We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine m...

The actin bundles in Drosophila bristles run the length of the bristle cell and are accordingly 65 microns (microchaetes) or 400 microns (macrochaetes) in length, depending on the bristle type. Shortly after completion of bristle elongation in pupae, the actin bundles break down as the bristle surface becomes chitinized. The bundles break down in a bizarre way; it is as if each bundle is sawed ...

After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose several problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical de...