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ECA 1:2 (2021) Interview #S3I7

Interview with Larry GuthToufik Mansour

Larry Guth received a B.S. in mathematics from Yale University in2000, and a Ph.D. in mathematics from the Massachusetts Institute ofTechnology (MIT) in 2005 under the supervision of Tomasz Mrowka.After getting his postdoctoral position at Stanford, he moved to theUniversity of Toronto as an assistant professor. In 2011, Guth wasappointed as professor of mathematics at the Courant Institute. Since2012, he is a professor of mathematics at MIT, and in 2020 he wasmade a Claude Shannon professor. Guth’s research interests are in

metric geometry, harmonic analysis and extremal combinatorics. In 2010, Guth won an AlfredP. Sloan Fellowship. In 2013, the American Mathematical Society (AMS) awarded him theSalem Prize in mathematics, for outstanding contributions to analysis. In 2014, he receiveda Simons Investigator Award. In 2015, Guth received the Research Prize of the Clay Mathe-matics Institute and was awarded the New Horizons in Mathematics Prize “for ingenious andsurprising solutions to long-standing open problems in symplectic geometry, Riemannian ge-ometry, harmonic analysis, and combinatorial geometry.” In 2020, Guth received the BocherMemorial Prize of the AMS and the Maryam Mirzakhani Prize in mathematics. He is a Fellowof the AMS.

Mansour: Professor Guth, first of all, wewould like to thank you for accepting this in-terview. Would you tell us broadly what com-binatorics is?

Guth: Combinatorics is about studying finiteobjects, like finite (abstract) sets, or a finite setof numbers or a finite set of circles in the plane.Questions from number theory, like questionsabout prime numbers or questions about in-teger solutions of equations, are finite in thissense, but partly for historical reasons, theyare considered a separate field. In the 20thcentury, people realized that there are manyother interesting questions about finite objectsoutside of number theory. These questions andthe community of people working on them arecombinatorics.

People in combinatorics have made a partic-ular effort to seek out questions that are easyto state. There are many really difficult openproblems in combinatorics which high school

students can understand. In my opinion, thisis an important part of the culture of the field.

Mansour: What do you think about the de-velopment of the relations between combina-torics and the rest of mathematics?

Guth: Personally, I did not start out study-ing combinatorics, and the only way I becameinvolved was because of some relations be-tween combinatorics and other parts of math.For instance, the Szemeredi-Trotter problemis a combinatorial problem about the inter-section patterns of finite sets of lines in theplane, and the solution to the problem turnsout to involve topology. This connection be-tween combinatorics and topology is one ofmy favorite parts of combinatorics. The prob-lem Szemeredi and Trotter solved is analogousto some problems in harmonic analysis. TomWolff discovered that analogy, and he learnedideas from the combinatorics community andapplied them to study solutions of the wave

The authors: Released under the CC BY-ND license (International 4.0), Published: January 15, 2021Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected]

Interview with Larry Guth

equation. Along the way, he introduced a cir-cle of problems and ideas from combinatorialgeometry to people working in harmonic anal-ysis. I learned about these problems and ideasfrom Nets Katz. That had a big impact onmy work. There are many other examples ofconnections between combinatorics and otherparts of math.

Connections between one field and anotherare always interesting and important, but Ithink this may be especially true for combi-natorics. On the one hand, combinatorics isdistinguished by searching for questions thatare very simple to pose. When those questionsget connected to other branches of math andto more abstract structures, then I think it isa good sign for both the question and the ab-stract structure. It shows that the questionleads us somewhere and it shows that the ab-stract structure connects to things.

Mansour: We would like to ask you aboutyour formative years. What were your earlyexperiences with mathematics? Did that hap-pen under the influence of your family, or someother people?

Guth: I was incredibly fortunate with thementors I had growing up. My father, AlanGuth, is a physicist and very interested inmath, and we would talk a lot. He is a wonder-ful teacher. He taught me many things, and hewas also happy to talk about questions he didnot know the answers to. We would guess whatthe answer might be, and then we would try toprove it. If that did not work, we would try todisprove it. If that did not work, we wouldtry to prove it again. Now I teach studentsmyself, and it is very common that someonetries to prove something, and it does not work,and they get frustrated. Of course, it happensto me too. But sometimes, when we are in theright mood, we can say to ourselves, “That didnot work. That is interesting. I wonder whyit did not work...” I learned this mood when Iwas a kid talking math with my father.

