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。代数几何研究具有重要意义的代数变异。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精确的近似，最重要的是，它们在理论物理、编码理论和计算机科学的很多学科中都很自然地出现。这就是为什么代数几何——代数变异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，是代数拓扑的一个重要分支，是众所周知的“橡胶板几何”的现代发展，即研究曲线、曲面和高维物体在弯曲和拉伸等操作下保持的性质；在同伦理论中，人们允许通过“连续变形”进行额外的修正。自同伦理论产生以来，已成为现代数学的重要组成部分。同伦理论在数学内外都有许多应用，包括理论物理和计算机科学。动机同伦理论是代数几何与同伦理论的融合。它的主要目的是从同伦理论的观点来研究代数变异。这门学科的许多基本思想和技术起源于代数拓扑学。在代数几何中，动机同伦理论导致了密尔诺猜想和布洛赫-加托猜想的解等引人注目的应用。除了这些相当引人注目的应用之外，人们可以使用同伦理论的思想和技术来解决代数几何中的问题，这一事实吸引了来自这两个领域的数学家，并导致了大量新的结构和应用。本课题的主要目的是发展整体动力同伦理论，研究动力等变谱和一系列代数变异的相关上同伦理论。它的研究将对动同伦理论中的一些经典问题有所启发。同时，我们希望将全局动力同伦理论的方法应用到经典的全局代数拓扑中。我们相信这些研究将具有重要的计算优势。我们发展的同伦理论和几何观点在代数拓扑、非交换几何和数学物理等其他数学领域也很有用。项目目标的有效发展需要代数几何、动力同伦理论、等变拓扑和表示理论的方法。

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.