Fourier-Mukai and Nahm transforms for holomorphic triples on elliptic   curves
  We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple -- vortices -- and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line. 
Oscar Garc\'ia-Prada (CSIC, Madrid), Daniel Hern\'andez Ruip\'erez   (Salamanca, Spain), Fabio Pioli (CSIC, Madrid), Carlos ...