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).

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.