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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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In this talk, basing ourselves on a new theory due to Francesca Da Lio, Matilde Gianocca, and
Tristan Rivière (ETH Zürich), we will explain how to show stability results for the Morse index and
the nullity of Willmore immersions of bounded energy (work in collaboration with Tristan Rivière).
The theory was since successfully applied to a variety of problems: biharmonic maps, p-harmonic
maps, CMC surfaces, Yang–Mills connections, Ginzburg–Landau-type approximations, and Ricci
Shrinkers.