Featureplaced by several applications of Cauchy inequality and ofthe q-van der Corput method).One portion of the argument that was still hurting the exponentsconcerned the “type III sums”. We 5 had realised atan early stage that our previous work on the distribution ofthe ternary divisor function in large arithmetic progressionsmight help the cause; however, it was only during the Fouvry60 conference at CIRM in the middle of June that wemade this concrete and decided that it could be worth contactingPolymath (through Terry Tao). It is highly probablethat without the existence of the Polymath8 project – its opennessand the chance for anybody to bring their own contribution– we would never have dared to make such a technicalimprovement public and, most likely, we would have forgottenabout it after some time! This work was also an occasionto develop material of broader interest: one can find there ageneral account on -adic trace functions which will hopefullybe useful to the working analytic number theorist; forinstance, we provide an easy to use presentation of the q-vander Corput method for general trace functions. Another interestingoutcome is the following: the quest for improvementson the numerical value of the distribution exponent has led toquite sophisticated transformations of the sums appearing inthe dispersion method. The fact that the resulting “complete”algebraic exponential sums and their associated sheaves havea nice geometrical structure is a pleasant surprise that triggersa lot of interesting questions on the -adic side.I very much like working in a collaborative manner, 6sometimes with fairly large groups of people, and, yet, thisfirst participation in a Polymath project was at an entirely differentscale and it took me some time to adapt. One issue wasto cope with the continuous flow of comments and new ideasmade by the many participants (in particular, I often wonderedwhether I was contributing enough by comparison withothers); another was to absorb the fact that the project wasperformed under the public eye (and the vertiginous feelingthat any mistake could be known to any mathematician in theworld and would stay forever). As for others, these concernseventually disappeared and I began to fully enjoy the spontaneityof having everybody working openly through a publicblog; at some point even this became slightly addictive:while attending a thesis defence (not in my area fortunately!),I found myself checking the latest development of the projectand sending an email to some of the Polymath people to testan idea I had just had on how to handle some unpleasant exponentialsum. All in all, this has been a fun and rewardingexperience and I am very thankful to Terry Tao for setting upand conducting the project; I am also thankful to the otherparticipants for their constructive and highly collaborative attitude.Bibliography[1] J. Andersson, Bounded prime gaps in short intervals, preprint.[2] W. Banks, T. Freiberg, C. Turnage-Butterbaugh, Consecutiveprimes in tuples, preprint.5 E. Fouvry, E. Kowalski, P. Nelson and I.6 The last non-survey paper I wrote alone was back in 2004.[3] W. D. Banks, T. Freiberg, J. Maynard, On limit points of thesequence of normalized prime gaps, preprint.[4] W. D. Banks, T. Freiberg, C. L. Turnage-Butterbaugh, Consecutiveprimes in tuples, preprint.[5] J. Benatar, The existence of small prime gaps in subsets of theintegers, preprint.[6] E. Bombieri, J. Friedlander, H. Iwaniec, Primes in arithmeticprogressions to large moduli, Acta Math. 156 (1986), no. 3–4,203–251.[7] E. Bombieri, J. Friedlander, H. Iwaniec, Primes in arithmeticprogressions to large moduli. II, Math. Ann. 277 (1987), no.3, 361–393.[8] E. Bombieri, J. Friedlander, H. Iwaniec, Primes in arithmeticprogressions to large moduli. III, J.Amer.Math.Soc.2 (1989), no. 2, 215–224.[9] R. de la Breteche, Petits écarts entre nombres premiers et polymath: une nouvelle maniére de faire de la recherche?, Gazettedes Mathématiciens, Soc. Math. France, Avril <strong>2014</strong>, 19–31.[10] A. Castillo, C. Hall, R. J. Lemke Oliver, P. Pollack, L. Thompson,Bounded gaps between primes in number fields and functionfields, preprint.[11] L. Chua, S. Park, G. D. Smith, Bounded gaps between primesin special sequences, preprint.[<strong>12</strong>] P. D. T. A. Elliott, H. Halberstam, A conjecture in prime numbertheory, Symp. Math. 4 (1968), 59–72.[13] B. Farkas, J. Pintz, S. Révész, On the optimal weight functionin the Goldston-Pintz-Yıldırım method for finding small gapsbetween consecutive primes, To appear in: Paul Turán MemorialVolume: Number Theory, Analysis and Combinatorics, deGruyter, Berlin, 2013.[14] É. Fouvry, H. Iwaniec, On a theorem of Bombieri-Vinogradovtype, Mathematika 27 (1980), no. 2, 135–152 (1981).[15] É. Fouvry, H. Iwaniec, Primes in arithmetic progressions,Acta Arith. 42 (1983), no. 2, 197–218.[16] T. Freiberg, A note on the theorem of Maynard and Tao,preprint.[17] H. Furstenberg, Y. Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64–119.[18] D. Goldston, J. Pintz, C. Yıldırım, Primes in tuples. I, Ann. ofMath. 170 (2009), no. 2, 819–862.[19] W. T. Gowers, gowers.wordpress.com.[20] W. T. Gowers, M. Nielsen, Massively collaborative mathematics,Nature, 15 October 2009.[21] S. W. Graham, C. J. Ringrose, Lower bounds for leastquadratic nonresidues, Analytic number theory (AllertonPark, IL, 1989), 269–309, Progr. Math., 85, BirkhäuserBoston, Boston, MA, 1990.