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CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity. For example, finding solutions to the vacuum Einstein constraint equations is simplified by assuming CMC initial data. In this talk, we review and extend some results concerning the existence (and nonexistence) of CMC Cauchy surfaces in cosmological spacetimes, i.e., Lorentzian manifolds with compact Cauchy surfaces satisfying the strong energy condition (Ricci nonnegative on timelike vectors). We will also consider the case of a positive cosmological constant. This talk is based on joint work with Greg Galloway and with Argam Ohanya

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In this talk, I will discuss the stability of Euclidean 3-space for the Positive Mass Theorem, and the stability of Schwarzschild 3-manifold for the Penrose inequality, in the pointed measured Gromov-Hausdorff topology modulo negligible spikes. This talk is based on joint work with Antoine Song.

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This talk will give an overview of the current landscape of research in Bryant's Laplacian flow, a geometric flow of closed $G_2$-structures in the direction of their Hodge Laplacian. This flow, it is hoped, will help us to understand manifolds with special holonomy and calibrated geometry. We will sketch a few of these motivating goals.

The rest of the talk will have two aims. The first will be to describe the features of Laplacian flow by direct comparison to other geometric flows, especially Ricci flow, and in this way to give a sense of the progress of Laplacian flow over the last few years. The second is to collect and summarize the various approaches taken to studying the flow, and to describe their relative successes and limitations. Finally, we plan to discuss open problems that are hopefully accessible to researchers in the two SLMath semester programs, without requiring deep expertise in special holonomy.

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Juan Meza will give a short talk followed by a panel discussion/Q&A session.

Please join us for a special afternoon tea.  The SLMath staff will take a few minutes to introduce themselves so that you can put faces to names.  We will have a special treat for a snack.

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Aligned to his macroscopic dimension conjecture for uniformly positive scalar curvature, Gromov also proposed a volume growth conjecture for manifolds with nonnegative Ricci curvature and uniformly positive scalar curvature. It says under such curvature conditions, the polynomial volume growth order is (dim-2). In this talk we will explore some analog "dim-2" upper bound for the first Betti number and dimension of linear growth harmonic functions under the same curvature assumptions. We will also discuss the rigidity when such an upper bound for the first Betti number is achieved. The splitting theorem and the theory of singular spaces with Ricci curvature lower bounds (Ricci limit spaces) will play a crucial role. A part of this talk is based on joint work with Jinmin Wang, Zhizhang Xie and Bo Zhu.

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