This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.

This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.

Holographic View of Singularities in General Relativity

 

Résumé : I will discuss new features which emerge when one studies several types of singularities present in General Relativity using methods stemming from the AdS/CFT correspondence. Some of the issues involved are the black hole information "paradox", complementarity and the nature and properties of space like singularities. I will attempt to present in each of the lectures problems which I feel need further study.

Holographic View of Singularities in General Relativity (Part II)

I will discuss new features which emerge when one studies several types of singularities present in General Relativity using methods stemming from the AdS/CFT correspondence. Some of the issues involved are the black hole information « paradox », complementarity and the nature and properties of space like singularities. I will attempt to present in each of the lectures problems which I feel need further study.

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.

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.