In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ||\cdot||_{T}$ where $||\cdot||_T$ is the Teichm\"uller metric. Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_{g}$ of a closed Riemann surface $S_{g}$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm\"uller metric, which implies a conjecture of Farb-Weinberger in \cite{Farb-prob}.

Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.

We introduce a combinatorial curvature flow for piecewise constant curvature metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important one is that the evolution of the combinatorial curvature satisfies a combinatorial heat equation. It implies that the total curvature decreases along the flow. The local convergence of the flow to the hyperbolic metric is also established if the triangulation is isotopic to a totally geodesic triangulation.

Based on the proof of Labastida-Mari {} o-Ooguri-Vafa conjecture\cite {lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N $ invariants. Some symmetry properties of the infinite product will also be discussed.

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$, independently of the genus. This provides a counterexample to some suggestions that the Weil-Petersson metric becomes asymptotically flat, as the genus $g$ goes to infinity, in the thick loci in \tec space. Adopting a different perspective on curvature, we also show that the minimal (most negative) eigenvalue of the curvature operator at any point in the \tec space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius.