Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

多面体边界中的格点与奇异卡比萨宇面在簇种的正式几何量子化

Lattice points in polytope boundaries and formal geometric quantization of singular Calabi Yau hypersurfaces in toric varieties

研究通过正式几何量子化方法，推导出多面体边界格点数的计数多项式，并揭示其空隙原理与卡比萨宇面的关联。

Exact Euler Maclaurin formulas for simple lattice polytopes

Euler Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler Maclaurin formulas [Khovanskii-Pukhlikov, Cappell-Shaneson, Guillemin, Brion-Vergne] apply to exponential or polynomial functions; Euler Maclaurin formulas with remainder [Karshon-Sternberg-Weitsman] apply to more general smooth functions. In this paper we review these results and present proofs of the exact formulas obtained by these authors, using elementa…

Lattice points in polytopes, box splines, and Todd operators

Let $X$ be a list of vectors that is totally unimodular. In a previous article the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by $X$ can be extended to a function on the whole zonotope of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\Pcal$-space. In this paper we construct an explicit solution to this interpolation problem in terms of Todd operators. As a corollary we obtain a slight generalisation of the Khovanskii-Pukhlikov formula that relates the volu…

论Arthur迹公式的组合学、凸多面体与环面簇

On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties

本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。