EPSRC Reference: 
EP/S024808/1 
Title: 
Moduli and boundedness problems in geometry 
Principal Investigator: 
Svaldi, Dr R 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
EPSRC Fellowship 
Starts: 
01 July 2020 
Ends: 
30 June 2023 
Value (£): 
298,286

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Algebraic geometry is a sophisticated area of mathematics dating back to the mid 19thcentury, that links algebra and geometry with many parts of mathematics and theoretical physics. The basic objects, called algebraic varieties, are the common zero sets of polynomial functions, which are higher dimensional analogs to the ellipses and hyperbolas of antiquity. The subject has key applications in very many branches of modern mathematics, science and technology.
One of the main goals in algebraic geometry is to classify algebraic varieties.
They can often be decomposed into simpler shapes that act as fundamental building blocks in the classification.
But how many different shapes appear in each class of building blocks?
CalabiYau varieties, characterised as flat from the point of view of Ricci curvature, are one of three types of fundamental building blocks of algebraic varieties. CalabiYau threefolds and fourfolds have formed the focus of interest of string theorists over recent decades. A better understanding of the geometry and the classification of CalabiYau varieties would advance string theory in fundamental ways, and would provide many new examples and models to study.
Since they are building blocks for constructions in geometry and theoretical physics, understanding how many CalabiYau varieties there are is a question of fundamental importance. The problem is to know whether the shapes of CalabiYau varieties come in just finitely many families  a property that goes under the name of boundedness. This very difficult question remains wide open already in dimension three.
While this problem has long been considered to be out of reach, recent developments make powerful techniques available to investigate new aspects of it. The aim of this research project is to show that there are essentially finitely many families of CalabiYau varieties with some extra piece of structure  an elliptic fibration  in any dimension. This would be a striking result as it implies that there are only finitely many possible geometrical shapes for physical systems that describe string theory. A CalabiYau variety with an elliptic fibration can be decomposed like a bundle of doughnutslike fibers over a smaller dimensional object. Thus, in classifying elliptic CalabiYau varieties, we should first look separately at the base and the fibers. While the geometric structure of the fibers is very wellunderstood, it is not clear how the bases behave and how to control the bundling behaviour.

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