Alfréd Rényi Institute of Mathematics – Key publications

2019

1. Outstanding basic research outcome

Automorphic Forms Lendület Research Group

Automorphic forms are harmonic waves with a rich symmetry, and the various harmonies are encoded by L-functions (the most classical L-function is the Riemann zeta function, fundamental in prime number theory). Many famous and extremely difficult problems are concerned with these objects (e.g. the generalized Riemann hypothesis, the Ramanujan-Selberg conjecture, the Langlands program), and number theory is connected to several seemingly distant mathematical areas through these objects (e.g. representation theory of Lie groups, global analysis, mathematical physics). In recent years, numerous Fields medals were awarded for the research of automorphic forms: Vladimir Drinfeld (1990), Richard Borcherds (1998), Laurent Lafforgue (2002), Ngô Bao Châu (2010), Elon Lindenstrauss (2010), Peter Scholze (2018), Akshay Venkatesh (2018).

A Maass form

A Maass form

A central theme of modern analytic number theory is the estimation of automorphic forms in various families in various norms. Such estimates are needed in concrete applications, but they also play a leading role in the development of the theory. The most classical automorphic forms are functions of the algebraic groups GL(1) and GL(2) over the adele ring of various number fields – these have been studied for over a century and a half. It is a logical step to extend the existing results to the groups GL(n) of higher rank, but despite intensive research efforts, there are few general theorems. Hence it was all the more surprising when Blomer-Maga (2014) managed to give a nontrivial pointwise bound for every spherical Hecke-Maass form on a fixed compact subset of the group PGL(n) over the rational numbers. Around the same time, Brumley-Templier (2014) showed that the restriction to compact subsets is essential, because without it the automorphic form assumes much larger values. Within the framework of the Automorphic Forms Momentum Grant, Blomer-Harcos-Maga [1, 2] proved that the lower bound of Brumley-Templier (2014) is close to the truth, because an upper bound of similar shape is also valid. On the classical group PGL(2), Blomer-Harcos-Maga-Milićević [3] established a strong and general bound, which concerns every number field and non-spherical forms, too. This theorem generalizes and sharpens several earlier results, such as the bound of Blomer-Harcos-Milićević (2016) on arithmetic hyperbolic 3-manifolds. The new results were accepted for publication by first-rate journals – after a lengthy review process as customary in mathematics.

  • [1] Blomer V, Harcos G, Maga P: Analytic properties of spherical cusp forms on GL(n), JOURNAL D ANALYSE MATHEMATIQUE (2020) REAL arXiv
  • [2] Blomer V, Harcos G, Maga P: On the global sup-norm of GL(3) cusp forms, ISRAEL JOURNAL OF MATHEMATICS 229 : 1 pp. 357-379., 23 p. (2019) DOI REALMathematical Reviews WoS Scopus arXiv
  • [3] Blomer V, Harcos G, Maga P, Milićević D: The sup-norm problem for GL(2) over number fields, JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY 22 : 1 pp. 1-53., 53 p. (2020) DOI REAL Egyéb URL arXiv

2. Outstanding applied research outcome
Artificial Intelligence Research Group

Automated Theorem Proving (ATP) and Deep Learning (DL) are two important branches of artificial intelligence both of which have undergone huge development over the past decade. A novel and exciting research direction is to find a synthesis of these two domains. One possible approach is to use an intelligent learning system to guide the theorem prover as it explores the search space of possible derivations. The research group of the Institute tackled the question of how to generalize from short proofs to longer ones with a strongly related structure. This is an important task since proving interesting problems typically requires thousands of steps, while current ATP methods are only finding proofs that are at most a couple dozens of steps long. The project has a homepage (http://bit.ly/site_atpcurr) and a public code repository (http://bit.ly/code_atpcurr). This work has been presented at the Bumerang Workshop, the Conference on Artificial Intelligence and Theorem Proving the Dagstuhl Logic and Learning Seminar and the NeurIPS 2019 workshop Knowledge Representation Meets Machine Learning (KR2ML2019). A paper about this project is currently under review.

