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GRANT SIGNED

Groups, Representations and Analysis in Number Theory

Total Cost €

EC-Contrib. €

Partnership

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Outcomes and
results

GRANT project word cloud

Explore the words cloud of the GRANT project. It provides you a very rough idea of what is the project "GRANT" about.

expression theorem groups least interests isolation form arose questions automorphic model roots actions analytic actually line relevance kloosterman clear theory stronger commutative context perspective additive hard diameter bounding shown played algorithms classification algorithmic truly give newer techniques central invite forms ought rank links infinite dependence progress amounts permutation generators plan families seem outstanding hyperlinear exponential remarkable subject group pursue bound combinatorial sofic framed geometric time matrix decade last combinatorics expansion area sums fruit started express bounds consists directions matters removal showing follow

Project "GRANT" data sheet

The following table provides information about the project.

Coordinator	GEORG-AUGUST-UNIVERSITAT GOTTINGENSTIFTUNG OFFENTLICHEN RECHTS Organization address address: WILHELMSPLATZ 1 city: GOTTINGEN postcode: 37073 website: http://www.uni-goettingen.de contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	Germany [DE]
Total cost	1˙374˙250 €
EC max contribution	1˙374˙250 € (100%)
Programme	1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
Code Call	ERC-2014-CoG
Funding Scheme	ERC-COG
Starting year	2015
Duration (year-month-day)	from 2015-09-01 to 2020-08-31

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	GEORG-AUGUST-UNIVERSITAT GOTTINGENSTIFTUNG OFFENTLICHEN RECHTS GEORG-AUGUST-UNIVERSITAT GOTTINGENSTIFTUNG OFFENTLICHEN RECHTS Organization address address: WILHELMSPLATZ 1 city: GOTTINGEN postcode: 37073 website: http://www.uni-goettingen.de contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	DE (GOTTINGEN)	coordinator	1˙374˙250.00

Map

Project objective

The last decade has seen remarkable progress in the study of growth in infinite families of groups. The main approach has its roots in additive combinatorics, but has truly given fruit in a non-commutative context. It is becoming clear that the central role is played not by groups in isolation, but by actions of groups. It is from this perspective that my plan addresses, at the same time, questions on growth in groups as such and hard problems in analytic number theory.

While this line of research on growth started with the study of matrix groups, it has now given strong results on permutation groups as well. Two outstanding matters are the control of dependence on rank in matrix groups, and the removal of the need for the Classification Theorem in permutation groups. Going beyond these questions on diameter and expansion, there are at least three new directions I propose to follow: towards algorithms, towards geometric group theory, and towards number theory.

Some of the main recent results in the area take the form of diameter bounds. Bounding a diameter amounts to showing that one can express any element of a group as a short product of generators. One of the main algorithmic questions consists in actually finding such an expression, and doing so rapidly. Links between geometric group theory (which studies growth in infinite groups) and the new combinatorial techniques ought to become stronger. Sofic and hyperlinear groups -- which arose in part from geometric group theory -- seem to invite a combinatorial approach.

Additive combinatorics has already shown its relevance to exponential sums, a key subject in analytic number theory. Can a newer perspective based on actions of groups give more general results? Short Kloosterman sums, which are particularly hard to bound, can be framed as a test case.

I also plan to pursue related interests in automorphic forms - which are a classical example of the relevance of group actions to number theory - and model theory.

Publications

List of publications.
year	authors and title	journal	last update
2018	Harald AndrÃ©s Helfgott Isomorphismes de graphes en temps quasi-polynomial (d\'aprÃ¨s Babai et Luks, Weisfeiler-Leman...) published pages: , ISSN: , DOI:	AstÃ©risque	2019-06-06
2018	Harald A. Helfgott, Kate Juschenko Soficity, short cycles and the Higman group published pages: , ISSN: 0002-9947, DOI:	Transactions of the American Mathematical Society	2019-06-06
2019	Harald A. Helfgott Growth in linear algebraic groups and permutation groups: towards a unified perspective published pages: , ISSN: , DOI:	Proceedings of Groups St Andrews 2017 in Birmingham	2019-06-06

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