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Motivic zeta functions and the monodromy conjecture

Objective

The monodromy conjecture, formulated in the seventies by the Japanese mathematician Igusa, is one of the most important open problems in the theory of singularities. It predicts a remarkable connection between certain geometric and arithmetic invariants of a polynomial f with integer coefficients. The conjecture describes in a precise way how the singularities of the complex hypersurface defined by the equation f = 0 influence the asymptotic behaviour of the number of solutions of the congruence f = 0 modulo powers of a prime. Some special cases have been solved, but the general case remains wide open. A proof of the conjecture would unveil profound relations between several branches of mathematics.
In the past years, we have developed a new interpretation of the monodromy conjecture, based on non-archimedean geometry, and we have generalized it to a larger framework. A significant success of this approach was our proof of the monodromy conjecture for one-parameter degenerations of abelian varieties. The aim of our proposal is to generalize this proof to degenerations of Calabi-Yau varieties, and to adapt the arguments to the local case of the conjecture (hypersurface singularities). Degenerations of Calabi-Yau varieties play a central role in Mirror Symmetry, a mathematical theory in full development that emerged from string theory. We will explore in detail the connections between the monodromy conjecture and recent breakthroughs in Mirror Symmetry (tropical constructions of degenerating Calabi-Yau varieties). We hope to achieve these goals by combining advanced tools from several research domains, in particular: motivic integration, non-archimedean geometry, Hodge theory, logarithmic geometry and tropical geometry. We are convinced that all these research domains will greatly benefit from the systematic exploration of their mutual interactions, and that the impact of our project will go far beyond the monodromy conjecture.

Call for proposal

ERC-2012-StG_20111012
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Host institution

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Address
South Kensington Campus Exhibition Road
SW7 2AZ London
United Kingdom

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Activity type
Higher or Secondary Education Establishments
Administrative Contact
Brooke Alasya (Ms.)
Principal investigator
Johannes Nicaise (Prof.)
EU contribution
€ 637 427,13

Beneficiaries (2)

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
United Kingdom
EU contribution
€ 637 427,13
Address
South Kensington Campus Exhibition Road
SW7 2AZ London

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Activity type
Higher or Secondary Education Establishments
Administrative Contact
Brooke Alasya (Ms.)
Principal investigator
Johannes Nicaise (Prof.)
KATHOLIEKE UNIVERSITEIT LEUVEN
Belgium
EU contribution
€ 407 552,87
Address
Oude Markt 13
3000 Leuven

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Activity type
Higher or Secondary Education Establishments
Administrative Contact
Stijn Delauré (Dr.)