Now I am a father. My children are fouryears old and two years old. When I watchthem building a block tower or working on ajigsaw puzzle, I often think about doing mathresearch and about the different moods that I

go through when I am doing it. Sometimes,they are in a mood where they really want thetower to be tall. In that mood, they get frus-trated when it falls down. But other timesthey build in a more curious mood, and whenthe tower falls down, they laugh and they startagain. In the more curious mood, their handsare steadier and the towers are taller.

I also learned a lot from my father aboutwriting and teaching. I remember, when wewere talking, some times he would write downwhat we were talking about, and sometimes Iwould write down what we were talking about.When he wrote down what we were talkingabout, you saw the first thought and then thesecond thought and then the third thought.The pages were numbered and you could readit over later and see just what we thought.When I wrote down what we were talkingabout, the second thought was in the bottomright corner and the third thought was diago-nally across the top left and the fourth thoughtwas sideways along the right margin... It tookme something like ten years to learn how towrite thoughts down one at a time.Mansour: Your father, Alan Guth, is aneminent theoretical physicist and cosmologist.Have you ever hesitated when choosing be-tween mathematics and physics to pursue yourresearch career?Guth: I did not actually. We talked aboutphysics too, but I remember our math conver-sations more vividly. Our conversations mostlyfollowed what I was interested in talking about,and my interests went in a more mathematicaldirection.Mansour: Were there specific problems thatmade you first interested in combinatorics?Guth: When I was a postdoc, I learned aboutthe Szemerdi-Trotter theorem1,2 from MattKahle, and I found it very interesting. I wasinterested in the Kakeya problem from har-monic analysis. Around that same time, ZeevDvir3 proved the finite field Kakeya conjec-ture. The argument used ideas from error-correcting codes, and it was connected to com-binatorics. I was very interested in that and Ispent a long time trying to adapt those ideasto other problems. I met Nets Katz around

1E. Szemeredi and W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3(3–4) (1983), 381–3922E. Szemeredi and W.T. Trotter, A combinatorial distinction between the Euclidean and projective planes, European J. Combin.

4(4) (1983), 385–394.3Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), 1093–1097.

ECA 1:2 (2021) Interview #S3I7 2


that time, and he introduced me to a bunchof problems in combinatorial geometry thathave some analogy or connection to the Kakeyaproblem. These problems included the jointsproblem, the unit distance problem, the dis-tinct distance problem, and analogues of theSzemeredi-Trotter theorem using other kindsof curves in place of lines. Those are the prob-lems that got me into combinatorics, and I stillthink they are very interesting.

Mansour: What was the reason you choseMIT for your Ph.D. and your advisor, TomaszMrowka?

Guth: I liked the atmosphere among the stu-dents at MIT, and I also liked being in Bostonand close to my family. Coming out of under-grad, I was most interested in Fourier analysis,which I had learned about from Peter Jonesand Ronald Coifman, but I really was not surewhat I wanted to work on. MIT was not ac-tually the best place for Fourier analysis, butthere was clearly a lot of good math and it feltlike a good place for me.

When I was a first-year Ph.D. student, Iwas not really able to understand what any ofthe professors were working on. Tom had agroup of six or seven students. I knew some ofthem and I liked the community of this group.We had a weekly seminar meeting where wetook turns reading papers and presenting themto each other, and that was a great experi-ence for me. I had never read a math paperas an undergrad, or as a first year grad stu-dent. Learning to read a paper and digest itand present it was challenging and exciting.My first presentation had about one mistakeevery ten minutes, and Tom pointed them outwith gentle questions. I had not learned howmuch work it took me to digest a piece of mathand understand it. After that, I wrote abouteight pages of notes for each presentation, andI wrote more than one draft of the trickiestparts.

Mansour: What was the problem you workedon in your thesis?

Guth: In Gromov and Lawson’s work onscalar curvature4, they bring up mappingsbetween n-dimensional Riemannian manifoldsthat decrease all the 2-dimensional areas. I gotvery interested in these mappings and I wrote

my thesis about them.

Mansour: What would guide you in your re-search? A general theoretical question or aspecific problem?

