Enriched motivic homotopy theory
Lead Research Organisation:
Swansea University
Department Name: College of Science
Abstract
The versatility of motivic homotopy theory and its associated range of cohomological techniques has made it an important branch of mathematics. Recently there have been several fundamental developments which have been used to solve a number of longstanding problems. The new strategically important developments are related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A. Suslin, I. Panin, M. Levine, F. Morel.
The principal aim of this project is to investigate enriched motivic homotopy theory. It is hoped that its study will shed light on some classical problems in A^1-topology. We propose to use its methods to construct various triangulated categories of motives, one of the central objects of study in modern A^1-topology. We believe that these constructions will have important computational advantages. The homotopy-theoretic outlook that we develop is likely to be useful in other areas of mathematics such as algebraic topology and non-commutative geometry.
The research will be undertaken in the Department of Mathematics, Swansea University.
The principal aim of this project is to investigate enriched motivic homotopy theory. It is hoped that its study will shed light on some classical problems in A^1-topology. We propose to use its methods to construct various triangulated categories of motives, one of the central objects of study in modern A^1-topology. We believe that these constructions will have important computational advantages. The homotopy-theoretic outlook that we develop is likely to be useful in other areas of mathematics such as algebraic topology and non-commutative geometry.
The research will be undertaken in the Department of Mathematics, Swansea University.
Planned Impact
The main beneficiaries will be the departments of pure mathematics of UK universities. The project will be of great interest for experts in algebraic geometry, algebraic topology, K-theory.
There is a growing interest in various parts of Mathematics to applications of contemporary algebraic geometry and Voevodsky motivic homotopy theory. This project will add new values to applications of enriched motivic homotopy theory to algebraic geometry and algebraic topology.
Besides of its research merits, this project is aimed to enhance the collaboration between UK-universities and their counterpart in Russia. Discussion and dissemination of the involved mathematical ideas will be undoubtedly beneficial for mathematical communities in both countries.
There is a growing interest in various parts of Mathematics to applications of contemporary algebraic geometry and Voevodsky motivic homotopy theory. This project will add new values to applications of enriched motivic homotopy theory to algebraic geometry and algebraic topology.
Besides of its research merits, this project is aimed to enhance the collaboration between UK-universities and their counterpart in Russia. Discussion and dissemination of the involved mathematical ideas will be undoubtedly beneficial for mathematical communities in both countries.
Organisations
Publications
Garkusha G
(2015)
ON THE MOTIVIC SPECTRAL SEQUENCE
in Journal of the Institute of Mathematics of Jussieu
Garkusha G
(2014)
The triangulated category of K-motives
in Journal of K-Theory
| Description | Thanks to this grant we solved a conjecture of Voevodsky (Fields Medal 2002) and a problem of Suslin (Cole Prize 2000). The machinery which is necessary to solve these two conjectures was introduced and studied thanks to the EPSRC award EP/H021566/1. Moreover, the technique and methods developed within this grant, EP/J013064/1, are of great importance for the further theory of framed motives of algebraic varieties, developed joint with Panin. The theory accomplishes ideas of Voevodsky and is now a mainstream in the motivic homotopy theory. |
| Exploitation Route | The results can be taken in related areas of topology and noncommutative geometry. |
| Sectors | Education,Other |