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SUMMARY:Jesús A. Álvarez López (University of Santiago de Compostela)
DTSTART:20240306T200000Z
DTEND:20240306T210000Z
DTSTAMP:20260420T053338Z
UID:NYC-NCG/158
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/158/
 ">A trace formula for foliated flows</a>\nby Jesús A. Álvarez López (Un
 iversity of Santiago de Compostela) as part of Noncommutative geometry in 
 NYC\n\n\nAbstract\nIn the lecture\, I will try to explain the ideas of a r
 ecent paper on the trace formula for foliated flows\, written in collabora
 tion with Yuri Kordyukov and Eric Leichtnam. Let $\\mathcal{F}$ be a trans
 versely oriented foliation of codimension one on a closed manifold $M$\, a
 nd let $\\phi=\\{\\phi^t\\}$ be a foliated flow on $(M\,\\mathcal{F})$ (it
  maps leaves to leaves). Assume the closed orbits of $\\phi$ are simple an
 d its preserved leaves are transversely simple. In this case\, there are f
 initely many preserved leaves\, which are compact. Let $M^0$ denote their 
 union\, $M^1=M\\setminus M^0$ and $\\mathcal{F}^1=\\mathcal{F}|_{M^1}$. We
  consider two locally convex Hausdorff spaces\, $I(\\mathcal{F})$ and $I'(
 \\mathcal{F})$\, consisting of the leafwise currents on $M$ that are conor
 mal and dual-conormal to $M^0$\, respectively. They become topological com
 plexes with the differential operator $d_{\\mathcal{F}}$ induced by the de
 ~Rham derivative on the leaves\, and they have an $\\mathbb{R}$-action $\\
 phi^*=\\{\\phi^{t\\\,*}\\}$ induced by $\\phi$. Let $\\bar H^\\bullet I(\\
 mathcal{F})$ and $\\bar H^\\bullet I'(\\mathcal{F})$ denote the correspond
 ing leafwise reduced cohomologies\, with the induced $\\mathbb{R}$-action 
 $\\phi^*=\\{\\phi^{t\\\,*}\\}$. $\\bar H^\\bullet I(\\mathcal{F})$ and $\\
 bar H^\\bullet I'(\\mathcal{F})$ are shown to be the central terms of shor
 t exact sequences in the category of continuous linear maps between locall
 y convex spaces\, where the other terms are described using Witten's pertu
 rbations of the de~Rham complex on $M^0$ and leafwise Witten's perturbatio
 ns for $\\mathcal{F}^1$. This is used to define some kind of Lefschetz dis
 tribution $L_{\\rm dis}(\\phi)$ of the actions $\\phi^*$ on both $\\bar H^
 \\bullet I(\\mathcal{F})$ and $\\bar H^\\bullet I'(\\mathcal{F})$\, whose 
 value is a distribution on $\\mathbb{R}$. Its definition involves several 
 renormalization procedures\, the main one is the b-trace of some smoothing
  b-pseudodifferential operator on the compact manifold with boundary obtai
 ned by cutting $M$ along $M^0$. We also prove a trace formula describing $
 L_{\\rm dis}(\\phi)$ in terms of infinitesimal data from the closed orbits
  and preserved leaves. Some term of the formula is related with Connes' No
 n-Commutative Geometry of foliations with a transverse measure. This solve
 s a conjecture of C. Deninger involving two leafwise reduced cohomologies 
 instead of a single one.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/158/
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