We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $n \geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2 $, a rotated normalized $L_p$-curvature image operator is an only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \neq =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.

In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.

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For a gerbe Y over a smooth proper Deligne-Mumford stack B banded by a finite group G, we prove a structure result on the Gromov-Witten theory of Y, expressing Gromov-Witten invariants of Y in terms of Gromov-Witten invariants of B twisted by various flat U(1)-gerbes on B. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.