Abstract: Given a compact Riemannian manifold $M$ and $D^{2}$ the Dirac Laplacian on the Clifford bundle, Bochner discovered the existence of a self-adjoint, non-negative Laplacian $\Delta$ such that the difference $D^{2} - \Delta$ is a zero-order operator that can be expressed in terms of the curvature tensor of $M$. In fact, combined with some harmonic theory, these operators allowed Bochner to obtain fundamental vanishing theorems involving the Betti numbers of $M$. In this talk, I will recall the Dirac and connection Laplacian operators and prove the general Bochner identity. Using this, I will prove the Weitzenbock formula and a vanishing theorem of the first Betti number $b_{1}(M) = dim (H^{1}(M, \mathbb{R}))$.

Abstract: Various results by Tannaka, Krein, Deligne, Lurie, Hall, Schaeppi, Chirvasitu and B. show that a scheme / algebraic stack can be recovered from its tensor category of quasi-coherent sheaves. This motivates to generalize several constructions from algebraic geometry to tensor category theory. I would like to illustrate this process for affine and projective morphisms, tangent bundles, and fiber products.