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Words and Waring type problems

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.

Field of science

  • /natural sciences/mathematics/pure mathematics/arithmetic
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry

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
Address
Edmond J Safra Campus Givat Ram
91904 Jerusalem
Israel
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 197 800
Principal investigator
Aner Shalev (Prof.)
Administrative Contact
Hani Ben-Yehuda (Mr.)

Beneficiaries (1)

THE HEBREW UNIVERSITY OF JERUSALEM
Israel
EU contribution
€ 1 197 800
Address
Edmond J Safra Campus Givat Ram
91904 Jerusalem
Activity type
Higher or Secondary Education Establishments
Principal investigator
Aner Shalev (Prof.)
Administrative Contact
Hani Ben-Yehuda (Mr.)