The topology and Hodge theory of algebraic maps

调查员建议继续他的调查拓扑结构，霍奇和循环理论的代数簇和地图，通过研究基本方面的一般理论以及重要的例子，通过工作：1。希钦纤维化上同调的对称性，2。双有理映射和复曲面映射的分解定理，3.在积极的特点，建设层，和4。由分解定理产生的循环的动机性质。研究者在代数几何领域工作。代数几何是专门研究多项式方程的学科。这是一个古老的主题，植根于人类早期的成就，就像车轮，埃及人的椭圆形插花和阿基米德的燃烧抛物面镜。圆、椭圆和抛物线都是从我们在高中学习的多项式方程中产生的。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。该资助项目强烈倾向于纯研究，并建议研究更复杂的代数方程（称为代数映射）的解的更深层次的性质。正如以往一样，纯数学和应用数学将相互影响，新的抽象思想将推动应用的进步。

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.