EPSRC Reference: 
EP/S025839/1 
Title: 
Mirror symmetry, Berkovich spaces and the Minimal Model Programme 
Principal Investigator: 
Nicaise, Dr J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Dept of Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research 
Starts: 
01 September 2019 
Ends: 
31 August 2023 
Value (£): 
463,465

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Algebraic geometry studies the shapes of geometric objects that can be defined by means of polynomials equations, leading to profound interactions between number theory, algebra and geometry. Since the 1990s, physics has exerted a major and surprising influence on algebraic geometry through the theory of mirror symmetry, which grew out of string theory, a proposal to describe fundamental particles mathematically as tiny vibrating strings. Models of the universe in string theory take as input a specific type of object in algebraic geometry: a socalled CalabiYau variety, named after the mathematicians Eugenio Calabi and ShingTung Yau. These objects attracted the attention of physicists because of their special symmetry properties. The CalabiYau variety is responsible for the 6 "hidden dimensions" that are postulated in string theory to explain the fundamental properties of particles.
Physicists soon realized that the CalabiYau variety is not uniquely determined by the physical model: rather, CalabiYau varieties seemed to come in "mirror pairs" giving rise to equivalent theories. Algebraic geometers were forced to take this idea very seriously after some spectacular applications to enumerative geometry (counting special types of curves living on CalabiYau varieties) around 1990. The main challenge for algebraic geometers was to provide mathematical foundations for these ideas, that is, give an exact definition of what it means to be a mirror pair, and devise techniques to construct such pairs. This is still an ongoing story, but much progress has been made. This project is concerned with one of the mathematical approaches to mirror symmetry, developed by Kontsevich and Soibelman: the nonarchimedean approach to the StromingerYauZaslow (SYZ) conjecture.
The SYZ conjecture is an ambitious attempt to give a geometric explanation of mirror symmetry, and it has been very influential in mathematics. Around 2000, Kontsevich and Soibelman had the groundbreaking insight that one should be able to find the structures predicted by the SYZ conjecture in a seemingly unrelated field: nonarchimedean geometry, a branch of geometry and analysis that was originally designed to solve problems in number theory.
In the last few years, I have realized an important part of Kontsevich and Soibelman's proposal, by introducing a new ingredient into the picture: the minimal model programme (MMP) in birational geometry. This programme is one of the most successful developments in algebraic geometry in the last 40 years; in 2018, the Cambridge mathematician Caucher Birkar received the Fields medal (the most prestigious award in mathematics) for his contributions to the MMP. The aim of the MMP is to classify all the geometric objects that arise in algebraic geometry. I have discovered that one can use nonarchimedean geometry as a dictionary to transfer questions and results back and forth between the field of mirror symmetry and the MMP, thus providing new tools to study both fields simultaneously. The goal of this project is to further exploit these interactions between mirror symmetry, nonarchimedean geometry, and birational geometry. In this way, I aim to prove some of the central conjectures in the nonarchimedean approach to mirror symmetry, and to develop new tools to understand the MMP.

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Organisation Website: 
http://www.imperial.ac.uk 