I will give a brief survey of some problems in curvature free geometry and sketch
a new proof of the following:
Theorem (Guth). There is some $\delta (n)>0$ such that if $(M^n,g)$ is a closed aspherical Riemannian manifold and $V(R)$ is the volume of the largest ball of radius $R$ in the universal cover of $M$, then $V(R)\geq \delta(n)R^n$ for all $R$.
I will also discuss some recent related questions and results.
- Topology Seminar