numerative ombinatorics

A pplications Enumerative 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 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 ; 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 . 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 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, 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 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. It is very much connected to com- before you have the proof? binatorics and group theory with algebraic ge- ometry, in very mysterious and poorly under- Adiprasito: Yes, but that feeling can be 12 treacherous. Intuition tends to misguide me stood ways. Specifically , it says that the on occasion, but in general, it is a rather good Euler characteristic χ(M) of a 2n-dimensional companion. I had cases where something that compact closed manifold satisfies intuitively should have been one way turned (−1)nχ(M) ≥ 0. out to be another, and these are usually the most interesting cases. I also think that several questions in graph Mansour: What three results do you consider theory need to be understood (better), such the most influential in combinatorics during as the strong perfect graph theorem13 or the the last thirty years? graph reconstruction conjecture14 (which asks Adiprasito: Probably, in no particular order if a graph can be reconstructed if we only (and I should say they are selected for their see all the versions with one removed.) implications and the work they spawned more Another reconstruction conjecture, whether a than the individual results): simple cellulation of a sphere can be recon- 1. Peter McMullen’s8 work on valuation and structed from its 1-skeleton, is equally fasci- algebras, which opened up a (convex- nating and shows how little we know about )geometric avenue to algebraic geometry. It is graph embeddings in higher . quite an ingenious perspective, which in partic- And finally, I am interested in the problem ular also leads to our solution9 of Rota’s con- of renormalization in general relativity, and jecture and the Elias-Williamson work on the models of quantum gravity. I only recently positivity of Kazhdan-Lusztig polynomials. got interested in mathematical physics, but the 2. Gromov’s10 geometric measure the- amount of mathematics that comes together ory approach to expansion and multi-covered in this topic is fascinating. It is also related points that ingeniously connected several mea- to combinatorial problems, such as the prob- 8P. McMullen, The polytope algebra, Adv. Math. 78 (1989), 76–130 9K. Adiprasito, J. Huh, and E. Katz, Hodge theory for combinatorial geometries, Ann. of Math. (2) 188:2 (2018), 381–452. 10M. Gromov, Singularities, expanders, and topology of maps. Part 2: From combinatorics to topology via algebraic isoperime- try, Geom. Funct. Anal. 20(2) (2010), 416–526. 11I. Kh. Sabitov, The volume as a metric invariant of polyhedra, Discrete Comput. Geom. 20 (1998), 405–425. 12See, for example, J. Cao and F. Xavier, K¨ahlerparabolicity and the Euler number of compact manifolds of non-positive sectional curvature, Math. of Ann. 319 (2001), 483–491. 13M. Chudnovsky, Maria, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. of Math. 164 (1) (2006), 51–229. 14F. Harary, On the reconstruction of a graph from a collection of subgraphs, In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague, 1964, 47–52. 15See, for example, https://arxiv.org/pdf/1904.01966.pdf.

