Abstract：

In the lecture, I will try to explain the ideas of a recent paper on the trace formula for foliated flows, written in collaboration with Yuri Kordyukov and Eric Leichtnam.

Let $\mathcal{F}$ be a transversely oriented foliation of codimension one on a closed manifold $M$, and 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 and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, and write $M^1=M\setminus M^0$ 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 conormal and dual-conormal to $M^0$, respectively. They become topological complexes 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 corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$. The spaces $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$ are shown to be the central terms of short exact sequences in the category of continuous linear maps between locally convex spaces, where the other terms are described using Witten's perturbations of the de~Rham complex on $M^0$ and leafwise Witten's perturbations for $\mathcal{F}^1$. This is used to define some kind of Lefschetz distribution $L_{\rm dis}(\phi)$ 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 obtained 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. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one.

Remark (by Deng): This talk is based on arXiv:2402.06671