The topology and Hodge theory of algebraic maps

研究者拟通过研究一般理论的基本方面和重要实例，继续对代数变异和代数映射的拓扑、Hodge和循环理论进行研究。希钦颤振上同调中的对称性，2。二分映射和环面映射的分解定理，3。3 .可施工的轴具有正特性；由分解定理引起的循环的动力性质。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。这是一门古老的学科，植根于人类早期的成就，就像轮子、埃及人的椭圆形插花和阿基米德的燃烧抛物面镜一样。圆、椭圆和抛物线都是我们在高中学过的多项式方程。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。该资助项目强烈倾向于纯粹研究，并提出研究更复杂代数方程（称为代数映射）解的更深层次性质。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 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.