Cel Regularity and irregularity plays a central role in mathematics. In the present research proposal we will select problems from combinatorics and number theory (including additive combinatorics), where regularity and irregularity appear. In some cases we have to deal, e.g. with arbitrary finite or infinite subsets of natural numbers, where the only information we have is their cardinality, namely, that they are of positive (lower asymptotic) density within the set of all natural numbers or within the interval [1,N] for a large N. In other cases we consider an arbitrary distribution of n points within the unit square, where all we know is the density of our point set. The goal is often to show that certain configurations appear within the arbitrary set of numbers or points. These configurations definitely appear in a random set of numbers or points, but we have to show this for an arbitrary set of numbers or points with certain general properties. In order to reach our goal one can use two well-known methods. The first one is deterministic, often some kind of greedy algorithm. The second is the probabilistic method of Erdős, which shows that almost all arrangements of the given points or numbers (or graphs) fulfill the wanted property. A third method, the so called pseudorandom method, was initiated by the PI (together with M. Ajtai and J. Komlós), uses a combination of these. In other cases we have a deterministic set of numbers with certain quasi-random properties, for example, the primes. Randomness was the key idea in the recent breakthrough of Green and Tao, in proving that primes contain arbitrarily long arithmetic progressions. We will deal with 6 groups of problems: (i) finite or infinite sequences of integers, (ii) difference sets and Fourier analysis, (iii) graph and hypergraph embedding theorems, (iv) Ramsey theory, (v) distribution of points in the plane and in the unit square, (vi) regularities and irregularities in the distribution of primes. Dziedzina nauki natural sciencesmathematicspure mathematicsmathematical analysisfourier analysisnatural sciencesmathematicspure mathematicsdiscrete mathematicsgraph theorynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatoricsnatural sciencesmathematicspure mathematicsarithmeticsprime numbers Program(-y) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Temat(-y) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Zaproszenie do składania wniosków ERC-2012-ADG_20120216 Zobacz inne projekty w ramach tego zaproszenia System finansowania ERC-AG - ERC Advanced Grant Instytucja przyjmująca HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET Wkład UE € 1 776 000,00 Adres REALTANODA STREET 13-15 1053 Budapest Węgry Zobacz na mapie Region Közép-Magyarország Budapest Budapest Rodzaj działalności Other Kontakt administracyjny Tiziana Del Viscio (Ms.) Kierownik naukowy Endre Szemeredi (Prof.) Linki Kontakt z organizacją Opens in new window Strona internetowa Opens in new window Koszt całkowity Brak danych Beneficjenci (1) Sortuj alfabetycznie Sortuj według wkładu UE Rozwiń wszystko Zwiń wszystko HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET Węgry Wkład UE € 1 776 000,00 Adres REALTANODA STREET 13-15 1053 Budapest Zobacz na mapie Region Közép-Magyarország Budapest Budapest Rodzaj działalności Other Kontakt administracyjny Tiziana Del Viscio (Ms.) Kierownik naukowy Endre Szemeredi (Prof.) Linki Kontakt z organizacją Opens in new window Strona internetowa Opens in new window Koszt całkowity Brak danych