**Speaker #1:** Renato Bettiol (University of Pennsylvania)

**Title: ** A Weitzenbock viewpoint on sectional curvature and application

**Abstract: ** In this talk, I will describe a new algebraic characterization of sectional curvature bounds that only involves curvature terms in the Weitzenboeck formulae for symmetric tensors. This characterization is further clarified by means of a symmetric analogue of the Kulkarni-Nomizu product, which renders it computationally amenable. Furthermore, a related application of the Bochner technique to closed 4-manifolds with indefinite intersection form and positive or nonnegative sectional curvature will be discussed, yielding some new nsight about the Hopf Conjecture. This is based on joint work with R. Mendes (Univ. Koln, Germany).

**Speaker #2:** Or Hershkovitzs (Stanford University)

**Title:** The topology of self-shrinkers and sharp entropy bounds

**Abstract:** The Gaussian entropy, introduced by Colding and Minicozzi, is a rigid motion and scaling invariant functional which measures the complexity of hypersurfaces of the Euclidean space. It is defined to be the supremal Gaussian area of all dilations and translations of the hypeprsurface, and as such, is well adapted to be studied by mean curvature flow. In the case of the n-th sphere in Rn+1, the entropy can be computed explicitly, and is decreasing as a function of the dimension n. A few years ago, Colding Ilmanen Minicozzi and White proved that all closed, smooth self-shrinking solutions of the MCF have larger entropy than the entropy of the n-th sphere. In this talk, I will describe a generalization of this result, which derives better (sharp) entropy bounds under topological constraints. More precisely, we show that if M is any closed self-shrinker in Rn+1 with a non-vanishing k-th homotopy group (with k less than or equal to n), then its entropy is higher than the entropy of the k-th sphere in Rk+1. This is a joint work with Brian White.