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.

In this course, we will present different techniques developed over the past few years, enabling mathematicians to prove that phase transitions are sharp. We will focus on a few classical models of statistical physics, including Bernoulli percolation, the Ising model and the random-cluster model.

In this course, we will present different techniques developed over the past few years, enabling mathematicians to prove that phase transitions are sharp. We will focus on a few classical models of statistical physics, including Bernoulli percolation, the Ising model and the random-cluster model.

In this course, we will present different techniques developed over the past few years, enabling mathematicians to prove that phase transitions are sharp. We will focus on a few classical models of statistical physics, including Bernoulli percolation, the Ising model and the random-cluster model.

In this course, we will present different techniques developed over the past few years, enabling mathematicians to prove that phase transitions are sharp. We will focus on a few classical models of statistical physics, including Bernoulli percolation, the Ising model and the random-cluster model.

Let X be a semi-stable arithmetic surface of  genus at least two and $omega$  the relative dualizing sheaf of X, equipped with the Arakelov metric. Parshin and Moret-Bailly have conjectured an upper bound for the arithmetic self-intersection of $omega$. They proved that a weak form of the abc conjecture follows from this inequality. We shall discuss a way of making their conjecture more precise in order that it implies the full abc conjecture (a  proof of which has been announced by Mochizuki).

Let X be a semi-stable arithmetic surface of  genus at least two and $omega$  the relative dualizing sheaf of X, equipped with the Arakelov metric. Parshin and Moret-Bailly have conjectured an upper bound for the arithmetic self-intersection of $omega$. They proved that a weak form of the abc conjecture follows from this inequality. We shall discuss a way of making their conjecture more precise in order that it implies the full abc conjecture (a  proof of which has been announced by Mochizuki).

Let X be a semi-stable arithmetic surface of  genus at least two and $omega$  the relative dualizing sheaf of X, equipped with the Arakelov metric. Parshin and Moret-Bailly have conjectured an upper bound for the arithmetic self-intersection of $omega$. They proved that a weak form of the abc conjecture follows from this inequality. We shall discuss a way of making their conjecture more precise in order that it implies the full abc conjecture (a  proof of which has been announced by Mochizuki).