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A Enumerative Combinatorics and Applicationsecajournal.haifa.ac.il

ECA 1:2 (2021) Interview #S3I10

Interview with Van VuToufik Mansour

Photo by Nam Long

Van Vu finished high school in Vietnam, where he was born.He received his Bachelor degree at Eotvos Lorand University inBudapest in 1994. He obtained his Ph.D. at Yale in 1998 underthe guidance of Laszlo Lovasz. He is currently the Percey F.Smith chair of mathematics at Yale University. Before his posi-tion at Yale, he spent some years at the Institute of AdvancedStudies at Princeton, Microsoft Research, University of Califor-nia San Diego, and Rutgers University. Professor Vu has giventalks at many conferences, including an invited talk at the In-ternational Congress of Mathematicians in 2014. Professor Vuhas received several awards, including the Sloan Fellowship in2002, Polya Prize in 2008 and Fulkerson Prize in 2012. In 2012,he became an inaugural fellow of the American Mathemati-cal Society, and in 2020, a fellow of Institute of MathematicalStatistics. He is on the editorial board of Combinatorica andJournal of Combinatorial Theory, Series A.

Mansour: Professor Vu, first of all, we wouldlike to thank you for accepting this interview.Would you tell us broadly what combinatoricsis?

Vu: To me, combinatorics is more of a stylerather than a separate area of mathematics.Roughly speaking, we think discretely. In thebeginning, this applies, naturally, only to dis-crete objects, such as graphs. Later, generalprinciples have been formed (a good exampleis Szemeredi’s regularity lemma1) which applyacross many fields.

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

Vu: In recent years, combinatorial thoughtshave been used in many branches of math-ematics, sometimes with astonishing efficacy.In many studies in different areas, it hasturned out that at the very core of the sub-ject there is a deep combinatorial problem,and ideas and tools from combinatorics be-came very essential. For instance, the notion ofpseudo-randomness (originated from works ofThomasson2 and Graham-Chung-Wilson3 ongraphs) is crucial in the Green4-Tao5 proof ofthe existence of long arithmetic progressions inprimes. Szemeredi’s lemma is used quite fre-quently in analysis and probability also. Manyof my own works on eigenvalues of random ma-trices and roots of random functions rely on

The authors: Released under the CC BY-ND license (International 4.0), Published: February 26, 2021Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is tmansour@univ.haifa.ac.il

1E. Szemeredi, Regular partitions of graphs, in: Proc. Colloque Inter. CNRS (J.-C. Bermond, J. -C. Fournier, M. Las Vergnasand D. Sotteau eds.), 1978, 399–401.

2A. Thomason, Pseudo-random graphs, Annals of Discrete Math. 33 (1987), 307–331.3F. R. K. Chung, R. L. Graham and R. M. Wilson, Quasi-Random Graphs, Comb. 9:4 (1989), 345–362.4B. Green, Long arithmetic progressions of primes, in W. Duke, Y. Tschinkel (eds.), Analytic number theory, Clay Mathematics

Proceeding 7, Providence, RI: American Math. Soc. (2007), 149–167.5T. Tao, Arithmetic progressions and the primes, Collectanea Mathematica, Vol. Extra (2006), 37–88.6G.A. Freiman, Inverse problems of additive number theory, VI. On the addition of finite sets, III, Izv. Vyss. Ucebn. Zaved.

Matematika 3 (1962), 151–157.

Interview with Van Vu

modern anti-concentration theory, which hasits roots in subset sums and Freiman’s inversetheorem6 in combinatorial number theory.

In the beginning, combinatorialists bor-rowed lots of tools from other areas to proveour theorems (probabilistic method, topologi-cal method etc). Now, with some pride, I cansay that we have started to return the favor.

Mansour: What have been some of the maingoals of your research?

Vu: Like everybody, I like to solve big/famousproblems. However, to me, it is more im-portant to develop new tools and methods ornotions along the way, which other researcherscan use for different purposes. This is whatreally makes mathematics move forward, notthe fact that certain conjectures are true.

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?

Vu: Like most mathematicians, I went to aspecial school for gifted children from an earlyage (around 10 or so). It was my elementaryschool teacher who recognized that I could begood at math and told my parents, who thenencouraged me to change schools.

