Topological 1D Gravity, KP Hierarchy, and Orbifold Euler Characteristics of $\overline{\mathcal M}_{g,n}$

$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

本文通过分解模空间、利用环面作用推导闭式公式，计算了Kontsevich模空间$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数，建立了其几何与对称函数、图染色计数的联系。

关于$\boldsymbol{\bar{\frak{M}}_{0,n}}$的Betti数、庞加莱多项式与欧拉示性数

On the Betti numbers, Poincaré polynomials, and Euler characteristics of $\overline{\mathcal M}_{0,n}$

本文推导了$\bar{\frak{M}}_{0,n}$的庞加莱多项式等的两个新闭式，得到Betti数新公式与欧拉示性数新递推，精炼了其渐近估计。

Local topological recursion governs the enumeration of lattice points in $\overline{\mathcal M}_{g,n}$

The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom degree coefficients store psi-class intersection numbers and orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$, respectively. Furthermore, they ask whether the enumeration is governed by the topological recursion and whether the intermediate coefficients also store algebro-geometric information. In this paper, we prove that the enumeration does …

Local Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group

We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…

黎曼-洛伦兹延拓的拓扑障碍

Topological Obstructions to Riemannian-Lorentzian Continuation

本文研究黎曼区域到洛伦兹区域延拓的拓扑障碍，给出相对欧拉类障碍消失的充要条件，并构造显式符号变化度量，揭示拓扑变化可局限于洛伦兹区域。

The Goulden-Harer-Jackson matrix model

An alternative formula for the partition function of the Goulden-Harer-Jackson matrix model is derived, in which the Penner and the orthogonal Penner partition functions are special cases of this formula. Then the free energy that computes the parametrized Euler characteristic $ξ^s_g(γ)$ of the moduli spaces as yet an unidentified, for $g$ is odd, shows that the expression for $ξ^s_g(γ)$ contains the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with $s$ punctures for all parameters $γ$. The other contributions are written as a linear combinations of Berno…