ECA 2:2 (2022) Interview #S3I6 3 Interview with Karim Adiprasito lem of counting the number of triangulations terests besides mathematics? of manifolds (so-called Regge calculus15). Adiprasito: I like literature, both in the sense Mansour: What kind of mathematics would that I write prose myself and read constantly. you like to see in the next ten-to-twenty years Whenever I do not read math, I usually have a as a continuation of your work? novel in my hand. Mostly though, I love cook- Adiprasito: I think one thing that I would ing. It is the mix of craft and creativity that is love to see is a deeper understanding of differ- only found again in mathematics, as it is both ential geometry at the singular limit. In par- an art and bound to real laws (after all, you ticular, there are invariants of manifolds that do not want to poison everyone at the table.) we know only to compute if we have a trian- Mansour: This is a quote by John Fraleigh gulation or a smooth structure, but no uni- that says “Never underestimate a theorem that fying theory of both. So geometric and com- counts something.” How far you agree with it? binatorial questions still feel kind of disjoint. To what extent do you agree with this quote? In particular, many things are proven twice in Adiprasito: I never was a good quotes per- very similar forms, once for combinatorics and son, honestly. It mostly feels as if they should once in geometry, which feels a little strange have stayed in their own time, as part of their at times. conversation, their context. The closest thing Mansour: Do you think that there are core or I like is, perhaps, poetry. And I think I like mainstream areas in mathematics? Are some Rilke best most of the time. The Panther in topics more important than others? particular has stayed with me for a long time. Adiprasito: I do not think so. On the con- Mansour: Recently, Professor Abigail trary, I think it rather frustrating if researchers Thompson, a Vice President of the AMS, ex- stay firmly rooted within their area, as it of- pressed her view on a mandatory “Diversity ten prevents them from seeing the bigger pic- Statement” for job applicants in a short es- ture. Still, to stay in their special corner is say. Apparently, and unfortunately, there were something mathematicians of many areas do, several attempts from members of the math- unfortunately. ematical community to intimidate her ‘voice’. Mansour: What do you think about the dis- You were among the undersigned mathemati- tinction between pure and applied mathemat- cian urging the AMS ‘to stand by the prin- ics that some people focus on? Is it mean- ciple that important issues should be openly ingful at all in your case? How do you see the discussed respectfully...’. Was your petition relationship between so-called “pure” and “ap- helpful? When should the issue of diversity be plied” mathematics? taken into consideration? How negatively can Adiprasito: I think the distinction is more such a mandatory act impact future research? one of “we stop once we have the solution to Adiprasito: I think I was mostly taken aback the technical problem posed” and “we want to by the tone of the discussion. That said, I do understand the bigger picture.” I think the for- not believe forcing faculty to write a diversity mer is certainly less exciting to me, but it is statement is useful, and it is the lazier and the less a distinction of pure vs. applied than an cheaper way of universities to feign activity in unwillingness to go on beyond what you were this issue. A more helpful measure would be to asked. make life easier for young families by providing Mansour: What advice would you give to childcare, parental leave, or to improve condi- young people thinking about pursuing a re- tions of students that do not have adequate search career in mathematics? financial means. Adiprasito: I think the best advice is to not Mansour: You recently started a blog. You confine yourself to a single area, and not be describe it as “A blog on mathematical mus- afraid to understand ideas regardless of where ings, nonsense, and observations that did not they are coming from. It gives a better scope of make it into a paper.” In what sense do you what you are doing and helps you understand use the word ‘nonsense’? How much time do the relevance of your research. And of course, you plan to dedicate to it? it gives you more tools at your disposal. Adiprasito: Well, I do expect that I might Mansour: Would you tell us about your in- also share ideas that do not work out or ran-

ECA 2:2 (2022) Interview #S3I6 4 Interview with Karim Adiprasito dom intuitions that might be wrong. In that orate on the result by emphasizing a little bit sense, nonsense. Mostly though, nonsense per- more on the combinatorial side of it? tained to little observations I had, that I felt Adiprasito: Well, the Hirsch conjecture, in a would not be worth a paper but that should more general setting, asks whether the diam- still be recorded somewhere. I do not know eter of the hypergraph of facets of a triangu- how much time will I dedicate to it. lated manifold is bounded, say polynomially, Mansour: Recently you were awarded the in terms of the number of vertices. This is a prestigious New Horizons Prize in Mathe- rather intriguing conjecture, but so far, poorly matics for the development of combinatorial understood. What I observed is that for hyper- Hodge theory leading to the resolution of the graphs arising as clique complexes (this is also log-concavity conjecture of Heron-Rota-Welsh. known as the no-triangle condition), one could What is Hodge theory? Especially, in joint use a method developed by Gromov to prove work with and Eric Katz9, you re- the strongest form of this conjecture. Essen- solved the Heron–Rota–Welsh conjecture on tially, one constructs the shortest path on ver- the log-concavity of the characteristic polyno- tices geometrically, and from this, one is able mial of . Would you tell us about this to construct a short path in the hypergraph. result, the crucial ideas behind the proof, and Mansour: You were awarded the European the possible future research directions? Mathematical Society prize for your work16, Adiprasito: Hodge Theory is a way to study Peter McMullen’s g-conjecture for simplicial the cohomology of manifolds and varieties, but spheres. We would like to hear your comments in this context, it should be taken more as a about the proof. What do you think about the way to study the signature of certain bilinear role of prizes such as this one in mathematical forms associated with matroids. This is re- research? ally a special part of Hodge theory though. Adiprasito: This is a far-reaching generaliza- The combinatorics mostly enters when prov- tion of the work of McMullen, as well my joint ing that the bilinear forms indeed say some- work with June and Eric on the Rota con- thing about the coefficients of the characteris- jecture. Essentially, when in the case of ma- tic polynomial. troids, we had a geometric framework to work The idea is quite simple: log-concavity of with, specifically a notion of convexity, played sequences ai can be restated as saying that a by submodular functions, the g-conjecture had certain matrix, the matrix to come from a direction entirely without ge-   ometry. I managed to do that by connecting ai+1 ai a a the problem to an algebraic version of Hall’s i i−1 marriage theorem, which yielded the desired has non-positive determinant, or equivalently, after much additional work. In particular, I it cannot be definite. To prove that, one needs noticed that what was needed is the fact that to establish that the matrix arises as a bilin- in a Poincar´eduality algebra associated with ear form that has a geometric meaning, in our the problem, the face ring, the pairing needs case, the Hodge-Riemann relations. Proving to stay non-degenerate when restricted to ide- them is the major feat of our joint work, as we als. This property, called biased pairing, as had to reprove a classical algebraic geometry central as it turned out. Recently, with Pa- result in a much larger generality than previ- padakis and Petrotou17 we gave another proof ously known. The limits of the latter are the based on perturbation of volume polynomials most interesting to me and remain to be ex- which seems more miraculous and remains to plored. be understood, but that again uses the biased Mansour: Another great piece of your work, pairing property critically. carried out together with Bruno Benedetti4, is Mansour: In a very recent paper, you and Ra- the resolution of the Hirsch conjecture for flag man Sanyal18, have solved two long-standing triangulations of manifolds. Would you elab- problems on the combinatorial complexity of 16K. Adiprasito, Combinatorial Lefschetz theorems beyond positivity, arXiv:1812.10454. 17See https://arxiv.org/pdf/2101.07245.pdf. 18K. A. Adiprasito and R. Sanyal, Relative Stanley-Reisner theory and upper bound theorems for Minkowski sums, Publications math´ematiquesde l‘IHES,´ (2016), 1–65.