Mansour: Were there specific problems thatmade you first interested in combinatorics?

Vu: After high school, I got a fellowship togo to study in Budapest, but in an engineer-ing school. There, one of my math teachers,Kati Vesztergombi (Laci Lovasz’s wife) ran amath circle and showed me an Erdos problem(I think it was the distinct distances problem).I really liked it (but could only achieve someprogress 10 years later). I also took part inthe famous Schweitzer competition and aftera year or so Kati and Laci convinced and sup-ported me to switch to studying mathematics.

Mansour: What was the reason you choseYale University for your Ph.D. and your advi-sor Laszlo Lovasz?

At the time, Eastern Europe had juststepped out of the iron curtain, and I didnot know much about research and Americanschools. I chose Lovasz as my advisor not only

because it was natural, given our history, butalso because his style of doing mathematics im-pressed me so much.

Combinatorics was new at Yale then (1994),and most of the time my fellow students didnot know what I was doing. When I applied, Ialso got an offer from Massachusetts Instituteof Technology (MIT). I remember that thechair of the department called me and askedif I wanted to come to MIT. He sounded veryconfused when I told him that I chose Yale tostudy combinatorics, so he asked who wouldbe my advisor. When I told him it was Lovasz,he seemed relieved. “Oh, you will be in goodhands,” he said.

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

Vu: There were two problems. The first is todetermine how big the inverse of a (0, 1) matrixof size n can be. This originated from a paper7

of Dmitry Kozlov and mine on a seemingly in-nocent coin weighing problem. We solved thisproblem with Noga Alon8. The second wasSerge’s arc problem in finite geometry. Whatis the minimal size of a maximal arc on a finiteprojective plane? (A maximal arc is a set ofpoints with no three on a line and maximalwith respect to that property.) I worked onthis with Jeong Han Kim9. I still like the re-sults we obtained back then.

Mansour: What would guides you in yourresearch? A general theoretical question or aspecific problem?

Vu: Mostly, I like specific problems which are,or seem to be, at the heart of a bigger theory.Of course, when one solves a good problem,one gets some recognition and that helps one’scareer. But the real value of the solution is thenew ideas and tools that were introduced andthat other people can use on other, sometimestotally different, problems.

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

Vu: Most of our conjectures are true. I donot doubt the Riemann hypothesis or the twinprime conjecture. The real challenge is to findout why. I guess at the end what really mattersis the new phenomenon or viewpoint that we

7D. N. Kozlov and V. H. Vu, Coins and Cones, J. Comb. Theory, Ser. A 78:1 (1997), 1–14.8N. Alon, D. N. Kozlov and V. H. Vu, The geometry of coin-weighing problems, FOCS (1996), 524–532.9J. H. Kim and V. H. Vu, Small complete arcs in projective planes, Comb. 23:2 (2003) 311–363.

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discover, which contains the problem (or thetechnical heart of it) as a special case. I men-tioned the work of Green4 and Tao5 on primesearlier. The stunning thing that they foundout is that this is a phenomenon about arith-metic progressions rather than about primes.Basically, they give a necessary condition forsets of integers to contain arbitrarily longarithmetic progressions. That was the realinnovation, done with the help of the notion ofpseudo-randomness from combinatorics. Thepart of verifying these conditions for primescan be done by standard tools from analyticnumber theory.

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

Vu: I should mention the Freiman inversetheorem6. The other is Szemeredi’s regularitylemma1. Both were proved more than 30 yearsago, but their impact is most profoundly felt inthe last 30 years and in so many different areasof math. The third one would be the develop-ment of the nibble method and its variants andrefinements (such as the differential method orabsorbing method). It started with a paperby Ajtai, Komlos, and Szemeredi10 (I think),but most leading researchers in probabilisticcombinatorics worked on it and added manyimportant new ideas to make it more powerful.

Mansour: What are the top three open ques-tions in your list?

