Periodic Reporting for period 1 - BoundModProbAG (Boundedness and Moduli problems in birational geometry)
Período documentado: 2019-07-01 hasta 2021-06-30
Algebraic geometry is a sophisticated area of mathematics dating back to the mid 19th-century, that links algebra and geometry with many parts of mathematics and theoretical physics.
The basic objects of study, 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?
Calabi-Yau varieties, characterised as flat from the point of view of curvature, are one of three types of fundamental building blocks of algebraic varieties.
Calabi-Yau 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 Calabi-Yau 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 Calabi-Yau varieties there are is a question of fundamental importance. The problem is to know whether the shapes of Calabi-Yau 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.
Recent developments make powerful techniques available to investigate new aspects of it. This action has focused on showing that there are essentially finitely many families of Calabi-Yau varieties with some extra piece of structure -- an elliptic fibration -- in any dimension. A Calabi-Yau variety with an elliptic fibration can be decomposed like a bundle of doughnuts-like fibers over a smaller dimensional object: hence, the goal is to show that their bases are themselves bounded and then to spread the boundedness from the bases of the elliptic fibrations back to elliptic Calabi-Yau varieties.This goal was avhieved in a wide number of cases across al dimensions.
The basic objects of study, 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?
Calabi-Yau varieties, characterised as flat from the point of view of curvature, are one of three types of fundamental building blocks of algebraic varieties.
Calabi-Yau 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 Calabi-Yau 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 Calabi-Yau varieties there are is a question of fundamental importance. The problem is to know whether the shapes of Calabi-Yau 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.
Recent developments make powerful techniques available to investigate new aspects of it. This action has focused on showing that there are essentially finitely many families of Calabi-Yau varieties with some extra piece of structure -- an elliptic fibration -- in any dimension. A Calabi-Yau variety with an elliptic fibration can be decomposed like a bundle of doughnuts-like fibers over a smaller dimensional object: hence, the goal is to show that their bases are themselves bounded and then to spread the boundedness from the bases of the elliptic fibrations back to elliptic Calabi-Yau varieties.This goal was avhieved in a wide number of cases across al dimensions.
Over the course of the fellowship, I have obtained the following results:
- I have shown that there indeed exists only finitely many families of shapes, up to possibly admitting that the shapes of the varieties in question are slightly modified, for elliptic Calabi-Yau varieties of any dimension once we have fixed a point of reference on each the doughnut-like fibre in the elliptic fibration.
- for elliptic Calabi-Yau varieties of dimension three, I have shown that the boundedness problem can be fully solved. We do not need the existence of a rational section on the elliptic fibration, and neither do we need to allow any modification of the original shapes;
-- essential to the proof of the previous results was to understand the boundedness problems for the bases of elliptic Calabi-Yau varieties. I have shown that there exists only finitely many families of shapes, again, up to admitting that the shapes of the varieties in question are slightly changed, also for the bases of elliptic Calabi-Yau varieties.
I have also studied the naturally associated problem of constructing a moduli space for such structures. A moduli space is an ”atlas” containing all distinct shapes of a given class of varieties/structures.
For moduli spaces of the bases of elliptic Calabi-Yau varieties, I have developed new techniques that improve our understanding of how we can achieve the construction of such spaces and also lay the foundations to extend such constructions to more general frameworks (generalized pairs and foliated varieties)
-- the ideas and techniques developed to obtain the above results have found natural application in nearby problems and fields.
For example, the study of boundedness has led me to show that there exists finitely many shapes also for foliated surfaces with (completely) positive or negative curvature. This type of of structures are of great importance not only for classifying algebraic varieties, since they captures a lot of information on the curvature of a variety, but also in many other branches of Mathematics.
The dissemination of the results obtained over the course of the fellowship has been achieved through a wide variety of measures.
I have engaged with the academic community, by participating in conferences and workshops, and organizing visits to and invitations of researchers with relevant knowledge to the research program. I have communicated and disseminated my results to a large audience, through numerous invited presentations to various events. I have engaged with students by teaching two courses on topics related to research that was explored in the fellowship and by supervising several study projects and bachelor/master's theses.
As the obtained results pertain to blue sky research in Pure Mathematics, there is no immediately obvious intellectual property outcome or potential for commercial exploitation.
Nonetheless, the findings of the action have enhanced the scientific expertise of the mathematical community in Europe, attracting new talent to Europe. The action has also implemented collaborations between several European groups, contributing to European excellence and more integrated research networks, a goal achieved also via the training of future talent.
- I have shown that there indeed exists only finitely many families of shapes, up to possibly admitting that the shapes of the varieties in question are slightly modified, for elliptic Calabi-Yau varieties of any dimension once we have fixed a point of reference on each the doughnut-like fibre in the elliptic fibration.
