Algebraic Maps: Topology and Algebraic Cycles

研究者和他的合作者研究复杂代数映射的拓扑，并将其应用于代数循环和非代数映射。研究区域拓扑与目标拓扑之间的关系是一个有趣的问题，在几何和拓扑领域有着广泛的应用。研究者的工作是：1)对所谓的代数映射分解定理的拓扑和霍奇理论证明，2)寻找定理对其他映射有效性的拓扑障碍，3)计算在重要情况下由映射产生的相交形式，以及4)分解定理在代数循环中的应用。作为这些研究的结果，一些新方法正在被引入。上面的“代数映射”一词是多项式方程的专业术语。代数几何是专门研究它们的学科。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解的更深层次性质。一直以来，纯粹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 and his collaborator work onthe topology of complex algebraic maps, with applicationsto algebraic cyclesand to non-algebraic maps.The study of the relations between the topologyof the domain and target of a mapis interesting and has broad applicationsin 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 DecompositionTheorem to algebraic cycles.Several new methods are being introduced as a result of these studies.The term ``algebraic maps" above is a technical termfor 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 beautifuland 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 researchand proposes to study the deeper propertiesof the solutions to more complicated algebraic equations.As it has always been the case, pure and applied mathematicswill influence each other and new abstract ideas will fuel the progress of applications.