The goal of the first part of the course is to describe and compare various cohomology theories for algebraic varieties endowed with global function. In the second part infinite-dimensional applications will be discussed, including non-perturbative quantization of algebraic symplectic varieties.

The goal of the first part of the course is to describe and compare various cohomology theories for algebraic varieties endowed with global function. In the second part infinite-dimensional applications will be discussed, including non-perturbative quantization of algebraic symplectic varieties.

The recent observation of gravitational wave signals from inspiralling and coalescing binary black holes has been significantly helped, from the theoretical side, by the availability of analytical results on the motion and gravitational radiation of binary systems.

The course will deal with the Effective One-Body (EOB) theory of the motion and radiation of binary systems, and explain the links between this formalism and various classical and quantum approaches to gravitationally interacting two-body systems, from traditional post-Newtonian computations of the effective two-body action to quantum gravitational scattering amplitudes.

The following analytical techniques will be reviewed ab initio:

Matched Asymptotic Expansions approach to the motion of black holes and neutron stars;
post-Newtonian theory of the motion of point particles;
Multipolar post-Minkowskian theory of the gravitational radiation of general sources;
Effective One-Body (EOB) theory of the motion and radiation of binary systems.

The EOB formalism was initially based on a resummation of post-Newtonian-expanded results. The post-Newtonian approach assumes small gravitational potentials and small velocities, and loses its validity during the last orbits before the merger of black holes. The resummed EOB approach was able to extend the validity of the post-Newtonian description of the motion and radiation of binary black holes to the strong-field, high-velocity regime reached during the last orbits, and the merger. EOB theory initially used a dictionary to translate post-Newtonian-expanded results on (slow-motion) bound states of gravitationally interacting binary systems into the (resummed) Hamiltonian of a particle moving in an effective external gravitational field.

The second part of the course will present the recent extension of EOB theory to the description of (classical) scattering states within the post-Minkowskian approach which does not assume that velocities are small. This led to new insights in the high-energy limit of gravitational scattering and opened the way to transcribe quantum gravitational scattering amplitudes into their EOB Hamiltonian description. For instance, some two-loop ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano could be transcribed into an improved knowledge of the high-energy limit of the classical gravitational interaction of two black holes. This leads also to interesting predictions about a linear-Regge-trajectory behavior of high-angular-momenta, high-energy circular orbits.

The recent observation of gravitational wave signals from inspiralling and coalescing binary black holes has been significantly helped, from the theoretical side, by the availability of analytical results on the motion and gravitational radiation of binary systems.

The course will deal with the Effective One-Body (EOB) theory of the motion and radiation of binary systems, and explain the links between this formalism and various classical and quantum approaches to gravitationally interacting two-body systems, from traditional post-Newtonian computations of the effective two-body action to quantum gravitational scattering amplitudes.

The following analytical techniques will be reviewed ab initio:

Matched Asymptotic Expansions approach to the motion of black holes and neutron stars;
post-Newtonian theory of the motion of point particles;
Multipolar post-Minkowskian theory of the gravitational radiation of general sources;
Effective One-Body (EOB) theory of the motion and radiation of binary systems.

The EOB formalism was initially based on a resummation of post-Newtonian-expanded results. The post-Newtonian approach assumes small gravitational potentials and small velocities, and loses its validity during the last orbits before the merger of black holes. The resummed EOB approach was able to extend the validity of the post-Newtonian description of the motion and radiation of binary black holes to the strong-field, high-velocity regime reached during the last orbits, and the merger. EOB theory initially used a dictionary to translate post-Newtonian-expanded results on (slow-motion) bound states of gravitationally interacting binary systems into the (resummed) Hamiltonian of a particle moving in an effective external gravitational field.

The second part of the course will present the recent extension of EOB theory to the description of (classical) scattering states within the post-Minkowskian approach which does not assume that velocities are small. This led to new insights in the high-energy limit of gravitational scattering and opened the way to transcribe quantum gravitational scattering amplitudes into their EOB Hamiltonian description. For instance, some two-loop ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano could be transcribed into an improved knowledge of the high-energy limit of the classical gravitational interaction of two black holes. This leads also to interesting predictions about a linear-Regge-trajectory behavior of high-angular-momenta, high-energy circular orbits.

The recent observation of gravitational wave signals from inspiralling and coalescing binary black holes has been significantly helped, from the theoretical side, by the availability of analytical results on the motion and gravitational radiation of binary systems.

The course will deal with the Effective One-Body (EOB) theory of the motion and radiation of binary systems, and explain the links between this formalism and various classical and quantum approaches to gravitationally interacting two-body systems, from traditional post-Newtonian computations of the effective two-body action to quantum gravitational scattering amplitudes.

The following analytical techniques will be reviewed ab initio:

Matched Asymptotic Expansions approach to the motion of black holes and neutron stars;
post-Newtonian theory of the motion of point particles;
Multipolar post-Minkowskian theory of the gravitational radiation of general sources;
Effective One-Body (EOB) theory of the motion and radiation of binary systems.