Guth: In that thesis research, I was guidedby analogies. There is a lot known about map-pings that decrease all 1-dimensional lengths.I would work through those ideas and try toadapt them to mappings that decrease all the2-dimensional areas. For example, there is afundamental theorem in the area called theLipschitz extension theorem, which says thatif S ⊂ Rn is any set, and f : S → R is adistance-decreasing map, then f extends to alength-decreasing map from Rn to R. I wantedto know if there was some analogue of this the-orem involving 2-dimensional areas instead of1-dimensional lengths.

I worked on that for a while, but did notmake any progress, and got frustrated. Even-tually, I decided to give up and just work outsome simple examples of area-decreasing maps.It seemed natural to me that among all themaps from one n-dimensional box to anothern-dimensional box, the linear map would havethe best area-decreasing properties. I did notthink it would be that hard or that interesting,but I figured I would work on that and at leastI would prove something. So I gave up on my“real” problem, and I tried to prove that littleconjecture about rectangular boxes. That lit-tle conjecture turned out to be false! My thesisproblem was figuring out which map has thebest area-decreasing properties. (I was able todo this in some dimensions – the general prob-lem is still open.)

The moment I figured out that my littleconjecture was false was an important mo-ment for me becoming a math researcher. Theanalogies I had started with were not all thathelpful. Once I discovered that my intuitionabout this little conjecture was wrong, therewere a lot of followup questions and I had alot of concrete things to do that helped me un-derstand the area better.

Sometimes in research, I am guided bysomething that bothers me. I once went toa panel discussion about mentoring gradu-ate students, and Ingrid Daubechies said thatshe teachers her students to “be ornery read-

4M. Gromov and H.B. Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111:2(1980), 209–230.



ers”. In other words, she encourages them toread papers and pay attention to what both-ers them. Recently, I have been working alot on decoupling theory in Fourier analysis,following the work of Bourgain and Demeter.Their proof of the decoupling conjecture forthe paraboloid5 came as a complete shock tome, and I have been studying it and work-ing in the area ever since. A key idea in theproofs is to combine information from differentscales. There are a bunch of different tricks fordoing this. The proof of decoupling involvesfour distinct tricks for combining informationfrom different scales. And in applications, thedecoupling theorem itself is used at differentscales. Combining information from many dif-ferent scales seems natural and important tome. But having five different tricks for doingso does not feel natural – it bothers me. I feellike we have not yet found the clearest way ofunderstanding it. I have been thinking aboutthat for the last several years.

Mansour: When you are working on a prob-lem, do you feel that something is true evenbefore you have the proof?

Guth: Yes, I do. And sometimes I am wrong.

Mansour: What three results do you considerthe most influential in combinatorics duringthe last thirty years?

Guth: I do not feel qualified to answer thisquestion. I only know a very narrow partof combinatorics. But I will mention a fewthings that I find exciting and that I wouldlike to learn. I would like to learn about Gow-ers’s work6, 7 on arithmetic progressions usinghigher-order Fourier analysis. I would like tolearn about the Green-Tao theoremm8. TheGreen-Tao theorem is related to a lot of things,but one, in particular, I would like to learnabout is the analogue of Szemeredi’s regular-ity lemma9 in the context of sparse graphs. Iam not sure if this counts as combinatorics ex-actly, but I would like to learn more about thePCP theorem and hardness of approximation.

Mansour: What are the top three open ques-

tions in your list?

Guth: The problems I have spent the mosttime wondering about are the Kakeya prob-lem10,11 and its cousins. In combinatorics, oneof my favorite questions is the inverse problemfor the Szemeredi-Trotter theorem. I recentlylearned about an open problem in additivecombinatorics about the size of the sum set ofa convex set. If f(x) is a convex function, thenthe sequence of numbers f(1), f(2), . . . , f(n) isa convex set of size n. If A is a convex set ofsize n, how big does the sum set A+A need tobe? There was some striking fairly recent workby Schoen and Shkredov12, but we are far fromfully understanding.

Mansour: What do you think about the dis-tinction between pure and applied mathemat-ics that some people focus on? Is it meaningfulat all in your own case? How do you see therelationship between so-called “pure” and “ap-plied” mathematics?

Guth: So far I have only done pure math. Iwould really like to learn more applied math,but it is hard to find time for everything...

Mansour: Would you tell us about your in-terests besides mathematics?

