We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume. This is a joint work with Chris Connell and Shi Wang (arXiv:2410.19981).

We develop a new Fréchet sufficient dimension reduction (FdSDR) framework tailored for high-dimensional functional data with complex, metric-space-valued responses. Our main contribution is a low-rank distance covariance criterion that enables scalable, model-free identification of low-dimensional predictor structures while capturing nonlinear dependence. The proposed method is computationally efficient in high dimensions and avoids restrictive distributional assumptions. We establish theoretical guarantees and demonstrate its effectiveness through simulations and real data, providing a practical and flexible approach for modern functional data analysis.