Algebraic Maps: Topology and Algebraic Cycles

The investigator and his collaborator work on the topology of complex algebraic maps, with applications to algebraic cycles and to non-algebraic maps. 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. The investigator works on: 1) a topological and Hodge-theoretic proof of the so-called Decomposition Theorem for algebraic maps, 2) finding the topological obstructions to the validity of the theorem for other maps, 3) computing intersection forms arising from maps in significant cases, and 4) applications of the Decomposition Theorem to algebraic cycles. Several new methods are being introduced as a result of these studies. The term ``algebraic maps" above is a technical term for polynomial equations. Algebraic geometry is the discipline devoted to their study. 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. 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.