# MODALAN — Result In Brief

Project ID:
272021

Funded under:
FP7-PEOPLE

Country:
Portugal

Domain:
Industry

## Model theory with applications

Model theory is a branch of mathematical logic dealing with abstract structures, historically connected to other fields of mathematics. EU-funded mathematicians worked on deepening and strengthening the connections with algebraic analysis, often leading to striking applications.

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The study of o-minimal structures is a part of model theory that deals with ordered and hence topological structures with particular tameness properties. It generalises piecewise linear geometry, semi-algebraic geometry and globally subanalytic geometry.

The most striking successes of this model-theoretic point of view of subanalytic geometry include results from work on the project MODALAN (Model theory and algebraic analysis). Nearly all the results are completely new and contribute to the development of a new formalism of Grothendieck's six operations.

This formalism, named after the German-born French mathematician Alexander Grothendieck, had originally sprung from relations in 'étale cohomologie' that arise from a morphism of schemes. In the past, the six operations formalism had been extended to semi-algebraic and subanalytic sheaves.

The MODALAN team showed that o-minimality does indeed realise Grothendieck's notion of 'topologie modérée'. Specifically, they extended the six operations formalism to o-minimal sheaves and obtained the cohomological ingredients required to prove Pillay's conjecture.

An increasing number of theorems reinforced the resemblance of o-minimal groups with real Lie groups, culminating in the proof of Pillay's conjecture. In the process, mathematicians did not assume that their o-minimal universe has order type similar to the real Lie ones.

MODALAN also encompassed the generalisation of o-minimal theory to cover the case of T-topologies. Mathematicians extensively studied microlocalisation and multi-specialisation of subanalytic sheaves, resulting in 12 papers published in high-impact peer-reviewed journals.

The nature of this project was strongly multidisciplinary since it involved algebra and analysis as well as geometry and logic. This kind of work will have interesting applications at the intersection of these branches of mathematics.

The most striking successes of this model-theoretic point of view of subanalytic geometry include results from work on the project MODALAN (Model theory and algebraic analysis). Nearly all the results are completely new and contribute to the development of a new formalism of Grothendieck's six operations.

This formalism, named after the German-born French mathematician Alexander Grothendieck, had originally sprung from relations in 'étale cohomologie' that arise from a morphism of schemes. In the past, the six operations formalism had been extended to semi-algebraic and subanalytic sheaves.

The MODALAN team showed that o-minimality does indeed realise Grothendieck's notion of 'topologie modérée'. Specifically, they extended the six operations formalism to o-minimal sheaves and obtained the cohomological ingredients required to prove Pillay's conjecture.

An increasing number of theorems reinforced the resemblance of o-minimal groups with real Lie groups, culminating in the proof of Pillay's conjecture. In the process, mathematicians did not assume that their o-minimal universe has order type similar to the real Lie ones.

MODALAN also encompassed the generalisation of o-minimal theory to cover the case of T-topologies. Mathematicians extensively studied microlocalisation and multi-specialisation of subanalytic sheaves, resulting in 12 papers published in high-impact peer-reviewed journals.

The nature of this project was strongly multidisciplinary since it involved algebra and analysis as well as geometry and logic. This kind of work will have interesting applications at the intersection of these branches of mathematics.