his undergraduate studies at TU Dortmund in 2010. He obtained a Ph.D. from Free University Berlin in 2013, under the supervision of G¨unter M. Ziegler. He has been an Assistant Professor at the Einstein Institute for Mathematics Hebrew University of Jerusalem since 2015, promoted to full professor in 2018. Professor Adiprasito has given numerous invited talks in conferences and seminars. He received several prizes, for example, European Prize in Combi- natorics, in 2015, for his work in discrete geometry; New Hori- zons Prize for Early-Career Achievement in Mathematics, as- sociated with the Breakthrough Prize in Mathematics, in 2019; Prize of the European Mathematical Society in 2020, and is the 2021 Hadamard Lecturer. Mansour: Professor Adiprasito, first of all, we torics and the rest of mathematics? would like to thank you for accepting this in- Adiprasito: I am not a fan of isolated combi- terview. Would you tell us broadly what com- natorics as a field or any field for that matter. binatorics is? It is to be primarily a place to meet and discuss Adiprasito: In my perspective, combinatorics mathematical problems from various areas. I is a common ground in mathematics. It is a do not feel I am particularly talented in math- place where everything comes together, which ematics overall, so translating a problem to a has limitless applications to other fields, as well combinatorial one is always helpful to me to as being the playground for other fields. What grasp it for myself, and then hopefully to solve is most fascinating to me though is its ubiq- it. uity in everything in daily life, as our lives are Mansour: What have been some of the main naturally discrete. As such, when we speak of goals of your research? a continuous process, we often mean a really Adiprasito: Good question (in the sense that large discrete one, and so the discrete and the I needed to think about it for a bit). I was singular is really the germane issue to look at always more guided by cool ideas rather than when we want to find out the limits of our rea- goals, and I am rather impulsive at that. So soning. if I find some interesting idea, have an inter- Combinatorics is also just how we like to esting thought, then I would just try it out. think, how we like to order things in our minds. But I am not someone that sits and broods It was always fascinating to me that the an- over a specific problem for a long time unless cient Greeks chose polyhedra to model nature, he sees a crack in the wall. I have a collection pointy and unwieldy shapes, instead of the cir- of problems in the back of my head, but I con- cle. sider this collection more as a playground to Mansour: What do you think about the de- test ideas on. velopment of the relations between combina- Mansour: We would like to ask you about The authors: Released under the CC BY-ND license (International 4.0), Published: June 18, 2021 Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected] Interview with Karim Adiprasito your formative years. What were your early Also, it has remained as my favorite book. experiences with mathematics? Did that hap- Mansour: What was the reason you chose the pen under the influence of your family or some Free University Berlin for your Ph.D. and your other people? advisor G¨unter M. Ziegler? Adiprasito: I think the earliest encounter Adiprasito: I was honestly supposed to go to with mathematics is from riddles my grandfa- the United States for graduate school (I had ther asked me. I must say, I never liked riddles. several offers, but ultimately hoped to work It always feels annoying to be put on the spot, with Gromov at Courant). But, for personal and be asked to come up with an idea right reasons, I decided to stay in Germany, so I con- there and then. It always feels more like a test tacted Guenter M. Ziegler while I was vacation- than an invitation to think. ing in Berlin. We talked about some work we So I think the first time I actually liked did, and to be honest, I think I selected him mathematics was when I discovered it myself. out of personal compatibility just as much as One instance from school I remember vividly I was interested in polytopes. He let me do was not actually in a mathematics class (which research mostly on my own terms, but we had I found to be mostly average and boring, es- some nice projects together as well. pecially towards the final years. The teachers Mansour: What was the problem you worked were just not competent) but from social sci- on in your thesis? ence class (or rather, the first course in eco- Adiprasito: