Homological and Homotopical Algebra

Lead Research Organisation: Swansea University
Department Name: College of Science


Algebra & Geometry. Homological and homotopical algebra is a spectacular area of Mathematics with many applications within and around mathematics. One of the ingenious discoveries were made by V Voevodsky, who was awarded the Fields medal in 2002. His work lead top solution of a number of longstanding problems with far reaching implications. Very recently, Dr G Garkusha and I Panin have accomplished a programme of Voevodsky, which began in 2001. G Garkusha and I Panin use enriched motivic homotopy theory. The project is to explore and develop further aspects of the enriched homotopy theory and enriched homological algebra which is of major importance in Garkusha-Panin's theory. Applications are expected in motivic homotopy theory, ring and module theory, homological algebra..


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509553/1 01/10/2016 30/09/2021
1817931 Studentship EP/N509553/1 01/10/2016 30/09/2019 Darren Alexander Jones
Description The derived category D[C, V] of the Grothendieck category of enriched functors [C, V], where V is a closed symmetric monoidal Grothendieck category and C is a small V-category, was studied. It was shown that if the derived category D(V) is a compactly generated triangulated category satisfying some reasonable assumptions on compact generators, then the derived category D[C, V] is also compactly generated triangulated with explicit compact generators. Further, a series of recollements for Grothendieck categories of enriched functors were established. It was also proven that these recollements can be extended to their derived categories. The notion of the (strict) V -property for localizing Serre subcategories was introduced. It is inspired by the fundamental theorem of Voevodsky on homotopy invariant presheaves with transfers. It was shown that the recollements for those derived categories can be further extended to the derived categories of associated Grothendieck categories, localized with respect to localizing Serre subcategories satisfying the strict V - property. An application was given for triangulated categories of motives with various transfers.
Exploitation Route Further developments are expected in motivic homotopy theory, homological algebra and ring/module theory.
Sectors Other