- for elliptic Calabi-Yau varieties of dimension three, I have shown that the boundedness problem can be fully solved. We do not need the existence of a rational section on the elliptic fibration, and neither do we need to allow any modification of the original shapes;
-- essential to the proof of the previous results was to understand the boundedness problems for the bases of elliptic Calabi-Yau varieties. I have shown that there exists only finitely many families of shapes, again, up to admitting that the shapes of the varieties in question are slightly changed, also for the bases of elliptic Calabi-Yau varieties.
I have also studied the naturally associated problem of constructing a moduli space for such structures. A moduli space is an ”atlas” containing all distinct shapes of a given class of varieties/structures.
For moduli spaces of the bases of elliptic Calabi-Yau varieties, I have developed new techniques that improve our understanding of how we can achieve the construction of such spaces and also lay the foundations to extend such constructions to more general frameworks (generalized pairs and foliated varieties)
-- the ideas and techniques developed to obtain the above results have found natural application in nearby problems and fields.
For example, the study of boundedness has led me to show that there exists finitely many shapes also for foliated surfaces with (completely) positive or negative curvature. This type of of structures are of great importance not only for classifying algebraic varieties, since they captures a lot of information on the curvature of a variety, but also in many other branches of Mathematics.
The dissemination of the results obtained over the course of the fellowship has been achieved through a wide variety of measures.
I have engaged with the academic community, by participating in conferences and workshops, and organizing visits to and invitations of researchers with relevant knowledge to the research program. I have communicated and disseminated my results to a large audience, through numerous invited presentations to various events. I have engaged with students by teaching two courses on topics related to research that was explored in the fellowship and by supervising several study projects and bachelor/master's theses.
As the obtained results pertain to blue sky research in Pure Mathematics, there is no immediately obvious intellectual property outcome or potential for commercial exploitation.
Nonetheless, the findings of the action have enhanced the scientific expertise of the mathematical community in Europe, attracting new talent to Europe. The action has also implemented collaborations between several European groups, contributing to European excellence and more integrated research networks, a goal achieved also via the training of future talent.
Results on boundedness of Calabi-Yau varieties are extremely rare. In dimension higher the tree our knowledge of the boundedness problem for Calabi-Yau varieties is very scarce.
The results that I obtained are the first of their kind in higher dimension. Moreover, they are relevant not just to the problem of classifying algebraic varieties, but they also imply that there are only finitely many possible geometrical shapes for physical systems that describe string theory. This is a powerful consequence, since it tells us that many characteristics quantities and qualities of such systems can only take finitely many different forms.
The results on the structure of the bases of elliptic Calabi-Yau varieties are also providing an important advancement of our understanding of such spaces which play a central role in the classification of algebraic varieties.
Moreover, the study of moduli problems is one of the main interests of birational geometers, as these geometric obejcts naturally parametrize the shapes of algebraic varieties. Hence, the new techniques and results that I have proved towards the construction of such spaces constitute a new advancement in the theory of algebraic varieties, even more so, as they can be applied to other classes of geometric objects as well.
The impact of the obtained results stems from the many ways in which they improve our understanding of geometric structures central to our comprehension not only of algebraic varieties, but also of those models that we use to describe the universe. The innovative nature of these projects and the untapped territory they delve into has already unveiled a large number of new questions and possible developments that are worth pursuing as they will increase our knowledge even further and will also have a wider societal impact, via the nurturing of a new generation of talented scientists that will be engaging in these new research directions.
The results that I obtained are the first of their kind in higher dimension. Moreover, they are relevant not just to the problem of classifying algebraic varieties, but they also imply that there are only finitely many possible geometrical shapes for physical systems that describe string theory. This is a powerful consequence, since it tells us that many characteristics quantities and qualities of such systems can only take finitely many different forms.
The results on the structure of the bases of elliptic Calabi-Yau varieties are also providing an important advancement of our understanding of such spaces which play a central role in the classification of algebraic varieties.
Moreover, the study of moduli problems is one of the main interests of birational geometers, as these geometric obejcts naturally parametrize the shapes of algebraic varieties. Hence, the new techniques and results that I have proved towards the construction of such spaces constitute a new advancement in the theory of algebraic varieties, even more so, as they can be applied to other classes of geometric objects as well.
The impact of the obtained results stems from the many ways in which they improve our understanding of geometric structures central to our comprehension not only of algebraic varieties, but also of those models that we use to describe the universe. The innovative nature of these projects and the untapped territory they delve into has already unveiled a large number of new questions and possible developments that are worth pursuing as they will increase our knowledge even further and will also have a wider societal impact, via the nurturing of a new generation of talented scientists that will be engaging in these new research directions.