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.

 

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).

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Cours des Professeurs permanents de l'IHES

 

There are two canonical « quantizations'' of symplectic manifolds:

begin{itemize}

item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)

symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras

$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.

item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,

with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.

This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]

end{itemize}

The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):

 

{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$

and $B$-field $Im(omega/hbar)$.}

 

 

The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.

As a corollary we obtain

the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.

 

Liouville conformal field theory (LCFT hereafter), introduced by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings", can be seen as a random version of the theory of Riemann surfaces. LCFT appears in Polyakov's work as a 2d version of the Feynman path integral with an exponential interaction term. Since then, LCFT has emerged in a wide variety of contexts in the physics literature and in particular recently in relation with 4d supersymmetric gauge theories (via the AGT conjecture). 

 

A major issue in theoretical physics was to solve the theory, namely compute the correlation functions. In this direction, an intriguing formula for the three point correlations of LCFT was proposed in the middle of the 90's by Dorn-Otto and Zamolodchikov-Zamolodchikov, the celebrated DOZZ formula. 

 

The purpose of the course is twofold (based on joint works with F. David, A. Kupiainen and R. Rhodes). First, I will present a rigorous probabilistic construction of Polyakov's path integral formulation of LCFT. The construction is based on the Gaussian Free Field. Second, I will show that the three point correlation functions of the probabilistic construction indeed satisfy the DOZZ formula. This establishes an explicit link between probability theory (or statistical physics) and the so-called conformal bootstrap approach of LCFT.

Liouville conformal field theory (LCFT hereafter), introduced by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings", can be seen as a random version of the theory of Riemann surfaces. LCFT appears in Polyakov's work as a 2d version of the Feynman path integral with an exponential interaction term. Since then, LCFT has emerged in a wide variety of contexts in the physics literature and in particular recently in relation with 4d supersymmetric gauge theories (via the AGT conjecture).

 

cA major issue in theoretical physics was to solve the theory, namely compute the correlation functions. In this direction, an intriguing formula for the three point correlations of LCFT was proposed in the middle of the 90's by Dorn-Otto and Zamolodchikov-Zamolodchikov, the celebrated DOZZ formula.

 

The purpose of the course is twofold (based on joint works with F. David, A. Kupiainen and R. Rhodes). First, I will present a rigorous probabilistic construction of Polyakov's path integral formulation of LCFT. The construction is based on the Gaussian Free Field. Second, I will show that the three point correlation functions of the probabilistic construction indeed satisfy the DOZZ formula. This establishes an explicit link between probability theory (or statistical physics) and the so-called conformal bootstrap approach of LCFT.