Global motivic homotopy theory

这项研究计划是在数学领域被称为代数几何和同伦理论。代数几何研究代数簇是主要的重要性。首先，它们相对容易理解，因为它们只是由多项式方程定义的，其次，它们通常对其他形状给出相当准确的近似，最重要的是，它们确实自然地出现在理论物理，编码理论和计算机科学的许多学科中。这就是为什么代数几何-代数簇的理论对数学的发展和应用如此重要。同伦理论是一个相当新的数学领域，是一个重要的分支代数拓扑，现代发展什么是俗称的“橡皮片几何”，即研究的性质曲线，曲面和物体的高维下保存的操作，如弯曲和拉伸;在同伦理论之一，允许额外的修改“连续变形”。自同伦理论创立以来，它已成为现代数学的重要组成部分。同伦理论在数学中和数学之外都有许多应用，包括理论物理和计算机科学。动机同伦理论是代数几何和同伦理论的混合。它的主要目的是从同伦理论的观点来研究代数簇。这门学科的许多基本思想和技巧都起源于代数拓扑学。动机同伦理论在代数几何中有着惊人的应用，例如解决了米尔诺猜想和布洛赫-加藤猜想。除了这些相当壮观的应用，一个事实，即可以使用同伦理论的思想和技术来解决问题的代数几何吸引了数学家从这两个领域，并导致了丰富的新的建设和应用。本项目的主要目的是发展全球motivic同伦理论，研究motivic等变谱和一系列相关的上同调理论的代数簇。它的研究将有助于揭示动机同伦理论中的一些经典问题。同时，我们也希望将整体运动同伦理论的方法应用到经典的整体代数拓扑中。我们相信，这些调查将有重要的计算优势。同伦理论和几何观，我们开发的也将是有用的数学的其他领域，如代数拓扑，非交换几何和数学物理。有效的发展项目的目标需要代数几何，motivic同伦理论，等变拓扑和表示论的方法。

This research proposal is in the areas of mathematics known as algebraic geometry and homotopy theory. Algebraic geometry studies algebraic varieties which are of principal importance. First they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects in theoretical physics, coding theory and computer sciences. That is why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. Homotopy theory is a considerably newer area of mathematics, being an important branch of algebraic topology, the modern development of what is popularly known as "rubber-sheet geometry", that is, the study of the properties of curves, surfaces and objects of higher dimension which are preserved under operations such as bending and stretching; in homotopy theory one allows additional modifications by "continuous deformation". Since its creation homotopy theory has become an essential component of modern mathematics. Homotopy theory has numerous applications both in and out of mathematics, including theoretical physics and computer sciences.Motivic homotopy theory is a blend of algebraic geometry and homotopy theory. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. Motivic homotopy theory led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture, in algebraic geometry. Besides these quite spectacular applications, the fact that one can use the ideas and techniques of homotopy theory to solve problems in algebraic geometry has attracted mathematicians from both fields and has led to a wealth of new constructions and applications.The principal aim of this project is to develop global motivic homotopy theory, investigate motivic equivariant spectra and a range of associated cohomology theories of algebraic varieties. Its study will shed light on some classical problems in motivic homotopy theory. Also, we want to apply methods of global motivic homotopy theory to classical global algebraic topology. We believe that these investigations will have important computational advantages. The homotopy-theoretic and geometric outlook that we develop will also be useful in other areas of mathematics such as algebraic topology, non-commutative geometry and mathematical physics.Effective developments of the project objectives require methods of algebraic geometry, motivic homotopy theory, equivariant topology and representation theory.