Guth: When I was younger I was interested inacting and in theater. When I was in college,I spent a semester studying acting in Moscowand I did another workshop the summer aftermy first year of graduate school. My familyused to go to the theater when I was grow-ing up and there were many plays around ourhouse that I used to read. For me, it was also away of exploring different life experiences andtrying out different ways of connecting withpeople. Once in a while I still have friendsover and read a play out loud.

I started taking some tai chi classes ingraduate school. I got interested in it partlythrough theater and partly because I used tohave a little knee pain and back pain, and Ithink it helped with that. I really like the med-itative side of it, and I still like to do a littlebit when I get into the office in the morning.

5J. Bourgain and C. Demeter, The proof of the l2 Decoupling Conjecture, Ann. of Math. 182 (2015), 351–389.6T. Gowers, A new proof of Szemeredi’s theorem, Geom. Funct. Anal. 11(3) (2001), 465–588.7T. Gowers, Hypergraph regularity and the multidimensional Szemeredi theorem, Ann. of Math. 166(3) (2007), 897–946.8B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481–547.9B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers Us+1[N ]-norm, Ann. of Math. 176 (2012), 1231–1372.

10A.S. Besicovitch, On Kakeya’s problem and a similar one, Math. Z. 27:1 (1928), 312–320.11T. Tao, From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis and PDE,

Notices of the AMS. 48(3) (2001), 297–303.12T. Schoen and I.D. Shkredov, On sumsets of convex sets, Combin. Probab. Comput. 20:5 (2011), 793–798.



Mansour: Before we close this interview, wewould like to ask some more specific math-ematical questions. You have worked onproblems from different fields of mathematicsthroughout your career. In your survey pa-per “Unexpected applications of polynomialsin combinatorics,” you write “I am a big ad-mirer of hard problems that are simple to state.The most exciting - in my opinion - is a sim-ply stated problem that is hard for a new rea-son.” Would you elaborate more about this andgive some examples of such problems? Are youworking on a problem recently which is hardfor a new reason?Guth: There are many hard questions aboutprimes that are simple to state, such as thetwin primes conjecture. I think that is a verystriking fact. In hindsight, because prime num-bers are defined by crossing out numbers withfactors and looking at what is left, it is hard toget a handle on what happens when we addprime numbers etc. Once one knows aboutthis, it is not so hard to produce other simplequestions about primes that are also hard toanswer. Even though the definition of a primenumber is simple, testing whether a numberis prime is quite complicated, and so in hind-sight, perhaps it is not so surprising that it ishard to know how often n and n + 2 are bothprime.

Erdos posed the unit distance problem13,14

in the 1940s, and it remains wide open andlooks hard. The reason it is hard is not re-lated to the reason the twin primes conjectureis hard. As far as I know, there was not aprevious problem that was hard for a similarreason. In fact, I have some trouble articulat-ing why I believe the problem is hard. ActuallyI think a hard problem is the most interestingin this situation – when it is not so clear whyit is hard. Then our research has two oppositegoals: either to make progress on the problemor to explain why it is hard to make progress.

The unit distance problem asks for the max-imum possible number of unit distances amongn points in the plane. Since we have 2n de-grees of freedom, it is not hard to get about2n unit distances. Examples with more thanthat many unit distances are rare, and they allhave some special lattice structure. Those ex-

amples only have slightly more unit distances,still less than n1+ε asymptotically. Valtr15 dis-covered that the unit distance conjecture isfalse for other smooth convex norms. Theexamples still have lattice structure! So per-haps there is some general principle that ex-amples with many unit distances must havesome kind of lattice structure. That wouldbe very striking – and reminiscent of severalother problems in combinatorics and Fourieranalysis where the known examples have a lotof lattice structure, such as examples for theSzemeredi-Trotter theorem and examples forthe Bourgain-Demeter decoupling theorem. Ifa set has a lot of unit distances, what kind ofstructure does it need to have, and how do weknow it is there?Mansour: Between 2006-2013, several verydifficult combinatorial problems have beensolved in an unexpected way using high de-gree polynomials. For instance, the finite fieldNikodym and Kakeya problems, and Erdos dis-tinct distance problem. You have already writ-ten a wonderful survey paper and a book onthe topic called “Polynomial methods in Com-binatorics.” Would you tell us about the mainideas behind the polynomial method? Why isit so powerful?Guth: In these problems and related prob-lems, the extreme examples have polynomialstructure. For example, a set of points is con-tained inside of a surface defined by (fairly lowdegree) polynomials. The polynomial methodis a set of tools that helps figure out when ex-treme examples for different types of problemsneed to have this kind of polynomial structure.Mansour: You, with Nets Katz, introduceda variation of the polynomial method, oftencalled polynomial partitioning, in your solu-tion to the Erdos distinct distances problem inthe plane. Would you tell us about the prob-lem, polynomial partitioning method, and thehigher dimensional cases?Guth: For example, suppose we have a set ofpoints P in 3-dimensional space and a set oflines L in 3-dimensional space, and we want toestimate the number of incidences of P and L(the number of pairs p ∈ P and ` ∈ L withp ∈ `). The worst case occurs when all thepoints and lines lie in a plane. Nets and I

