Complex geometry of noncommutative tori and t-structures on derived categories

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The proposed research will focus mainly on two topics:1) holomorphic bundles on noncommutative tori, and 2) t-structures and stability conditions on derived categories.Noncommutative tori play an important role in noncommutative geometry being the simplest examples of noncommutative manifolds. However, putting the complex structure and considering corresponding holomorphic objects on them is a recently new idea. The first project is devoted to the study of the categories of holomorphic bundles on noncommutative tori generalizingknown results in the two-dimensional case. The second project is concerned withinteresting new structures on derived categories of coherent sheaves on algebraic varietiesdiscovered by Bridgeland (motivated by some ideas from physics). Namely, the goalis to attack a number of problems related to stability conditions on such derived categories.Two more projects in the proposal are concerned with the study of tautological cycles on Jacobian varieties and with A-infinity algebras arising in algebraic geometry.The proposed research is in the fields of noncommutative geometry and algebraic geometry.Noncommutative geometry is a relatively new field originally developed by Connes that combines ideas from noncommutative algebra, differential geometry and functional analysis. Algebraic geometry is a classical branch of mathematics studying geometric objects defined by polynomial equations and related mathematical concepts. Many recent advances in both fieldswere motivated by their use in physics.