Vu: Well, all of us would love to see why Rie-mann’s hypothesis is true, I guess. Could therebe a different approach to the Four color theo-rem (and many other problems of this nature)?An approach that convinces us that the theo-rem is true is based on some fundamental fact.Finally, as it brings back fond memories (frommy Budapest days), a solution to the distinctdistances in all dimensions. (The dimension 2case was solved recently by Guth and Katz11.)

Mansour: What kind of mathematics wouldyou like to see in the next ten-to-twenty yearsas the continuation of your work?

Vu: As I mentioned earlier, combinatorics hasstarted to produce methods and tools whichare of interest to researchers from many otherareas. I would really like to see this trend con-

tinue. Not so long ago, most people viewedcombinatorics as a collection of clever, but ad-hoc, problems, and ideas. It should no longerbe the case. We must build more theories,methods, and tools, which can have influenceacross the boundaries of mathematics.

Compared to other areas of mathematics,combinatorics has a big advantage in that itis quite close to real-life applications. Whena practitioner explains his/her problem to amathematician, my bet is that a combinato-rialist would have a better chance of under-standing or even proposing a solution than aresearcher from a more abstract area. I wouldlike to see more penetrating applications ofcombinatorics in areas outside mathematics.The application of the de Brujin graph ongenome ensembles is a wonderful example.

Mansour: Do you think that there are core ormainstream areas in mathematics? Are sometopics more important than others?

Vu: It is a matter of taste, I guess, and everybranch has its own merit. From the appli-cations’ point of view, however, I think thatprobability and statistics have become moreand more important. With respect to generalscience and industry, they probably play therole of analysis in Newton’s time.

Mansour: What do you think about the dis-tinction between pure and applied mathemat-ics that some people focus on? Is it mean-ingful at all in your case? How do you seethe relationship between so-called “pure” and“applied” mathematics?

Vu: No, I am not even sure how one definesthe distinction. Personally, I do appreciate thekind of mathematics which can be explainedand makes sense (at least the motivation andthe result) to graduate students, regardless ofthe area or the label of “pure” or “applied”.When mathematics makes sense, it is intrigu-ing and beautiful. Unfortunately, quite a fewbranches have become more and more abstract,and we now have far too many talks wheremost of the audience get lost after the first3 minutes. As DGS of Yale’s mathematicsdepartment, I need to address this as it is aserious concern for our students.

Mansour: What advice would you give toyoung people thinking about pursuing a re-

10M. Ajtai, J. Komlos and E. Szemeredi, A note on Ramsey numbers, J. Combin. Theory Ser. A 29:3 (1980), 354–360.11L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, Ann. Math. 181 (2015), 155–190.

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search career in mathematics?

Vu: Well, most mathematicians love whatthey do, because they can do what they love.This is our most valuable reward. If one is in itfor the glory, then there is a high chance thatone will be disappointed.

Some time ago, Terry Tao wrote an inter-esting essay “Does one has to be a genius todo Math.” Coming from him, the title soundsa bit funny. (The answer is No, but it wouldbe way more convincing if it came from, say,me.) But I totally agree with his points andhighly recommend young researchers to take alook.

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

Vu: I like to travel and I am a fan of sportsand movies. About ten years ago, I started towrite a blog (in Vietnamese). I have found itto be quite relaxing and entertaining.

Mansour: Before we close this interview withone of the foremost experts in combinatorics,we would like to ask some more specific math-ematical questions. What does universalitymean in the context of random matrix theory?What are the major results in this regard?Are there still challenging open questions inthis direction?

Vu: Actually, the story around this notion isinteresting. There are two different interpre-tations. In the beginning, random matrix the-ory was studied mostly by physicists, who wereparticularly interested in the interaction be-tween nearby eigenvalues. This interaction canbe expressed through a parameter called thecorrelation function, which they could com-pute precisely for matrices with Gaussian en-tries, thanks to special properties of the Gaus-sian distribution. The original universalityconjecture says that the correlation functiondoes not depend on the distribution of the en-tries. For instance, if we replace the Gaussianwith a different distribution, we should obtain(asymptotically) the same correlation function.

This was the main conjecture from Mehta’sbook “Random Matrices”12, which was consid-ered, for several decades, the key reference inthe area.

