Motivic Category Theory and Rational Motivic Gamma-spaces

动同伦理论的主要目的是从同伦理论的角度研究代数变异。这门学科的许多基本思想和技术起源于代数拓扑学。该理论是由Morel和Voevodsky在20世纪90年代发明的，现在是代数几何和代数拓扑中最活跃的领域之一，汇集了拓扑，代数和表示理论的研究人员。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。这个项目将研究各种三角分类的动机和理性动机伽玛空间的分类方面。本课题的主要目的是促进我们对丰富动机范畴理论、理性连通动机谱和动机无限环空间的认识。期望在一系列相关的代数变异上同伦理论和经典稳定同伦理论中得到重要的应用。

The primary goal of motivic homotopy theory is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. The theory was invented by Morel and Voevodsky in the 90s and it is now one of the most active areas in algebraic geometry and algebraic topology, bringing together researchers in topology, algebra and representation theory. 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.This project will investigate categorical aspects of various triangulated categories of motives and rational motivic Gamma-spaces. The main purpose of this project is to advance our understanding of enriched motivic category theory, rational connected motivic spectra and motivic infinite loop spaces. Important applications for a range of associated cohomology theories of algebraic varieties and for the classical stable homotopy theory are expected as well.