The topology and Hodge theory of algebraic maps

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The investigator proposes to continue his investigations on the topology, Hodge and cycle theory of algebraic varieties and maps by studying fundamental aspects of the general theory as well as important examples by working on: 1. symmetries in the cohomology of the Hitchin fibration, 2. decomposition theorem for birational maps and for toric maps, 3. constructible sheaves in positive characteristic, and 4. motivic properties of cycles arising from the decomposition theorem.The investigator works in the field of algebraic geometry. Algebraic geometry is the discipline devoted to the study of polynomial equations. It is an ancient subject rooted in the early achievements of humanity, like the wheel, the Egyptians' elliptical flowers arrangements and Archimedes' burning parabolic mirrors. Circles, ellipses and parabolas arise from the polynomial equations we study in high school. They are both beautiful and ubiquitous in nature as they describe many natural phenomena, from the motion of planets to the shape of leaves and flowers, to the behavior of microscopic particles. The funded project is strongly inclined towards pure research and proposes to study the deeper properties of the solutions to more complicated algebraic equations (called algebraic maps). As it has always been the case, pure and applied mathematics will influence each other and new abstract ideas will fuel the progress of applications.