The topology and Hodge theory of algebraic maps

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator works on the topology of algebraic maps, with applications to algebraic cycles. The study of the relations between the topology of the domain and target of a map is interesting and has broad applications in the fields of geometry and topology. This project aims at1) Furthering the study and the understanding of the Hodge and Chow-theoretic properties of complex algebraic maps and 2) Explicitly studying these properties in significant cases, such as the Hitchin systems and the system of hyperplane sections.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.