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.
Talk IV: Aditya Kumar (University of Maryland), October/23/2026, 4:00-4:50 pm
Title: Metric invariants and scalar curvature
Abstract: We will discuss scalar curvature from the point of view of metric invariants. In particular, we will describe a new estimate for the codimension two width of circle bundles, and explain how it leads to counterexamples to the macroscopic version of Gromov’s Urysohn width conjecture, in dimensions four and higher. Based on joint work with Balarka Sen.
Talk V: Dongyeong Ko (MIT), November/13/2026, 4:00-4:50 pm
Title: Capillary minimal slicing and scalar curvature rigidity
Abstract: We develop minimal slicing via capillary hypersurfaces to understand scalar curvature metric on manifolds bounded below with boundary. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain. Moreover, we prove a comparison and rigidity result of manifolds with Listing type scaling invariant scalar and mean curvature comparison: $R g \ge -12 g_{\mathbb{H}^{4}}$ and $\hat{H}^{2} g \ge \hat{H}{0} g{\mathbb{H}^{4}}$, where $\hat{H}$ is a shifted mean curvature depending on the scale of manifold. This proves the traced version of Gromov's conjecture on comparison and rigidity of hyperbolic convex domains. This is based on the joint work with X. Yao.