13P. Brass, W.O. Moser and J. Pach, Research problems in discrete geometry, Springer, 2005.14P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.15P. Valtr, Strictly convex norms allowing many unit distances and related touching questions, preprint, 2005.



proved that if not too many points or lines lie ina single plane, then there are significantly bet-ter estimates. It turns out that to analyze thisproblem, one should first check that the worstcase occurs when most of the points and lineslie in a low degree algebraic surface. Then oneanalyzes the different surfaces, and one findsthat only planes and degree 2 surfaces matter.Since only planes and degree 2 surfaces matter,why introduce all the higher degree surfaces inthe proof? I think the reason is that there is amore robust phenomenon – for a broad rangeof problems in this spirit, the worst case occurswhen most of the objects involved lie in an al-gebraic variety of fairly low degree. It so hap-pens that for points and lines in 3-space, onlyplanes and sometimes degree 2 surfaces mat-ter, but this last point is a little bit of a coinci-dence. The more general robust phenomenondoes involve higher degree varieties, and themore general and robust phenomena are ofteneasier to prove.

This estimate about lines in 3-space was themain step in realizing an approach to the dis-tinct distance problem that was invented bySharir and Elekes16.

There are many higher dimensional versionsof these problems. One can ask about inci-dences between lines in n-space and more gen-erally between low degree varieties in n-space.The higher dimensional distinct distance prob-lem leads to incidence problems of this kind.Most of these higher dimensional versions areopen, although Miguel Walsh recently made alot of progress on them.

Mansour: Is there a connection between NogaAlon’s combinatorial Nullstellensatz and thepolynomial method?

Guth: They have a somewhat similar flavorbecause they both involve doing linear alge-bra on the space of polynomials. There are anumber of other cool theorems and proofs incombinatorics that also involve linear algebraon the space of polynomials – one good sourcefor them is the book ‘Linear algebra methodsin combinatorics’ by Babai and Frankl. Therecent breakthrough in the cap set problem17

by Croot, Lev, and Pach18 is another example.Mansour: Would you list a few open ques-tions from combinatorics such that in theirsolutions you expect that polynomial methodwill play an important role?Guth: For several years, people used to askme if this or that problem might be related tothe polynomial method. I was not very goodat guessing the answer – and the answer wasusually no. Unless the problem was very sim-ilar to a problem that had been solved usingthe polynomial method, I had no special in-sight. I would tell people that a problem wasa good candidate if the worst known examplehad some polynomial structure.

In the last ten years, two of the most ex-citing applications of the polynomial methodwere the work on finite field sum-product esti-mates using Rudnev’s19 point-plane incidencebound and the work of the cap set problem byCroot, Lev, and Pach. In these cases, thereis not any interesting example with polyno-mial structure, so my criterion would not haveguessed that the polynomial method should beuseful!Mansour: Terence Tao has the following com-ment on the polynomial method: “What Iwould like to see more of in the future is moredevelopment of the somewhat vague idea of the“Zariski complexity” of various sets, by which Imean something like the least degree of a non-trivial polynomial which vanishes on that set.One can view the polynomial method as thestrategy of comparing upper and lower boundson the Zariski complexity of sets to obtain non-trivial combinatorial consequences. I have thevague feeling that ultimately, such notions ofcomplexity should play as prominent a role inthese sorts of combinatorial problems as exist-ing notions of “size” for such sets, such as car-dinality, dimension, or Fourier uniformity.”20

