1. Ricci Flow

Consider $(S^3,g)$ with scalar curvature $R\geq 6$. The Ricci Flow exists on the maximal interval $(0,T)$. The scalar curvature evolves by $$\partial_t R = \Delta R + 2 Rc^2.$$ One can estimate that $$\partial_t R = \Delta R + 2 Rc^2 = \Delta R + \frac 2 3 R^2 + \mathring{Rc}^2 \geq \Delta R + \frac 2 3 R^2 .$$ It follows from the maximum principle that $$R\geq \frac{6}{14t} \mbox{ and } T \leq \frac 1 4.$$
Problem 1.05.
[D. Maximo, R. Bamler] If $T$ is close to $\frac 1 4$, is $g(0)$ close to the round metric in the GromovHausdorff sense and in the $C^0$ sense? One can also ask the same question for other flows such as mean curvature flow. What would be a suitable topology?
Remark. ndavj


Problem 1.1.
[B. Wilking] Let $K$ be a simplicial complex with metric $g$ such that each simplex has a metric with positive (or flat) curvature and $K$ has positive (or nonnegative) curvature in the Alexander sense. Does a Ricci flow $(K, g(t))$ exist such that for each $t>0$, $g(t)$ is a smooth orbifold metric and $$\lim_{t\to 0} g(t) =g $$ in the GromovHausdorff topology. Can one get rigidity in the nonnegative case? 
Problem 1.15.
[B.Kotschwar] Does there exist a complete Ricci Flow that starts with uniformly bounded curvature that instantly has unbounded curvature? 
Problem 1.2.
[J. Streets] Given a hypersurface $M\subset \mathbb{R}^{n+1}$, define $$F(M,x_0,t_0)=(4\pi t_0)^{\frac n 2} \int_M e^{\frac{xx_0^2}{4t_0}} dH^{n},$$ $$\lambda(M) = \sup_{(x_0,t_0)} F(M,x_0,t_0).$$ Question: Is there a useful notion of $\mu$stability for Ricci Flow selfshrinkers analogous to ColdingMinicozzi for the $\lambda$ entropy in mean curvature flow? What are the stable solitons? 
Problem 1.25.
[R. Bamler] Understand singularities in Ricci Flow in dimension $n\geq 4$. Examples are $S^3\times \mathbb{R}$, $S^2\times \mathbb{R}^2$, $4d$ Bryant soliton, $3d \mbox{Bryant soliton} \times \mathbb{R}$ and FIK solitons. 
Problem 1.3.
[R. Bamler] In dimension $4$, can there be an ALE space as singularity for Ricci Flow? 
Problem 1.35.
[R. Bamler] Does $\max_M R(\cdot,t)$ blow up as $t\to T$ the singular time? 
Problem 1.4.
[R. Bamler] Let $(M^n,g_t)_{t\in[0,T)}$ be a Ricci flow with $T<\infty$. Assume $R <1$ on $M\times [0,T)$. Characterize $g_t$ as $t\to T$. 
Problem 1.45.
[R. Bamler] Let $(M^n,g_t)_{t\in[0,T)}$ be a Ricci flow with $T<\infty$. Assume $R< \frac{C}{Tt}$. Characterize $(Tt)^{1} g_t$ as $t\to T$. This is related to the HamiltonTian conjecture. Note that the Kahler Ricci flow on Fano manifolds satisfies $R< \frac{C}{Tt}$. 
Problem 1.5.
[R. Bamler] Classify singularities modulo singularities that in bounded scalar curvature setting. 
Problem 1.55.
[R. Bamler] R. Bamler and Q. Zhang proved a backwards pseudolocality theorem: Assume $R<1$. For $r \in (0,1)$, if $Rm < r^{2} $ on $B(x,t,r)$, then $$Rm < (\epsilon r)^{2}$$ on $P(x,t,\epsilon r,(\epsilon r)^2 ) =B(x,t,\epsilon r) \times [t(\epsilon r)^2, t)$. Can we remove $R<1$ assumption in the backwards pseudolocality theorem? 
Problem 1.6.
[R. Bamler] Can we remove $R<1$ assumption in R. Bamler and Q. Zhang’s heat kernel bound and distance bound? For more information, see their papers on arxiv: HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE, and HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATUREPART II.
Cite this as: AimPL: Geometric flows and Riemannian geometry, available at http://aimpl.org/flowriemannian.