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A brief introduction to Massive GravityRecent years have seen the development of a range of modified gravity theories to tackle the dark energy and cosmological constant problem. An interesting class of such models are those in which the graviton effectively becomes massive, modifying gravity at large (cosmological) distances without changing physics at solar system and smaller scales. Such effective field theories have the Vainshtein mechanism built in, and are closely associated with Galileon theories.

In these lectures I will give a brief general overview of both soft and hard massive gravity theories, their origin from extra dimensional models, and the special class of ghost-free massive gravity models and their extensions with multiple massive (and massless) spin-2 states. I will review the Vainshtein mechanism, and the decoupling limits of these theories and how they are related to Galileons. I will then discuss a variety of implications of such theories.

• lecture 1: I will introduce the description of massless and massive spin-2 states, and explain how theseemerge naturally in extra dimensional models, and go on to give the description of so-called ghost-freeor $\Lambda_{3}$ theories of interacting massive spin-2 fields, aka massive gravity and its multi-gravity extensions.

• lecture 2: I will review some aspects of the phenomenology of the general class of massive gravity theories: screening - cosmology - black hole solutions, and if time go on to discuss more recent extensions which attempt to raise the cutoff, connections with UV completions such as TTbar deformations, and the significance of ‘positivity bounds’ applied to these effective theories.
Bacterial movement by run and tumble: models, patterns, pathways, scalesAt the individual scale, bacteria as E. coli move by performing so-called run-and-tumble movements. This means that they alternate a jump (run phase) followed by fast re-organization phase (tumble) in which they decide of a new direction for run. For this reason, the population is described by a kinetic-Botlzmann equation of scattering type. Nonlinearity occurs when one takes into account chemotaxis, the release by the individual cells of a chemical in the environment and response by the population.

These models can explain experimental observations, fit precise measurements and sustain various scales. They also allow to derive, in the diffusion limit, macroscopic models (at the population scale), as the Flux-Limited Keller-Segel system, in opposition to the traditional Keller-Segel system, this model can sustain robust traveling bands as observed in Adler’s famous experiment.

Furthermore, the modulation of the tumbles, can be understood using intracellular molecular pathways. Then, the kinetic-Boltzmann equation can be derived with a fast reaction scale. Long runs at the individual scale and abnormal diffusion at the population scale, can also be derived mathematically.
Table ronde : impacts de la crise sanitaire et économique sur le cyberespace
Cosmological effects in waveform modelingIn this course I will give an overview of different line of sight and environmental effects that are expected to modify/distort a gravitational wave (GW) signal emitted by an astrophysical source at cosmological distance. After an overview of the state of the art of GW observations, I will review the derivation of the waveform emitted by a binary system of compact objects in a flat background, in the newtonian approximation. I will then consider a cosmological context and introduce the concept of standard siren. Finally I will derive which is the expected distortion of an emitted waveform induced by the presence of peculiar velocities and (strong) gravitational lensing.

Lecture 1 — Gravitational waves-introduction
Lecture 2 — Gravitational waves-propagation effects in waveform modeling
Spectrum of the Möbius strip: true, fake and not-so-fakeThe Laplace-Beltrami operator in the curved Möbius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are approximated well by an unconventional flat model whose spectrum can be computed explicitly in terms of Mathieu functions. Contrary to the traditional flat Möbius strip, our effective model contains a geometric potential. A comparison of the three models is made and analytical results are accompanied by numerical computations.
Theory of gravitational waves and approximation methods in general relativityIn these lectures we give a general introduction to the theory of gravitational waves and the analytic approximation methods in general relativity. More precisely we focus on the theory which is necessary to accurately and reliably predict the gravitational waves generated by compact binary systems, made of black holes or neutron stars. The predictions are used in the form of gravitational-wave “templates” in the data analysis of the detectors LIGO, VIRGO, ... LISA. In particular we present the state-of-the-art on the post-Newtonian approximation in general relativity, which is the main tool for describing the famous gravitational wave “chirp” of compact binary systems. The outline of the lectures is :
1. Gravitational wave events
2. Methods to compute gravitational waves
3. Einstein quadrupole formalism
4. Post-Newtonian parameters
5. Finite size effects in compact binaries
6. Synergy with the effective field theory
7. Radiation reaction and balance equations.
Sharp inequalities for solutions to elliptic problems with mixed boundary conditionsWe show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on L1 comparison) between solutions to an elliptic partial differential equation on a smooth bounded set Ω with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as Ω. This includes for instance mixed problems.
The candidate EUCC certification scheme

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