##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry Seminar

## Mat Langford

#### University of Tennessee, Knoxville

## Ancient solutions out of polytopes

##### Abstract:

I will show how to construct a very large family of new examples of convex ancient and translating solutions to mean curvature flow in all dimensions. At $t=-\infty$, these examples resemble a family of standard Grim hyperplanes of certain prescribed orientations. The existence of such examples has been suggested by Hamilton and Huiskenâ€”Sinestrari. Our examples include solutions with symmetry group $D\times \mathbb Z_2$, where $D$ is the symmetry group of any given regular polytope, and, surprisingly, many examples which admit only a single reflection symmetry. We also exhibit a family of eternal solutions which do not evolve by translation, settling a conjecture of Brian White in the negative. Time permitting, I will present further structure and partial classification results for this class of solutions, as well as some open questions and conjectures. \\ \\ Joint with T. Bourni and G. Tinaglia.

Hosts: Bennett Chow and Luca Spolaor

### February 17, 2021

### 3:00 PM

### Zoom Meeting ID: 988 8132 1752

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