Terry Tao and I started to work on spectralproperties of random matrices about 15 yearsago. We actually did not know much aboutthe mathematical physics literature. Our view-point was purely linear algebraic. In general,we thought that all limiting distributions con-cerning spectral parameters of random ma-trices (eigenvalues, eigenvectors, determinantetc) are universal, namely, they do not dependon the distribution of the entries. This moregeneral “universality” belief has turned out tobe true in most cases considered so far, includ-ing the special case of the correlation function.

The last 20 years saw an enormous amountof work on random matrices, with severalbreakthroughs leading to solutions of long-standing, famous, problems. In my opinion,most major questions about spectral limit dis-tributions, such as the Circular Law conjec-ture13 (the non–hermitian analog of Wignersemi-circle law14,15) or Mehta’s conjecture12

(mentioned above), has been settled (for themost studied models of random matrices, atleast). There are still very important ques-tions left open, such as the Localization con-jecture16. But for this, one needs to under-stand an entirely different model of randommatrices, which, I would say, appeals more toa physicist than to a combinatorialist.

Mansour: The study of random polynomi-als, initiated by Marc Kac, has also motivatedmany research programs and currently rich lit-erature accumulated on the topic. Would youtell us briefly about the milestone results inthe historical context of this topic? What arethe major open problems?

Vu: Actually, Kac did not initiate the studyof random polynomials. The notion of randompolynomials was considered earlier by War-ing and Sylvester17. The first official result

12M. L. Mehta, Random Matrices, 3rd edition, Academic Press, New York, 2004.13T. Tao and V. Vu, Random Matrices: Universality of ESDs and the circular law, Ann. Prob. 38:5 (2010), 2023–2065.14E. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62 (1955), 548–564.15E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958), 325–328.16G. Stolz, An introduction to the mathematics of Anderson localization, arXiv:1104.2317v1 [math-ph], 2011.17I. Todhunter, A history of the mathematical theory of probability, Stechert, New York, 1931.18A. Bloch and G. Polya, On the roots of certain algebraic equations, Proc. London Math. Soc. 33 (1932), 102–114.19M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49, (1943), 314–320.

Erratum: Bull. Amer. Math. Soc. 49, (1943), 938.20M. Kac, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. 50 (1949), 390–408.

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was proved by Bloch and Polya18 about 10years before Kac19,20 entered the game. It wastrue, however, that the results of Littlewood-Offord21,22,23 and Kac (all in the 1940s) drewthe attention of the math community to thesubject and made it a focus of both analysisand probability.

Briefly speaking, a random polynomial is ofthe form Pn(x) =

∑ni=1 ciξix

i, where ci are de-terministic parameters which may depend onboth i and n, and ξ are iid random variables.The most natural problem is to study the num-ber and distribution of the real roots. This wasthe subject in all the papers mentioned above,restricted to the special case when ci = 1 forall i. In this case, the polynomial is referredto as Kac polynomials. There are other ensem-bles, such as the Weyl polynomial (ci = 1/

√i!),

which have also been studied intensively. Dif-ferent ensembles usually lead to totally differ-ent behaviors (for instance, the number of realroots is of different orders of magnitude).

Similar to the situation with random ma-trices, precise formulae can be obtained forthe Gaussian case (when all ξi are iid Gaus-sian). Again, one would make a universalityconjecture. This has been settled in variouscases but very basic questions remain open.For instance, is it true that the number of realroots follows a central limit theorem? The an-swer is Yes for Kac polynomials, as a result ofMaslova24,25 from the 1970s, almost 50 yearsago. But we still do not know the answer forthe Weyl ensemble and many other ensembles.

Mansour: In one of your influential pa-pers, coauthored with Terence Tao, InverseLittlewood-Offord theorems and the conditionnumber of random discrete matrices26, pub-

lished at Annals of Mathematics, you de-veloped the Inverse Littlewood-Offord theory.What is this theory about? Why were theseresults important?

Vu: In the series of papers on random polyno-mials from the 1940s (mentioned above), Lit-tlewood and Offord21,22,23 developed their fa-mous anti-concentration result. In the simplestform, it says that if ai are non-zero integers,and ξi are iid ±1 random variables, then thesum S =

∑i aiξi does not have too much mass

on any point. Technically speaking,

ρ := maxaP (S = a) = O(log n/

√n).