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Minkowski sums of polytopes. Would you tell torial problem would look if looked at continu- us about this work and point out some possible ously, and then try to see where the boundary future research directions? in the behavior is. Adiprasito: The problem is rather simple to Mansour: Would you tell us about your state: given polytopes, how complex can their thought process for the proof of one of your Minkowski sum be, and how complex does it favorite results? How did you become inter- have to be. For instance, we could fix the num- ested in that problem? How long did it take ber of vertices of each summand and could ask you to figure out a proof? Did you have a “eu- for the number of vertices in the Minkowski reka moment”? sum. That problem sounds rather innocent, Adiprasito: I usually do have several eureka but solving it requires a rather powerful geo- moments, few of which actually work out in the metric construction, the Cayley polytope, plus end. My proof of g-conjecture is a bad example a lot of commutative algebra and combina- probably, as several were needed to complete torics, to relate the complexity of summands to the proof. I think I needed a year to work out the complexity of the sum. But of course, one all the steps. On the other hand, in the case of can ask many other questions concerning the the Rota conjecture, once the idea was there, complexity of Minkowski sums. In general, it it took like a week to have everything together is a broader problem of interest to understand in my head. how combinatorial objects mix, and I feel that I think I most vividly remember filling in often algebraic methods are the right way to a specific step in the proof of the g-conjecture understand it. though when I realized that a lemma of Kro- Mansour: You have worked on outstanding necker I used works together beautifully with and solved some of them. What Poincar´e duality algebras (you can find the is your initial approach towards conjectures? more technical details in my recent newslet- Would you describe some of your experiences? ter of the EMS survey). I was walking home Adiprasito: I usually try to get a feeling for from a meeting with my friend and colleague conjectures by connecting to my knowledge of Eran Nevo, and decided to have some wonder- techniques and intuitions from different areas ful ice cream instead, and realized the connec- of mathematics. That usually gets me to a tion when enjoying some saffron ice. point where I feel I understand the conjecture Mansour: Is there a specific problem you well enough to at least be able to say some- have been working on for many years? What thing relevant. However, that is just where the progress have you made? process starts, and it takes a lot of work after Adiprasito: I worked for a long time on the that. g-conjecture, but currently I do not have a Mansour: In your work, you have extensively big problem that occupies me constantly. It is used combinatorial reasoning to address im- more the case that I am trying to understand portant problems. How do enumerative tech- some fundamental issues in graph theory a bit niques engage in your research? better. Adiprasito: Actually, I often do not use so Mansour: Professor Karim Adiprasito, I much combinatorial reasoning, but reasoning would like to thank you for this very interesting from various areas. Indeed, the contrary often interview on behalf of the journal Enumerative helps me, as I try to imagine how a combina- Combinatorics and Applications.

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