| Date: | February 25, 2019 |
| Time: | 4:00pm |
| Location: | Blocker 117 |
| Speaker: | Panagiota Daskalopoulos, Columbia University |
| Title: | Nonlinear Geometric Flows |
| Abstract: | We will give an overview on the development of extrinsic Nonlinear Geometric Flows, emphasizing both the analytical and geometric point of views. Special emphasis will be given to the Mean curvature flow (an example of quasilinear diffusion), the Inverse Mean curvature flow (an example of ultra-fast diffusion) and to the Gauss curvature flow (an example of slow-diffusion). We will address the questions of the long time existence of the flow, regularity of solutions, asymptotic behavior and singularities. |
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| Date: | February 26, 2019 |
| Time: | 4:00pm |
| Location: | Blocker 117 |
| Speaker: | Panagiota Daskalopoulos, Columbia University |
| Title: | Ancient Solutions to Geometric Flows |
| Abstract: | Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time $-infty < t leq T$ for some $T leq +infty$. We refer to them as ancient if $T < +infty$ and eternal if $T=+infty$. The classification of such solutions often sheds new insight to the singularity analysis. In some flows it is also important for performing surgery near a singularity. In this lecture we will give an overview of uniqueness theorems for ancient solutions to geometric partial differential equations such as the Mean Curvature flow, the Ricci flow and the Yamabe flow. This often involves the understanding of the geometric properties of such solutions. We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons. |
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| Date: | February 28, 2019 |
| Time: | 4:00pm |
| Location: | Blocker 117 |
| Speaker: | Panagiota Daskalopoulos, Columbia University |
| Title: | Uniqueness of Ancient Solutions to the Mean Curvature flow and Ricci flow |
| Abstract: | This is a continuation of Lecture 2. We will discuss new results regarding the classification of ancient non-collapsed solutions to the Mean Curvature flow and the Ricci flow, answering well known conjectures in the field. |
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