This result generated a whole area of researchin combinatorics, with many influential re-sults from leading researchers such as Erdos(who reproved and sharpened the above resultwith lovely use of Sperner lemma), Kleitman27,Sarkozy-Szemeredi28, Stanley29, Halasz30 etc.The general direction is to make more assump-tions on the ai, and prove sharper bounds. Forinstance, a famous result of Sarkozy-Szemerediand Stanley showed that the bound can be im-proved to O(1/n3/2) if we assume that ai aredifferent.

Terry and I brought a different viewpoint tothe problem26. We assumed that ρ is relativelylarge (say ρ ≥ n−100), and tried to character-ize all the sets of ai which make ρ this large.This idea is motivated by the Freiman inversetheorem6 on sumsets, and that was the reasonwe named it Inverse Littlewood-Offord theory.It turned out that such a characterization ispossible. We showed that the ai must have avery strong additive structure in order to makeρ large. Later, many researchers found differ-ent characterizations. One particularly useful

21J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. I, J. London Math. Soc. 13(1938), 288–295.

22J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. II, Proc. Cambridge Philos.Soc. 35 (1939), 133–148.

23J. E. Littlewood and A. C, Offord, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik]N.S. 54 (1943), 277–286.

24N. B. Maslova, The variance of the number of real roots of random polynomials, Teor. Verojatnost. i Primenen. 19 (1974),36–51.

25N. B. Maslova, The distribution of the number of real roots of random polynomials, Theor. Probability Appl. 19 (1974),461–473.

26T. Tao and V. H. Vu, Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. Math.169:2 (2009), 595–632.

27D. Kleitman, On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors, Advances in Math.5 (1970), 155–157.

28A. Sarkozy and E. Szemeredi, Uber ein problem von Erdos und Moser, Acta Arithmetica 11 (1965), 205–208.29R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1:2 (1980),

168–184.30G. Halasz, Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar.

8:3-4 (1977), 197–211.

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characterization is due to Rudelson and Ver-shynin31 who introduced the notion of asymp-totic least common divisor.

The new results have found a number of ap-plications. First, it implies most of the classical“forward” results, such as the result of Sarkozyand Szemeredi mentioned above. More impor-tantly, it plays a decisive role in the study ofrandom matrices and random functions. Letme single out an application in random matrixtheory. Here, a parameter that appears fre-quently is the distance from a random vectorX to a hyperplane H. If the coordinates of Xare xi and the coordinates of the normal vectorofH are ai, then this distance is exactly the ab-solute value of S. Usually, we know somethingabout H that ensures that the ai do not haveany additive structure. Thus, by the Inversetheory, we can conclude that with high prob-ability, the distance in question is not zero, orX does not belong to H. In fact, the Inversetheory also allows us to replace P (S = a) bythe probability that S belongs to a short inter-val centered at a. This way, we can bound thedistance from below. This fact plays a criticalrole in the study of the least singular value andthe circular law.

This is one of the papers I really enjoyed,as while the applications seem to belong en-tirely to probability, the key new element reallycomes from combinatorics.

Mansour: Would you briefly explain the no-tion of anti-concentration? In what type ofproblems is this new phenomenon observed?Would you point out some future research di-rections?

Vu: Let ξ1, . . . , ξn be iid real random variablesand F = F (ξ1, . . . , ξn) a real function. A typ-ical anti-concentration result asserts that theprobability that F belongs to a short inter-val I is small. The Littlewood-Offord theo-rem above is one example when F is a linearfunction. The Inverse theory provides a fairlysatisfactory understanding in this linear case.However, for other functions (such as higherdegree polynomials), we are far from having acomplete picture.

Mansour: In your paper Finite and infinitearithmetic progressions in sumsets32, coau-

thored by Endre Szemeredi, published at An-nals of Mathematics, you proved an old conjec-ture of Folkman, by showing that if A is a setof natural numbers of asymptotic density atleast cn1/2 for sufficiently large constant c thenthe collection of all subset sums of A containsan infinite arithmetic progression. What werethe main new ideas behind the solution of thislong-standing conjecture?

