Toward a minimal model for $H_\ast(\overline{\mathcal{M}})$

Relative Recognition Principle

In this paper the relative recognition principle will be proved. It states that a pair of spaces $(X_o,X_c)$ is weakly equivalent to $(Ω^N_\text{rel}(ι:B\hookrightarrow Y),Ω^N(Y))$ if and only if $(X_o,X_c)$ are grouplike $\overline{\mathcal{SC}^N}$-spaces, where $\overline{\mathcal{SC}^N}$ is any cofibrant resolution of the Swiss-cheese 2-operad $\mathcal{SC}^N$. This principle will be proved for connected $\overline{\mathcal{SC}^N}$-spaces, and also for grouplike $\overline{\mathcal{SC}^N}$-spaces for $2<N\leq\infty$, in the form of an equivalence of homotopy categories.

$\overline{\mathcal{M}}_{0,n}$上的cobordism值交理论

Cobordism-valued intersection theory on $\overline{\mathcal{M}}_{0,n}$

本文通过细化弦方程，计算了$\overline{\mathcal{M}}_{0,n}$上cobordism值Gromov-Witten不变量，并给出了相关psi类交的显式公式。

引力运算子与超交换运算子之间的链级Koszul对偶

Chain level Koszul duality between the Gravity and Hypercommutative operads

本文构造超交换运算子的链模型$C_*^{dual}(\overline{\mathcal{M}})$，证明其为引力运算子链模型$grav$的bar构造线性对偶，细化Getzler结果，表明两者在链级为Koszul对偶。

Homotopy invariants for $\overline{\mathcal{M}}_{0,n}$ via Koszul duality

We show that the integral cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.

实有理曲线Deligne-Mumford空间的爆破

Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves

本文构造了实有理曲线Deligne-Mumford模空间的一列光滑商爆破，给出Keel爆破构造的实类比，并显式描述了中间空间及三类爆破，为后续完全确定实模空间的有理（上）同调环奠定基础。