Algebraic Maps: Hodge Theory and Algebraic Cycles

这位研究者研究复代数映射的拓扑学，并将其应用于代数圈。研究映射域的拓扑和目标的拓扑之间的关系是一个有趣的问题，在几何和拓扑学领域有着广泛的应用。这个项目的目的是：1)进一步研究和理解复代数映射的Hodgeand Chow理论性质；2)在有意义的情况下明确地研究这些性质。代数几何是致力于他们研究的学科。它是一个古老的主题，植根于人类早期的成就，比如轮子、埃及人的椭圆花插图和阿基米德燃烧的抛物面镜子。圆、椭圆和抛物线源于我们在高中学习的多项式方程。它们在自然界中既美丽又无处不在，因为它们描述了许多自然现象，从行星的运动到树叶和花朵的形状，再到微观粒子的行为。受资助的项目强烈倾向于纯研究，并建议研究更复杂的代数方程的解的更深层次的性质。一如既往，纯数学和应用数学将相互影响，新的抽象概念将推动应用的进步。

The investigator works on the topology of complex 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 Hodgeand Chow-theoretic properties of complex algebraic maps and at2) Explicitly studying these properties in significant cases.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.