[22] A. Granville, Bounded gaps between primes, preprint.[23] G. H. Hardy, J. E. Littlewood, Some problems of “Partitio Numerorum”,III: On the expression of a number as a sum ofprimes, Acta Math. 44 (1923), 1–70.[24] D. Hensley, I. Richards, On the incompatibility of two conjecturesconcerning primes, Analytic number theory (Proc. Sympos.Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo.,1972), pp. <strong>12</strong>3–<strong>12</strong>7. Amer. Math. Soc., Providence, R.I., 1973.[25] H. Li, H. Pan, Bounded gaps between primes of the specialform, preprint.[26] R. Lipton, K. Regan, People, Problems, and Proofs, Essaysfrom Gödel’s Lost Letter: 2010, Theoretical Computer Science,2013 XVIII, Springer-Verlag.[27] J. Maynard, Bounded length intervals containing two primesand an almost-prime II, preprint.[28] J. Maynard, Small gaps between primes, preprint.[29] J. Maynard, Dense clusters of primes in subsets, preprint.[30] S. Morrison, “I just can’t resist: there are infinitely many pairsof primes at most 59, 470, 640 apart”, sbseminar.wordpress.com/2013/05/30.22 EMS Newsletter December <strong>2014</strong>
Feature[31] A. Pease, U. Martin, Seventy-four minutes of mathematics: Ananalysis of the third Mini-Polymath project, In Proceedingsof AISB/IACAP 20<strong>12</strong>, Symposium on Mathematical Practiceand Cognition II.[32] J. Pintz, Polignac Numbers, Conjectures of Erdős on Gapsbetween Primes, Arithmetic Progressions in Primes, and theBounded Gap Conjecture, preprint.[33] J. Pintz, A note on bounded gaps between primes, preprint.[34] J. Pintz, On the ratio of consecutive gaps between primes,preprint.[35] J. Pintz, On the distribution of gaps between consecutiveprimes, preprint.[36] P. Pollack, Bounded gaps between primes with a given primitiveroot, preprint.[37] P. Pollack, L. Thompson, Arithmetic functions at consecutiveshifted primes, preprint.[38] D. H. J. Polymath, polymathprojects.org[39] D. H. J. Polymath, A new proof of the density Hales-Jewetttheorem, Ann. of Math. (2) 175 (20<strong>12</strong>), no. 3, <strong>12</strong>83–1327.[40] D. H. J. Polymath, Density Hales-Jewett and Moser numbers,An irregular mind, 689–753, Bolyai Soc. Math. Stud., 21,János Bolyai Math. Soc., Budapest, 2010.[41] D. H. J. Polymath, New equidistribution estimates of Zhangtype, and bounded gaps between primes, submitted.[42] D. H. J. Polymath, Variants of the Selberg sieve, and boundedintervals containing many primes, to appear, Research in theMathematical Sciences.[43] B. Siudeja, On the hot spots conjecture for acute triangles,preprint. arXiv:1308.3005.[44] T. Tao, E. Croot, H. Helfgott, Deterministic methods to findprimes, Math. Comp. 81 (20<strong>12</strong>), no. 278, <strong>12</strong>33–<strong>12</strong>46.[45] J. Thorner, Bounded gaps between primes in Chebotarev sets,preprint.[46] T. S. Trudgian, A poor man’s improvement on Zhang’s result:there are infinitely many prime gaps less than 60 million,preprint.[47] Y. Zhang, Bounded gaps between primes, Annals of Mathematics179 (<strong>2014</strong>), 1<strong>12</strong>1–1174.D. H. J. Polymath is the pseudonym for research papers arisingfrom online collaborative “Polymath” mathematical researchprojects. Correspondence concerning this particularPolymath project can be directed to Terence Tao at tao@math.ucla.edu and information about Polymath projects canbe found at http://michaelnielsen.org/polymath1/index.php?title=Main_Page.Faculty Positions in Statisticsor Computational Applied Mathematicsat Ecole polytechnique fédérale de Lausanne (EPFL)The School of Basic Sciences at the EPFL invites applicationsfor up to two professorial positions. The search isopen to all sub-domains of statistics and computationalapplied mathematics, but with a particular interest inapplications to data sciences, life sciences, optimisation,control and inverse problems. Current EPFL-wideresearch initiatives include neuroscience and materialsscience, and applications from mathematicians andstatisticians working in these areas are also encouraged.We seek candidates with an outstanding researchrecord and a strong commitment to excellence in teachingat all levels. While appointments are foreseen at thetenure-track assistant professor level, in exceptionalcases an appointment at a more senior level may beconsidered.Substantial start-up resources and research infrastructurewill be available.Applications should include a letter of motivation,curriculum vitae, publication list, concise statement ofresearch and teaching interests, as well as the namesand addresses (including email) of at least five refereesand should be submitted via the website:https://academicjobsonline.org/ajo/jobs/4249The evaluation process will start on November 1st,<strong>2014</strong>, but later applications may also be considered.Further enquiries should be made to:Prof. Philippe MichelChairman of the Search Committeee-mail: appliedmath<strong>2014</strong>@epfl.chThe School of Basic Sciences actively aims to increasethe presence of women amongst its faculty, and femalecandidates are strongly encouraged to apply.EMS Newsletter December <strong>2014</strong> 23
- Page 7 and 8: EMS News- mini-symposia;- film and
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- Page 45 and 46: Interview with Fields MedallistMart
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