Non-commutative fundamental groups in Diophantine geometry
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
This proposal is concerned primarily with Diophantine equations in two variables, i.e., polynomial relations with integers coefficients for which one seeks to understand the collection of integer solutions. The history of such investigations reaches back to the tradition of Greek mathematics, while the twentieth century has seen spectacular applications of abstract modern machinery to the resolution of difficult old questions, such as Wiles' proof of Fermat's last theorem. The investigator proposes a new approach to studying these classical problems by incorporating fundamental ideas of topology and geometry that go beyond the principal developments of the twentieth century in that the relevant structures are, in the main, non-commutative and non-linear. An eventual goal is to construct methods for effectively resolving Diophantine equations in two-variables.
People |
ORCID iD |
Minhyong Kim (Principal Investigator) |
Publications
Kim M
(2012)
Tangential localization for Selmer varieties
in Duke Mathematical Journal
Kakde M
(2012)
The main conjecture of Iwasawa theory for totally real fields
in Inventiones mathematicae
Balakrishnan J
(2010)
Appendix and erratum to "Massey products for elliptic curves of rank 1"
in Journal of the American Mathematical Society
Coates J
(2010)
Selmer varieties for curves with CM Jacobians
in Kyoto Journal of Mathematics
Kim M
(2010)
Fundamental groups and Diophantine geometry
in Open Mathematics
Kim M
(2012)
Number Theory, Analysis and Geometry
Description | This project has led to a number of publications providing: -New proofs of Diophantine finiteness theorems for a wide class of hyperbolic curves including generalised Fermat curves; -Deep connections to non-commutative Iwasawa theory, in particular, the non-commutative main conjecture; -A new conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves formulated in terms of non-abelian cohomology. |
Exploitation Route | There are potential applications to high energy physics that are currently being investigated, especially in relation to topological quantum field theory and string theory. |
Sectors | Digital/Communication/Information Technologies (including Software) |
Description | International collaboration |
Organisation | Ben-Gurion University of the Negev |
Country | Israel |
Sector | Academic/University |
PI Contribution | During the course of this project, international collaboration has developed with Dr. Jennifer Balakrishnan (Harvard) Dr. Amnon Besser (Ben-Gurion) Professor Stefan Wewer (Hannover) Dr. Ishai Dan-Cohen (Essen) |
Description | International collaboration |
Organisation | Harvard University |
Country | United States |
Sector | Academic/University |
PI Contribution | During the course of this project, international collaboration has developed with Dr. Jennifer Balakrishnan (Harvard) Dr. Amnon Besser (Ben-Gurion) Professor Stefan Wewer (Hannover) Dr. Ishai Dan-Cohen (Essen) |
Description | International collaboration |
Organisation | University Duisburg-Essen |
Country | Germany |
Sector | Academic/University |
PI Contribution | During the course of this project, international collaboration has developed with Dr. Jennifer Balakrishnan (Harvard) Dr. Amnon Besser (Ben-Gurion) Professor Stefan Wewer (Hannover) Dr. Ishai Dan-Cohen (Essen) |
Collaborator Contribution | Co-work on a paper. |
Impact | Publication currently submitted to Mathmatische Annalen. |
Start Year | 2012 |