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SUMMARY:Shifted Symplectic Structures on Derived Foliations
DTSTART:20260629T120000Z
DTEND:20260629T153000Z
DTSTAMP:20260707T001900Z
UID:indico-event-16908@indico.math.cnrs.fr
DESCRIPTION:Speakers: Victor Alfieri\n\nPrincipal link: https://rendez-vou
 s.renater.fr/private/Victor_Alfieri_thesis_id_20cbd2eb90e34da8efc21f59eec8
 ba25ee2a4838e60dccef1d631c916d8af955__w46vhpkfdz8q_f40e35-542c2e-aa381a Pa
 ssword: 796206 Jury members: M. Pavel SAFRONOV\, Rapporteur\, The Universi
 ty of Edinburgh M. Joost NUITEN\, Examinateur\, Université de Toulouse Mm
 e Wendy LOWEN\, Examinatrice\, Universiteit Antwerpen M. Jean-Baptiste TEY
 SSIER\, Examinateur\, Sorbonne Université M. Frédéric DÉGLISE\, Examin
 ateur\, CNRS Rhône Auvergne M. Bertrand TOEN\, Directeur de thèse\, CNRS
  Occitanie Ouest Invited member: M. Marco ROBALO\, Co-encadrant de thèse\
 , Sorbonne Université Abstract: In this thesis\, we work in derived algeb
 raic geometry in characteristic zero and study derived foliations and shif
 ted symplectic structures on such objects. Our first main result is an exi
 stence theorem for derived foliations on mapping stacks. More precisely\, 
 if S is a base derived stack\, X is a proper-schematic\, flat\, and local 
 complete intersection derived stack over S\, and Y is a relative derived D
 eligne-Mumford stack over S equipped with a derived foliation\, then the d
 erived mapping stack Map_S(X\,Y) itself inherits\, under natural represent
 ability hypotheses\, a derived foliation. In order to establish this resul
 t\, we first show that the infinity-category of relative derived foliation
 s can be realized as a full subcategory of an infinity-category of derived
  stacks equipped with extra structure\, and then prove that the Weil restr
 iction along a proper-schematic\, flat\, and local complete intersection m
 orphism preserves this structure. This yields\, more generally\, a push-fo
 rward functor for derived foliations\, together with an explicit descripti
 on of their tangent complexes after push-forward. As applications\, we obt
 ain derived foliations on certain derived moduli spaces\, including derive
 d moduli spaces of stable curves and derived Hilbert schemes. The second p
 art of the thesis introduces the new notions of shifted symplectic structu
 res on derived foliations and Lagrangian derived foliations. We then prove
  existence theorems for these structures: an intersection theorem for Lagr
 angian derived foliations in an n-shifted symplectic derived stack\, produ
 cing an (n-1)-shifted symplectic derived foliation\, and a symplectic push
 -forward theorem\, showing that the push-forward of an n-shifted symplecti
 c derived foliation along a d-oriented morphism naturally carries an (n-d)
 -shifted symplectic structure.\n\nhttps://indico.math.cnrs.fr/event/16908/
LOCATION:Salle Johnson (Institut de Mathématiques de Toulouse)
URL:https://indico.math.cnrs.fr/event/16908/
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