Objective Hilbert's solution to Waring problem in Number Theory shows that every positive integer is a sum of g(n) nth powers. Surprising non-commutative analogues of this phenomenon were discovered recently in Group Theory, where powers are replaced by general words. Moreover, the study of group words occurs naturally in important contexts, such as the Burnside problems, Serre's problem on profinite groups, and finite simple group theory. We propose a systematic study of word maps on groups, their images and kernels, as well as related Waring type problems. These include a celebrated conjecture of Thompson, problems regarding covering numbers and mixing times of random walks, as well as probabilistic identities in finite and profinite groups. This is a highly challenging project in which we intend to utilize a wide spectrum of tools, including Representation Theory, Algebraic Geometry, Number Theory, computational group theory, as well as probabilistic methods and Lie methods. Moreover, we aim to establish new results on representations and character bounds, which would be very useful in various additional contexts. Apart from their intrinsic interest, the problems and conjectures we propose have exciting applications to other fields, and the project is likely to shed new light not just in group theory but also in combinatorics, probability and geometry. Fields of science natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatoricsnatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) 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) Topic(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Call for proposal ERC-2009-AdG See other projects for this call Funding Scheme ERC-AG - ERC Advanced Grant Host institution THE HEBREW UNIVERSITY OF JERUSALEM EU contribution € 1 197 800,00 Address EDMOND J SAFRA CAMPUS GIVAT RAM 91904 Jerusalem Israel See on map Activity type Higher or Secondary Education Establishments Administrative Contact Hani Ben-Yehuda (Mr.) Principal investigator Aner Shalev (Prof.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all THE HEBREW UNIVERSITY OF JERUSALEM Israel EU contribution € 1 197 800,00 Address EDMOND J SAFRA CAMPUS GIVAT RAM 91904 Jerusalem See on map Activity type Higher or Secondary Education Establishments Administrative Contact Hani Ben-Yehuda (Mr.) Principal investigator Aner Shalev (Prof.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data
