Schedule, Notes/Handouts & Videos

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The celebrated positive mass theorem, first proven by Schoen and Yau and then by Witten, states that the ADM energy-momentum vector of an asymptotically flat initial data set with the dominant energy condition must be either future timelike or null. This led to a natural conjecture characterizing initial data sets with null ADM energy-momentum, known as the equality case. It turns out that this conjecture is interconnected with the problem of mass minimization in the context of the Bartnik quasi-local mass, known as the stationary conjecture. I will describe a variational approach to advance both conjectures, as well as counterexamples in higher dimensions without the optimal decay rate for asymptotically flatness. These talks are based on a series of joint works with Dan Lee.

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The nontriviality of the normal bundle of a minimal submanifold of codimension greater than one makes the second variation difficult to study because it is not clear how to choose variations that reduce the area of the submanifold. For a 2-D minimal surface, a partial averaging technique is encoded by a complex-valued version of the second variation formula and this has the benefit of bringing complex analytic techniques into play. I will survey the achievements of this technique and discuss open problems in this area. Other results, including ones for minimal hypersurfaces, that place this work in a broader context will also be mentioned.

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Ricci curvature lies between the sectional curvature and scalar curvature. It occurs naturally in Einstein equation, Ricci flow, optimal transport. We will review the basic tools (Bochner's formula, comparison inequalities) and survey many geometric inequalities (Neumann eigenvalue comparison, Brunn-Minkowski, Sobolev inequalities etc.), topological properties and examples in the smooth and nonsmooth settings.

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We will discuss the construction of certain interesting solutions to curve shortening and related flows. Some of these lead to classification results for ancient solutions.

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In this talk, I will outline a theory that provides a framework for taking geometric limits of Ricci flows. This theory can be viewed as a parabolic analog of Gromov-Hausdorff compactness for metric spaces. More specifically, we will examine how any sequence of Ricci flows, when appropriately pointed, converges subsequentially to a synthetic flow known as a metric flow. 

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We examine harmonic maps into non-positively curved metric spaces, with a focus on their regularity when mapping into Euclidean buildings. We extend the regularity results of Gromov and Schoen by considering buildings that are not necessarily locally finite. As an application, we prove a superrigidity theorem for algebraic groups, which generalizes the rank 1 p-adic superrigidity results of Gromov and Schoen. This work also provides a geometric framework for Bader and Furman's generalization of Margulis' higher rank superrigidity theorem, bridging the gap between geometric and algebraic approaches to rigidity phenomena.

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I will  review the basic principles regarding the basic techniques to prove existence of  closed minimal hypesurfaces. I will talk about some  current state of the art and finish by pointing out some possible new direction.

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A PDE on a cohomogeneity one manifold becomes and ODE and hence more accessible to obstructions and constructions of examples. At a singular orbit the smoothness conditions are complicated and we will discuss existence and uniqueness to solutions near the singular orbit.

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Building on Einstein’s equivalence principle, general relativity achieves coordinate‬ independence in its core equations. However, this also presents a challenge: it implies the‬ absence of a localized gravitational density. Consequently, defining fundamental quantities such‬ ‭ such as mass, and angular momentum becomes intricate. These talks will explore the‬ ongoing efforts over the past decades to understand these concepts in general relativity.‬

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The lecture discusses the relation of other parabolic geometric flows such as mean curvature flow and Ricci-flow to questions in GR.