In this paper, we give a simultaneous vanishing principle for the $v$-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where $v$ is a finite place of the rational function field over a finite field. This principle establishes the fact that the $v$-adic vanishing of CMPLs at algebraic points is equivalent to its $\infty$-adic counterpart being Eulerian. This reveals a nontrivial connection between the $v$-adic and $\infty$-adic worlds in positive characteristic.

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This paper studies affine Deligne-Lusztig varieties $X_{w}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{w}(b)$ for a minimal length element $w$ in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in \cite{HL} to the affine case. We then provide a reduction method that relates the structure of $X_{w}(b)$ for arbitrary elements $w$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of G\"ortz, Haines, Kottwitz and Reuman in \cite{GHKR}.

Wedge product on deRham complex of a Riemannian manifold M can be pulled back to H^∗(M) via explicit homotopy, constructed using Green’s operator, to give higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of M. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.