Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics.  

 

We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $gamma=sqrt{8/3}$.  

 

Based on joint work with Scott Sheffield.

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics.  

 

We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $gamma=sqrt{8/3}$.  

 

Based on joint work with Scott Sheffield.

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).

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 the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights.
 
Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties.  Many Feynman amplitudes in quantum field theories are of this type.  P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'.  One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power.
 
Topics to be addressed will include:
 
1)  A Galois theory of periods, multiple zeta values.
2)  Parametric representation of Feyman integrals and their mixed Hodge structures.
3)  Operads and the principle of small graphs.
4)  The cosmic Galois group: results, counterexamples and conjectures.

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights.
 
Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties.  Many Feynman amplitudes in quantum field theories are of this type.  P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'.  One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power.
 
Topics to be addressed will include:
 
1)  A Galois theory of periods, multiple zeta values.
2)  Parametric representation of Feyman integrals and their mixed Hodge structures.
3)  Operads and the principle of small graphs.
4)  The cosmic Galois group: results, counterexamples and conjectures.

A la fin des années 90, les liens entre transport optimal, entropie et courbure de Ricci étaient mis au jour (Jordan-Kinderlehrer-Otto, Otto-Villani); quelques années plus tard, ce liens étaient exploités pour démarrer l’étude systématique du « point de vue synthétique » de la courbure de Ricci (Lott-Sturm-Villani), un domaine en progression constante depuis lors. La résolution récente de plusieurs questions ouvertes majeures suggère que le moment est venu de faire un bilan; c’est l’objectif de ce cours. On y trouvera notamment une nouvelle preuve du théorème d’isopérimétrie de Lévy-Gromov (Cavalletti-Mondino).

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights.
 
Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties.  Many Feynman amplitudes in quantum field theories are of this type.  P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'.  One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power.
 
Topics to be addressed will include:
 
1)  A Galois theory of periods, multiple zeta values.
2)  Parametric representation of Feyman integrals and their mixed Hodge structures.
3)  Operads and the principle of small graphs.
4)  The cosmic Galois group: results, counterexamples and conjectures.

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights.
 
Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties.  Many Feynman amplitudes in quantum field theories are of this type.  P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'.  One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power.
 
Topics to be addressed will include:
 
1)  A Galois theory of periods, multiple zeta values.
2)  Parametric representation of Feyman integrals and their mixed Hodge structures.
3)  Operads and the principle of small graphs.
4)  The cosmic Galois group: results, counterexamples and conjectures.

Le cours sera consacré à la dérivation de modèles non linéaires (de type Schrödinger) à partir de la théorie quantique linéaire, dans une limite de type « champ moyen » où l’interaction entre les particules est faible et le nombre de particules est grand. La plus grande partie du cours concernera les états invariants du système (minimiseurs et, surtout, mesures de Gibbs), mais je ferai également des commentaires sur le problème dépendant du temps. Nous utiliserons des outils très variés, allant du calcul variationnel à la physique statistique, en passant par l’analyse semi-classique, les inégalités fonctionnelles et les propriétés fines de l’entropie.

Le cours sera consacré à la dérivation de modèles non linéaires (de type Schrödinger) à partir de la théorie quantique linéaire, dans une limite de type « champ moyen » où l’interaction entre les particules est faible et le nombre de particules est grand. La plus grande partie du cours concernera les états invariants du système (minimiseurs et, surtout, mesures de Gibbs), mais je ferai également des commentaires sur le problème dépendant du temps. Nous utiliserons des outils très variés, allant du calcul variationnel à la physique statistique, en passant par l’analyse semi-classique, les inégalités fonctionnelles et les propriétés fines de l’entropie.

Le cours sera consacré à la dérivation de modèles non linéaires (de type Schrödinger) à partir de la théorie quantique linéaire, dans une limite de type « champ moyen » où l’interaction entre les particules est faible et le nombre de particules est grand. La plus grande partie du cours concernera les états invariants du système (minimiseurs et, surtout, mesures de Gibbs), mais je ferai également des commentaires sur le problème dépendant du temps. Nous utiliserons des outils très variés, allant du calcul variationnel à la physique statistique, en passant par l’analyse semi-classique, les inégalités fonctionnelles et les propriétés fines de l’entropie.