Research

Overview

My field of research includes singular foliations, Lie algebroids, and their ‘higher’ counterparts such as Lie ∞-structures on graded manifolds, and their applications in mathematical physics. Not only such higher structures regularly appear in higher gauge theories, but they also form an asset for the understanding of invariants of singular foliations. Indeed, to a singular foliation one may associate a Lie ∞-algebroid which ‘desingularizes’ the singular foliation and makes tractable some of its properties. This algebraic replacement is justified by the fact that singularities in foliation theory carry informations that may not be easily caught by a pure geometric analysis. Replacing the singular foliation by a Lie ∞-algebroid makes things much easier to work with as the latter is a linear structure, but as a trade-off this comes at the cost of working with a much bigger space.

In mathematical physics, differential graded Lie algebras and Lie ∞-algebroids naturally appear in gauging procedures in supergravity, providing an exciting field of investigations which generalizes classical gauge theories. Regarding this topic, I study embedding tensors and tensor hierarchies from the point of view of Leibniz algebras, which are mild generalizations of Lie algebras. Here, one finds that the tensor hierarchy associated to an embedding tensor can be considered as a replacement and a ‘lie-fication’ of the latter. This provides a mathematical framework to study embedding tensors in more general context, as in Double and Exceptional field theory.

Recently, after an additional curriculum in sociology and in philosophy of science, I got caught in the philosophy of mathematical practice. In contrast with the foundationalist and the logicist traditions that have been prevalent in the philosophy of mathematics during the XXth century, this intellectual movement advocates the study of mathematical practice in the philosophical inquiry. On this side of my research work, I am currently striving to develop stronger ties between the sociology of science and the philosophy of mathematical practice.

 

Publications
Maths and Physics

S. L., The modular class of a singular foliation. arXiv:2203.10861

S. L. and J. Stasheff, From differential crossed modules to tensor hierarchies. arXiv:2003.07838

S. L. and J. Palmkvist, Infinity-enhancing of Leibniz algebras, Lett. Math. Phys. (2020) 110(11):3121-3152. arXiv:1907.05752

C. Laurent-Gengoux, S. L. and T. Strobl, The universal Lie ∞-algebroid of a singular foliation. Doc. Math. (2020) 25:1571-1652. arXiv:1806.00475

S. L., Tensor hierarchies and Leibniz algebras, J. Geom. Phys. (2019) 144:147-189. arXiv:1708.07068

S. L., A short guide through integration theorems of generalized distributions, Differ. Geom. Appl. (2018) 61:42-58. arXiv:1710.01627

S. L., H. Samtleben and T. Strobl, Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions, J. Geom. Phys. (2014) 86:497-533. arXiv:1403.7114

Philosophy and Sociology of science

S.L., What is a theorem (in practice)?, Trilogía Ciencia Tecnología Sociedad (2021) 13(25):e1765.
trilogia:1765

S. L., Qu’est-ce qu’un théorème (en pratique) ?, Rev. Anthropol. Connaiss. (2021) 15(2). open-edition:rac/22479

J. Larregue, S. L. and M. Khelfaoui, La sociobiologie est morte, vive la psychologie évolutionniste ! Zilsel (2021) 8:104-143. HAL-SHS:03201759

 
Sylvain Lavau | Research