  • Zombori Zs, Csiszárik A, Michalewski H, Kaliszyk C, Urban J: Towards Finding Longer Proofs (2019) arXiv

The group has built a system, called FLoP that searches for proofs using reinforcement learning. FLoP has been tested using a set of simple arithmetic theorems, which has many convenient properties for comparing machine learning methods: proofs are long (even thousands of steps) and repetitive, however, they often share a general structure. Despite their apparent simplicity, these arithmetic problems are already hard for the strongest automated theorem proving systems (E, Vampire), mostly due to the length of the proofs. The chart below compares our system (FLoP) with other methods that rely on search space exploration. FLoP performs reasonably well on these datasets.

From January 1 – August 31, 2019

  1. Pach J, Rubin N, Tardos G: Planar point sets determine many pairwise crossing segments, In: Charikar M, Cohen E (szerk.) STOC 2019 Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing New York (NY), Amerikai Egyesült Államok: Association for Computing Machinery (ACM), 1158-1166. (2019) http://real.mtak.hu/101917/
  2. Böröczky KJ, Ludwig M: Minkowski valuations on lattice polytopes, JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY 21:(1) 163-197. (2019) http://real.mtak.hu/60186/
  3. Titkos T: Arlinskii’s iteration and its applications, PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY 62:(1) 125-133. (2019) http://real.mtak.hu/60444/
  4. Rössler D, Szamuely T: Cohomology and torsion cycles over the maximal cyclotomic extension, JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 752: 211-227. (2019) http://real.mtak.hu/73294/
  5. Berkes I, Borda B: On the law of the iterated logarithm for random exponential sums, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 371: 5 pp. 3259-3280., 22 p. (2019) http://real.mtak.hu/83658/
  6. Blomer V, Harcos G, Maga P: On the global sup-norm of GL(3) cusp forms, ISRAEL JOURNAL OF MATHEMATICS 229:(1) 357-379. (2019) http://real.mtak.hu/85308/
  7. Halasi Z, Liebeck MW, Maróti A: Base sizes of primitive groups: Bounds with explicit constants, JOURNAL OF ALGEBRA 521: 16-43. (2019) http://real.mtak.hu/89801/
  8. Roche-Newton O, Ruzsa IZ, Shen C-Y, Shkredov ID: On the size of the set AA+A, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY 99:(2) 474-494. (2019) http://real.mtak.hu/103296/
  9. Darji UB, Elekes M, Kalina K, Kiss V, Vidnyánszky Z: The structure of random automorphisms of countable structures, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 371:(12) 8829-8848. (2019) http://real.mtak.hu/103298/
  10. Bianchi G, Böröczky KJ, Colesanti A, Yang D: The Lp-Minkowski problem for −n < p < 1, ADVANCES IN MATHEMATICS 341: 493-535. (2019) http://real.mtak.hu/89748/
  11. Halasi Z, Maróti A, Pyber L, Youming Q: An improved diameter bound for finite simple groups of Lie type, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY 51:(4) pp. 645-657. (2019) http://real.mtak.hu/92174/
  12. Backhausz A, Szegedy B: On the almost eigenvectors of random regular graphs, ANNALS OF PROBABILITY 47:(3) 1677-1725. (2019) http://real.mtak.hu/103299/
  13. Aceto P, Alfieri A: On sums of torus knots concordant to alternating knots, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY 51:(2) 327-343. (2019) http://real.mtak.hu/98398/
  14. Juhász P: Talent Nurturing in Hungary: The Pósa Weekend Camps, NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 66:(6) 898-900. (2019) http://real.mtak.hu/101971/
  15. Farkas Á: Dimension approximation of attractors of graph directed IFSs by self-similar sets, MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 167:(1) 193-207. (2019) http://real.mtak.hu/103300/