The EOB formalism was initially based on a resummation of post-Newtonian-expanded results. The post-Newtonian approach assumes small gravitational potentials and small velocities, and loses its validity during the last orbits before the merger of black holes. The resummed EOB approach was able to extend the validity of the post-Newtonian description of the motion and radiation of binary black holes to the strong-field, high-velocity regime reached during the last orbits, and the merger. EOB theory initially used a dictionary to translate post-Newtonian-expanded results on (slow-motion) bound states of gravitationally interacting binary systems into the (resummed) Hamiltonian of a particle moving in an effective external gravitational field.

The second part of the course will present the recent extension of EOB theory to the description of (classical) scattering states within the post-Minkowskian approach which does not assume that velocities are small. This led to new insights in the high-energy limit of gravitational scattering and opened the way to transcribe quantum gravitational scattering amplitudes into their EOB Hamiltonian description. For instance, some two-loop ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano could be transcribed into an improved knowledge of the high-energy limit of the classical gravitational interaction of two black holes. This leads also to interesting predictions about a linear-Regge-trajectory behavior of high-angular-momenta, high-energy circular orbits.

The recent observation of gravitational wave signals from inspiralling and coalescing binary black holes has been significantly helped, from the theoretical side, by the availability of analytical results on the motion and gravitational radiation of binary systems.

The course will deal with the Effective One-Body (EOB) theory of the motion and radiation of binary systems, and explain the links between this formalism and various classical and quantum approaches to gravitationally interacting two-body systems, from traditional post-Newtonian computations of the effective two-body action to quantum gravitational scattering amplitudes.

The following analytical techniques will be reviewed ab initio:

Matched Asymptotic Expansions approach to the motion of black holes and neutron stars;
post-Newtonian theory of the motion of point particles;
Multipolar post-Minkowskian theory of the gravitational radiation of general sources;
Effective One-Body (EOB) theory of the motion and radiation of binary systems.

The EOB formalism was initially based on a resummation of post-Newtonian-expanded results. The post-Newtonian approach assumes small gravitational potentials and small velocities, and loses its validity during the last orbits before the merger of black holes. The resummed EOB approach was able to extend the validity of the post-Newtonian description of the motion and radiation of binary black holes to the strong-field, high-velocity regime reached during the last orbits, and the merger. EOB theory initially used a dictionary to translate post-Newtonian-expanded results on (slow-motion) bound states of gravitationally interacting binary systems into the (resummed) Hamiltonian of a particle moving in an effective external gravitational field.

The second part of the course will present the recent extension of EOB theory to the description of (classical) scattering states within the post-Minkowskian approach which does not assume that velocities are small. This led to new insights in the high-energy limit of gravitational scattering and opened the way to transcribe quantum gravitational scattering amplitudes into their EOB Hamiltonian description. For instance, some two-loop ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano could be transcribed into an improved knowledge of the high-energy limit of the classical gravitational interaction of two black holes. This leads also to interesting predictions about a linear-Regge-trajectory behavior of high-angular-momenta, high-energy circular orbits.

Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.

Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.

Paraphrasing Alexander Polyakov, « Conformal Field Theory is a way to learn about elementary particles by studying boiling water ».
There is a technical statement behind this joke: Euclidean Conformal Field Theory, under certain conditions, can be rotated to the Lorentzian signature, and vice versa. This means that even statistical physicists studying finite-temperature phase transitions on a lattice should learn about the Minkowski space! The goal of this course will be to explain various classical and recent results pertaining to this somewhat surprising conclusion.

Plan of the course:
– elementary introduction to Euclidean CFT in d>2 dimensions
– the Osterwalder-Schrader theorem about the Wick rotation of general reflection-positive Euclidean Quantum Field Theories, and its limitations
– the Luescher-Mack theorem about continuation of CFT correlation functions to the Lorentzian cylinder, and its limitations
– recent results about the analytic structure of Lorentzian CFT correlators

Paraphrasing Alexander Polyakov, « Conformal Field Theory is a way to learn about elementary particles by studying boiling water ».
There is a technical statement behind this joke: Euclidean Conformal Field Theory, under certain conditions, can be rotated to the Lorentzian signature, and vice versa. This means that even statistical physicists studying finite-temperature phase transitions on a lattice should learn about the Minkowski space! The goal of this course will be to explain various classical and recent results pertaining to this somewhat surprising conclusion.

Plan of the course:
– elementary introduction to Euclidean CFT in d>2 dimensions
– the Osterwalder-Schrader theorem about the Wick rotation of general reflection-positive Euclidean Quantum Field Theories, and its limitations
– the Luescher-Mack theorem about continuation of CFT correlation functions to the Lorentzian cylinder, and its limitations
– recent results about the analytic structure of Lorentzian CFT correlators

Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.

Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.