In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

The gravitational waves detected by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which generic solutions of the Einstein vacuum equations can be described, asymptotically, by a finite number of Kerr solutions moving away from each other. Though the conjecture is so very easy to formulate and happens to be validated by both astrophysical observations as well as numerical experiments, it is far beyond our current mathematical understanding. In fact even the far simpler and fundamental question of the stability of one Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity of black holes and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of geometric and analytic techniques which I will try to outline in my lectures.

The gravitational waves detected by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which generic solutions of the Einstein vacuum equations can be described, asymptotically, by a finite number of Kerr solutions moving away from each other. Though the conjecture is so very easy to formulate and happens to be validated by both astrophysical observations as well as numerical experiments, it is far beyond our current mathematical understanding. In fact even the far simpler and fundamental question of the stability of one Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity of black holes and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of geometric and analytic techniques which I will try to outline in my lectures.

The gravitational waves detected by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which generic solutions of the Einstein vacuum equations can be described, asymptotically, by a finite number of Kerr solutions moving away from each other. Though the conjecture is so very easy to formulate and happens to be validated by both astrophysical observations as well as numerical experiments, it is far beyond our current mathematical understanding. In fact even the far simpler and fundamental question of the stability of one Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity of black holes and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of geometric and analytic techniques which I will try to outline in my lectures.

 

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

 

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

 

The aim of this course is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. After recalling basics on o-minimal structures and their tameness properties, I will discuss:
– the use of tame geometry in proving algebraization results (Pila-Wilkie theorem; o-minimal Chow and GAGA theorems in definable complex analytic geometry);
– the tameness of period maps; algebraicity of images of period maps;
– functional transcendence results: Ax-Schanuel conjecture from abelian varieties to Shimura varieties and variations of Hodge structures. Applications to atypical intersections (André-Oort conjecture and Zilber-Pink conjecture);
– the geometry of Hodge loci and their closures.

 

 

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

 

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

 

The aim of this course is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. After recalling basics on o-minimal structures and their tameness properties, I will discuss:
– the use of tame geometry in proving algebraization results (Pila-Wilkie theorem; o-minimal Chow and GAGA theorems in definable complex analytic geometry);
– the tameness of period maps; algebraicity of images of period maps;
– functional transcendence results: Ax-Schanuel conjecture from abelian varieties to Shimura varieties and variations of Hodge structures. Applications to atypical intersections (André-Oort conjecture and Zilber-Pink conjecture);
– the geometry of Hodge loci and their closures.

 

In the first lecture, I will talk about some general conjectures about Faltings heights including ABC conjecture, André-Oort conjecture, and Landau-Siegel zero conjecture, etc.

 

In the second lecture, I will talk about some decompositions of Faltings heights of CM abelian varieties into sums of small pieces and their relations with heights of CM points on Shimura curves.

 

In the third lecture, I will talk about Xinyi Yuan’s recent work on Faltings of CM points on Shimura curves as an extension of Gross-Zagier formula.

 

In the last lecture, I will talk about Yun-Zhang’s recent work on Faltings heights of CM points on the moduli of Shtukas.

In this series of 4 lectures, I will survey some recent work on Faltings heights of CM abelian varieties and applications.

 

In the first lecture, I will talk about some general conjectures about Faltings heights including ABC conjecture, André-Oort conjecture, and Landau-Siegel zero conjecture, etc.

 

In the second lecture, I will talk about some decompositions of Faltings heights of CM abelian varieties into sums of small pieces and their relations with heights of CM points on Shimura curves.

 

In the third lecture, I will talk about Xinyi Yuan’s recent work on Faltings of CM points on Shimura curves as an extension of Gross-Zagier formula.

 

In the last lecture, I will talk about Yun-Zhang’s recent work on Faltings heights of CM points on the moduli of Shtukas.

In this series of 4 lectures, I will survey some recent work on Faltings heights of CM abelian varieties and applications.

 

In the first lecture, I will talk about some general conjectures about Faltings heights including ABC conjecture, André-Oort conjecture, and Landau-Siegel zero conjecture, etc.

 

In the second lecture, I will talk about some decompositions of Faltings heights of CM abelian varieties into sums of small pieces and their relations with heights of CM points on Shimura curves.

 

In the third lecture, I will talk about Xinyi Yuan’s recent work on Faltings of CM points on Shimura curves as an extension of Gross-Zagier formula.

 

In the last lecture, I will talk about Yun-Zhang’s recent work on Faltings heights of CM points on the moduli of Shtukas.