2018

  1. Aceto P, Larson K: Knot Concordance and Homology Sphere Groups. INTERNATIONAL MATHEMATICS RESEARCH NOTICES 201:(23) 7318-7334. (2018) http://real.mtak.hu/89976/
  2. Backhausz Á, Gerencsér B, Harangi V, Vizer M: Correlation bound for distant parts of factor of IID processes. COMBINATORICS PROBABILITY & COMPUTING 27:(1) 1-20. (2018) http://real.mtak.hu/67348/
  3. Backhausz Á, Szegedy B: On large-girth regular graphs and random processes on trees. RANDOM STRUCTURES & ALGORITHMS 53:(3) 389-416. (2018) http://real.mtak.hu/89854/
  4. Böröczky KJ, Henk M, Pollehn N: Subspace concentration of dual curvature measures of symmetric convex bodies. JOURNAL OF DIFFERENTIAL GEOMETRY 109:(3) 411-429. (2018) http://real.mtak.hu/89749/
  5. Domokos M: Polynomial bound for the nilpotency index of finitely generated nil algebras. ALGEBRA AND NUMBER THEORY 12:(5) 1233-1242. (2018) http://real.mtak.hu/89757/
  6. Duyan H, Halasi Z, Maróti A: A proof of Pyber’s base size conjecture. ADVANCES IN MATHEMATICS 331 pp. 720-747. (2018) http://real.mtak.hu/89799/
  7. Banerjee A, Khaled M: First order logic without equality on relativized semantics. ANNALS OF PURE AND APPLIED LOGIC 168:(11) 1227-1242. (2018) http://real.mtak.hu/90014/
  8. Bárány B, Kiss G, Kolossváry I: Pointwise regularity of parameterized affine zipper fractal curves. NONLINEARITY 31:(5) 1705-1733. (2018) http://real.mtak.hu/89791/
  9. Maga P: The spectral decomposition of shifted convolution sums over number fields. JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 744: 1-27. (2018) http://real.mtak.hu/49321/
  10. Kolountzakis M, Matolcsi M, Weiner M: An application of positive definite functions to the problem of MUBs. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 146: 1143-1150. (2018) http://real.mtak.hu/71573/
  11. Miklós I: Computational Complexity of Counting and Sampling. Boca Raton (FL), Amerikai Egyesült Államok: CRC Press – Taylor and Francis Group (2018), 378 p. ISBN: 9781138035577
  12. Gorsky E, Némethi A: On the set of L-space surgeries for links. ADVANCES IN MATHEMATICS 333: 386-422. (2018) http://real.mtak.hu/89809/
  13. Fox J, Pach J, Suk A: More distinct distances under local conditions. COMBINATORICA 38:(2) 501-509. (2018) http://real.mtak.hu/82746/
  14. Garban C, Pete G, Schramm O: The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane. ANNALS OF PROBABILITY 46:(6) 3501-3557. (2018) http://real.mtak.hu/90613/
  15. Soukup DT, Soukup L: Infinite combinatorics plain and simple. JOURNAL OF SYMBOLIC LOGIC 83:(3) 1247-1281. (2018) http://real.mtak.hu/89850/
  16. Sheffer A, Szabó E, Zahl J: Point-curve incidences in the complex plane. COMBINATORICA 38:(2) 487-499. (2018) http://real.mtak.hu/89852/
  17. Szamuely T, Zábrádi G: The p-adic Hodge decomposition according to Beilinson. In: Tomasso, de Fernex; Brendan, Hassett; Mircea, Mustaţă; Martin, Olsson; Mihnea, Popa; Richard, Thomas (szerk.) Algebraic Geometry: Salt Lake City 2015 Providence (RI), Amerikai Egyesült Államok: American Mathematical Society, (2018) pp. 495-572. http://real.mtak.hu/73295/