Would you elaborate on this comment to makeit more accessible to a researcher who is newto the field? What are your thoughts in thisdirection?Guth: Suppose F is a field. If X ⊂ Fn isa set, we can define Deg(X) as the smallest

16G. Elekes and M. Sharir, Incidences in three dimensions and distinct distances in the plane, Combin. Probab. Comput. 20:4(2011), 571–608

17J.A. Grochow, New applications of the polynomial method: The cap set conjecture and beyond Bull. AMS 56:1 (2019), 29–64.18E. Croot, V.F. Lev and P.P. Pach, Progression-free sets in Zn

4 are exponentially small, Ann. of Math. (2) 185:1 (2017),331–337.

19M. Rudnev, On the number of incidences between points and planes in three dimensions, Combinatorica 38 (2018), 219–254.20See, https://mathoverflow.net/a/43549



degree of a non-zero polynomial that vanisheson X. This definition plays a big role in thepolynomial method, especially in some of theearly work. For a random set X ⊂ Fn, Deg(X)is approximately |X|1/n. If Deg(X) is muchsmaller than this, then we can say informallythat X has some algebraic structure. Manyproofs in the polynomial method begin by sup-posing that X has some interesting combinato-rial geometric property, and then showing thatDeg(X) is small. Once this happens, we knowthat X has some algebraic structure, and wecan often learn more about X by applying toolsfrom algebraic geometry.

So Deg(X) is an important player in thepolynomial method. But it is not the finalword, especially when n is large. If X is con-tained in an (n − 1)-plane in n-space, thenDeg(X) = 1. But a set contained in a 9-planecan still be pretty complicated. If a set is con-tained in a 2-plane, should not it be simplerthan a set that is just contained in a 9-plane?They both have Deg(X) = 1. So we shouldrefine Deg(X) by taking into account not justalgebraic hypersurfaces but algebraic varietiesof all dimensions. This is done in a somewhatad hoc way in various papers. Walsh’s21 recentwork is a big step towards studying varieties ofall dimensions in a systematic way.Mansour: You are the first winner of theMaryam Mirzakhani Prize in mathematics formid-career mathematicians given by the Na-tional Academy of Sciences. The citationstates that the award is for “developing sur-prising, original, and deep connections betweengeometry, analysis, topology, and combina-torics, which have led to the solution of, ormajor advances on, many outstanding prob-lems in these fields.” What do you think aboutthe importance of being competent in differ-ent branches of mathematics? Do you have aconcrete example in your research career whereyou made a significant progress on a researchproblem by using ideas or techniques from acompletely different field?Guth: On the one hand, dividing math into

fields is somewhat artificial, and there aremany relations between “different” fields. Soknowing different parts of math is certainlyhelpful. On the other hand, learning thingswell takes time, and none of us has time foreverything.

The idea of polynomial partitioning that wementioned above was partly suggested by workof Gromov in metric geometry, a rather differ-ent part of math from combinatorics or fromharmonic analysis. If you take two open sets inthe plane, then there is always a line that bi-sects both sets. You can kind of convince your-self of that by imagining a line continuouslyrotating and shifting. At each angle, there is ashift that bisects the first set, and as we changethe angle, the line sweeps across the second setuntil it bisects it as well. This idea is a spe-cial case of the ham sandwich theorem. Fol-lowing Gromov, I was using variations on theham sandwich theorem to study problems inmetric geometry. Then the idea played a rolein my work on harmonic analysis and combi-natorics, especially in the work with Nets onpolynomial partitioning.Mansour: Your initial work was on the sys-tolic geometry. Is there a connection betweencombinatorics and systolic geometry?Guth: Well, I was very excited by the appli-cation of ham sandwich theorems in combina-torics. I learned the ham sandwich theorem inthe context of metric geometry (a broader areathat includes systolic geometry). After that, Ilooked for more connections between the fields.I personally did not find any other connection.

Gromov found another interesting con-nection between metric geometry and com-binatorics: he found an analogy between“point-selection” theorems in combinatorics byBarany and others and the waist theorem inRiemannian geometry proven by Almgren andGromov.Mansour: Professor Larry Guth, I would liketo thank you for this very interesting interviewon behalf of the journal Enumerative Combi-natorics and Applications.

21M.N. Walsh, The polynomial method over varieties, Invent. Math. 222 (2020), 469–512.


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