Talk I: Runjie Hu (Texas A&M University), August/26
Title: Q_p-Homotopy Theory, and Applications to Topology and Algebraic Geometry
Abstract: We develop a Q_p-homotopy theory for p-complete spaces, which is an analogue of Sullivan's rational homotopy theory. We apply the Q_p-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for p-complete spaces, finiteness properties of étale homotopy types, formality of smooth proper varieties, Galois representations on étale homotopy groups, and constraints on étale fundamental groups.
Talk II: Simone Cecchini (Texas A&M University), September/02
Title: Topology, Singularities, and Positive Scalar Curvature
Abstract: Scalar curvature is a local geometric quantity, yet topology can prevent a closed manifold from carrying a metric of positive scalar curvature. What happens when the manifold is punctured, or when the metric is allowed to become singular on a small set? I will compare three settings: smooth closed manifolds, complete metrics on punctured manifolds, and bounded metrics with finite-distance singularities. Completeness preserves strong topological obstructions, but isolated or high-codimension singularities can lead to surprising flexibility. In dimensions at least eight, a singular set, sometimes consisting of a single point, can hide positive scalar curvature on a manifold that admits no smooth metric of positive scalar curvature. I will describe the geometry behind this contrast and explain how Dirac operators and index theory help distinguish rigidity from flexibility.
Talk III: Qiaochu Ma (Texas A&M University), September/16
Title: Gromov’s Simplicial Volume Vanishing Conjecture
Abstract: Gromov’s simplicial volume vanishing conjecture predict that if a closed oriented manifold admits a positive scalar curvature metric, then its simplicial volume vanishes. In this talk, we present a proof of this conjecture under a spin condition. Our proof is based on a quantitative refinement of the Connes-Moscovici higher index theorem. This is joint work with Jinmin Wang, Zhizhang Xie, Guoliang Yu, and Bo Zhu.