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Global motivic homotopy theory

全局动机同伦理论

基本信息

批准号：
EP/W012030/1
负责人：
Grigory Garkusha
金额：
$ 7.41万
依托单位：
Swansea University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW012030%2F1
关键词：
Global motivic homotopy theory

项目摘要

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.

这项研究是在被称为代数几何和同伦理论的数学领域进行的。代数几何研究具有重要意义的代数族。首先，它们是相对容易理解的，因为它们只是由多项式方程定义的；其次，它们通常给出了与其他形状相当准确的近似；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相当多的学科中。这就是为什么代数几何--代数变元理论对数学的发展和应用是如此重要的原因。同伦理论是一个相当新的数学领域，是代数拓扑学的一个重要分支，是对通常所说的“橡皮片几何”的现代发展，即研究在弯曲和拉伸等操作下保持不变的高维曲线、曲面和物体的性质；在同伦理论中，允许通过“连续变形”来进行额外的修改。同伦理论自诞生以来，已成为现代数学的重要组成部分。同伦理论在数学内外有着广泛的应用，包括理论物理和计算机科学。运动同伦理论是代数几何和同伦理论的混合体。它的主要目的是从同伦理论的观点研究代数簇。这门学科的许多基本思想和技巧都起源于代数拓扑学。基元同伦理论在代数几何中得到了诸如Milnor猜想和Bloch-Kato猜想的解这样引人注目的应用。除了这些相当壮观的应用外，一个人可以使用同伦理论的思想和技巧来解决代数几何中的问题的事实吸引了这两个领域的数学家，并导致了丰富的新的构造和应用。这个项目的主要目的是发展整体动机同伦理论，研究代数簇的动机等变谱和一系列相关的上同调理论。它的研究将有助于揭示动机同伦理论中的一些经典问题。此外，我们还希望将全局动机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。我们发展的同伦论和几何观也将用于其他数学领域，如代数拓扑、非交换几何和数学物理。要有效地发展项目目标，需要代数几何、动机同伦理论、等变拓扑和表示理论的方法。