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Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...

Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...

Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory. I found the ...