Vu: First we made really long arithmetic pro-gressions (APs) and then merged them to-gether. The initial approach to this, in spirit,is somewhat similar to the Green4-Tao5 ap-proach to long APs in primes. We tried tocharacterize all large sets whose collection ofsubset sums do not contain a (sufficiently) longAP and found out that the main reason is thatthe set itself is the sum of two original APs. Inthis case, the collection of subset sums has thesame structure. Thus, the problem becameactually a two-dimensional problem. You canthink of the collection of subset sums as arectangle that does not contain any AP longerother than its two sides. However, if c is suf-ficiently large, the two-dimensional object hasto “wrap around” itself and create a long AP.One can achieve this via a random tiling argu-ment, but the details are sort of complicated.It took us several months to write it down af-ter having a fairly convincing sketch. This wasthe paper where I learned a good deal aboutadditive combinatorics from Endre.

Mansour: In your work, you have extensivelyused combinatorial reasoning to address im-portant problems. How do enumerative tech-niques engage in your research?

Vu: Yes, very frequently. However, for us itis more important to have a “soft” estimateon a parameter (like determining its order ofmagnitude) than to compute it exactly or toachieve a closed formula (which could be veryhard to evaluate). As a matter of fact, one ofthe main purposes of the Inverse theory is togive us a characterization of some “bad” sets sothat we can estimate their size (or probability).

Mansour: When we read your papers, wesee that combinatorial and probabilistic argu-ments play an important role in your research.Would you comment on the interplay between

31M. Rudelson and R. Vershynin, The Littlewood-Offord Problem and invertibility of random matrices, Advances in Math. 218(2008), 600–633.

32E. Szemeredi and V. H. Vu, Finite and infinite arithmetic progressions in sumsets, Ann. Math. 163 (2006), 1–35.

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combinatorics and probability?

Vu: On one hand, the probabilistic method isone of the most efficient and powerful methodsin combinatorics. On the other hand, combi-natorial ideas can be used to build new proba-bilistic tools or lead to new ideas in probabilitytheory.

At a deeper level, I feel that people in cer-tain areas of analysis, combinatorics, and prob-ability are frequently interested in essentiallythe same things. It is very hard to put nameson these things, but usually, they are gen-eral phenomena (such as pseudo-randomnessor large deviation or anti-concentration or hy-percontractivity or expansion). With someexperience, one can get a sense of the rele-vance even when the results appear in ratherunfamiliar forms.

Mansour: In your works, you have severalreferences to Paul Erdos. Have you ever methim? Which of his results fascinate you most?There is a published book about him titledas The Man Who Loved Only Numbers. Doyou think saying The Man Who Loved Graphswould be a more precise description of him?

Vu: Yes. I met Erdos twice. Both times werememorable. The first time, I was a student inBudapest. Kati (Vesztergombi) introduced meto Erdos while he was visiting the Renyi in-stitute (maybe around 1992). I did not knowabout the meeting in advance and did not pre-pare any math problems, so I blew my chanceto have an Erdos number 1 (my number is still

2). We did talk a bit about the Vietnam warinstead.

The second time I was a graduate studentat Yale. He gave a title lecture there. Thenwe talked a bit about my thesis. He refusedto believe our result with Noga Alon8 on coinweighing. The problem was this: given n coinsof two possible weights. How many weight-ings does one need to be sure that they are allthe same (or not)? (As usual, at one weight-ing, one can weigh any k coins against anotherk). The answer was log n/ log log n; but he in-sisted it should be log n, so strongly that atsome point I started to panic and thought thatsomething was really wrong.

Erdos wrote so many influential papers inso many different areas of mathematics, notonly number theory and graph theory. If onelooks at logic or probability or analysis, onefinds fundamental results bearing his name.What about the Man who loves mathematics?

Mansour: Is there a specific problem youhave been working on for many years? Whatprogress have you made?

Vu: I am working on some basic questionsconcerning random polynomials and randommatrices. So basic that it is embarrassing thatwe cannot settle them. But maybe one day...

Mansour: Professor Van Vu, I would like tothank you for this very interesting interviewon behalf of the journal Enumerative Combi-natorics and Applications.

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