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Ricci curvature can be expressed as a sum of sectional curvatures, so lower sectional curvature bounds imply lower Ricci curvature bounds. By computing only partial sums, one obtains a ladder of curvature conditions that interpolate between, for example, positive sectional and positive Ricci curvature.

Recent joint work with Lawrence Mouillé seeks to extends results from the Grove Symmetry Program from the case of positive sectional curvature to the case of positive intermediate curvature in the above sense. Of special interest is the fact that one finds new examples at each step up the ladder from sectional to Ricci curvature.

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In this talk, we will discuss some recent results about translating solitons for the mean curvature flow. We will start by describing some examples, with finite topology and finite entropy, recently obtained by Hoffman, Martín, and White. Then move on to discuss some structural results for this type of flows, obtained in collaboration with Gama and Moeller.

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The spectrum of the Laplace-Beltrami operator can be computed using Freudenthal's formula for normal homogeneous spaces $(G/K,g)$, i.e. the metric $g$ is the restriction of a biinvariant metric of $G$. We extend this formula to naturally reductive homogeneous spaces and provide naturally reductive realizations of deformations of normal homogeneous metrics along commuting subalgebras. This method can be applied to $3$-Sasaki manifolds together with their canonical deformation, which are positive $3$-$(\alpha,\delta)$-Sasaki manifolds. We obtain a formula to compute the spectrum of $3$-$(\alpha,\delta)$-Sasaki manifolds. Our approach has been inspired by Wilking's normal homogeneous realization of the Aloff-Wallach manifold $W^{1,1}=SU(3)/S^{1}$. We compute its full $3$-$(\alpha,\delta)$-Sasaki spectrum and related its spectrum to its geometry.

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We relate the geometry of complete Calabi-Yau metrics and singular Kahler-Einstein metrics to their tangent cones.  Building on Donaldson-Sun’s 2-step degeneration theory, we proved an asymmetric phenomenon between the global and local setting. Part of the talk is based on joint work with Song Sun.

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We will discuss a 4th order flow in an almost Kahler manifold. The flow stays within the Hamiltonian isotopy class of the initial Lagrangian submanifold, it is volume non-increasing and the stationary solutions are the Hamiltonian stationary Lagrangian submanifods. The talk is based on joint work with Micah Warren.

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Siobhan Roberts will recount her trajectory from history major to math journalist, and give a behind-the-scenes look at writing about mathematics for the New York Times.

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