Several, actually. First, there nomics) where I learned, though in a hidden was a postdoc present there in my first year, way, about the connection between economic Bruno Benedetti, with whom I met out of prin- (or rather, game-theoretic) equilibria and fixed ciple in some Italian restaurant in Berlin in- point theorems. stead of the office. This was quite convenient, Mansour: Were there specific problems that as I was honestly not a good office student, and made you first interested in combinatorics? never became one. Anyway, we worked on vari- Adiprasito: I think the first encounter with ous questions in geometric topology, mostly re- combinatorics that made me go \huh" was lated to the question of how difficult geometric reading about spaces of bounded curvature, in triangulations can be. One cute fact that came relation to hyperbolic groups and Alexandrov's out of that was that at some point I realized 4 5 curvature notions1. That does not seem com- how to prove the Hirsch conjecture for flag binatorial at first, but the remarkable thing complexes using Gromov's work. was that Gromov2 really turned a geometric And then I studied a problem concerning ar- property into a combinatorial one, what he rangement theory that had been looked at by called the no-triangle condition (graph theo- Salvetti before, describing complements of ar- rists would call it the clique complex). That rangements. Essentially, I wanted to show that stuck with me for a long time, because it was in they are minimal in a homotopic sense. I real- geometry textbooks and contexts that I really ized that there was a simple way to do it via encountered the first (specifically, Gromov's combinatorial trick and in particular proved6 textbook3 on Metric Structures on Riemannian the minimality of some arrangement comple- and Non-Riemannian spaces). I noticed for ments. myself that Gromov was really being a com- And then finally, the \magnum opus" was binatorialist at heart when he wrote this, and something that came out of a discussion with I found the connection immensely impressive. G¨unter Ziegler, where we studied a very old 1 A. D. Alexandrov, V. N. Berestovskii and I. G. Nikolaev, Generalized Riemannian spaces, Russian Math. Surveys 41 (1986), 1{54. 2M. Gromov, Hyperbolic groups, In S. Gersten, editor, Essays in Group Theory, Volume 8 of Mathematical Sciences Research Institute Publications, 75{263, Springer, New York, 1987. 3M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkh¨auser,Boston, 1999. 4K. A. Adiprasito and B. Benedetti, The Hirsch conjecture holds for normal flag complexes, Mathematics of Operations Re- search 39:4 (2014), 1340{1348. 5G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, N.J., 1963. 6K. A. Adiprasito, Combinatorial stratifications and minimality of 2-arrangements, J. Topology 7:4 (2014), 1200{1220. 7E. Steinitz, Polyeder und Raumeinteilungen, Encyklop¨adieder mathematischen Wissenschaften, Dritter Band: Geometrie, III.1.2., Heft 9, Kapitel III A B 12 (W. Fr. Meyer and H. Mohrmann, eds.), B. G. Teubner, Leipzig, 1922, 1{139. xi, 53. ECA 2:2 (2022) Interview #S3I6 2 Interview with Karim Adiprasito conjecture of Legendre and Steinitz7 on the sures of complexity for hypergraphs. It is one dimension of deformation space of polytopes. aspect of the story of high-dimensional expan- The construction is a \Berlin" one, in the sense sion that I find very fascinating, and it master- that we realized that certain discretizations fully uses rather difficult geometric arguments of partial differential equations (used for in- to achieve it. stance in architecture) could be used to solve 3. I very much love Sabitov's proof11 of Bel- the problem for us. low's conjecture, that is, that the volume of a Mansour: What would guide you in your re- flexible polyhedron is constant. It is a very search?, a general theoretical question or a spe- beautiful result connecting geometry, algebra, cific problem? and combinatorics, and several questions re- Adiprasito: I mostly said this already: nei- main open. It is elegant and uses several inter- ther. It is mostly a neat idea that I obsess esting ideas masterfully. over, and it can be motivated by either gen- Mansour: What are the top three open ques- eral or specific problems. Having understood tions in your list? something about the inner workings though is Adiprasito: I am very much interested at what is leads my research. the moment in the Hopf conjecture, that de- Mansour: When you are working on a prob- scribes the Euler characteristic of aspherical lem, do you feel that something is true even manifolds.