Abstract: In this talk, we investigate the formation of singularities in mean curvature flow. Specifically, we study ancient asymptotically cylindrical flows, i.e. ancient solutions whose tangent flow at $-\infty$ is a round shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. While in the neck case, i.e. for $k=1$, a complete classification has been obtained in several breakthroughs, a classification for  the case $2\leq k\leq n-1$ until recently seemed out of reach.

To analyze ancient asymptotically cylindrical flows for $2\leq k\leq n-1$, we consider the cylindrical profile function $u$ that measures the deviation of the renormalized flow from the round cylinder. We prove that for $\tau\to -\infty$ we have the asymptotics  $u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$, where $Q$ is a constant symmetric $k\times k$-matrix whose eigenvalues are quantized to be either 0 or $-\sqrt{(n-k)/8}$.

We then focus on the extremal rank cases. Under the natural noncollapsing condition, we obtain a classification of all solutions with $\textrm{rk}(Q)=0$, and establish $\textrm{SO}(n-k+1)$-symmetry and unique asymptotics in the case $\textrm{rk}(Q)=k$, also known as the $k$-oval case.

Next, we confirm a conjecture by Angenent-Daskalopoulos-Sesum about uniqueness of $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient ovals and more generally classify all $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient noncollapsed solutions.  On the other hand, for every $2\leq k\leq n-1$ we construct a $(k-1)$-parameter family of  ancient ovals  that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n-k+1)$-symmetric, giving counterexamples to another conjecture of Daskalopoulos. We then investigate ancient ovals without any symmetry assumption. Specifically, we prove that any $2$-oval in $\mathbb{R}^4$, up to scaling and rigid motion, either is the unique $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by White and Haslhofer-Hershkovits, or belongs to our new one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

Passcode: 911245