Motivated by the question, « what coordinates choices are `good’ for general relativity? » I will discuss gauge theories from the from the free evolution point of view, in which initial data satisfying constraints of a theory are given, and because the constraints are compatible with the field equations they remain so. I will present a model constrained Hamiltonian theory and identify a particular structure in the equations of motion from which statements can be made about the status of the initial value problem, just by examining a subset of the equations, called the pure gauge subsystem. I will then present the application of these ideas to GR, which results in a five parameter generalization of the `good’ gauge used in Choquet-Bruhat’s original treatment of the initial value problem.

Massive neutrinos make up a fraction of the dark matter, but due to their large thermal velocities, cluster significantly less than cold dark matter (CDM) on small scales. An accurate theoretical modelling of their effect during the non-linear regime of structure formation is required in order to properly analyse current and upcoming high-precision large-scale structure data, and constrain the neutrino mass. Taking advantage of the fact that massive neutrinos remain linearly clustered up to late times, we treat the linear growth of neutrino overdensities in a non-linear CDM background. The evolution of the CDM component is obtained via N-body computations. The smooth neutrino component is evaluated from that background by solving the Boltzmann equation linearised with respect to the neutrino overdensity. CDM and neutrinos are simultaneously evolved in time, consistently accounting for their mutual gravitational influence. This method avoids the issue of shot-noise inherent to particle-based neutrino simulations, and, in contrast with standard Fourier-space methods, properly accounts for the non-linear potential wells in which the neutrinos evolve. Inside the most massive late-time clusters, where the escape velocity is larger than the neutrino thermal velocity, neutrinos can clump non-linearly, causing the method to formally break down. It is shown that this does not affect the total matter power spectrum, which can be very accurately computed on all relevant scales up to the present time.

: I will discuss some examples where differential geometry, topology and combinatorics of polyhedra can benefit from each other, and review some tools that make this possible. In particular, I will focus on the following two topics

— the Hirsch conjecture, concerned with bounding the running time of the simplex algorithm, and an application of CAT(1)-geometry to this problem.

— two problems of Legendre–Steinitz and Perles–Shephard concerning realization spaces of polytopes, and their solution based on solving discrete PDEs.

All notions will be introduced in the talk; I intend to make a friendly introduction to the theory, so non-experts are welcome.

Since the double helical structure of DNA was discovered in 1953, decades of research into the behaviour of DNA have revealed many other fascinating phenomena about of the molecule of life. In 1965, Jerome Vinograd discovered that DNA in the polyoma virus is naturally found in a circular form. This work opened the gates to a new interdisciplinary field that studies the topology of DNA and its biological implications for the functionality of the molecule.

In this talk I will introduce topological aspects of

(i) bacterial site-specific recombination – an important cellular reaction that exchanges segments of DNA and is capable of creating (topological) knots and links;

(ii) developmental genome rearrangements in ciliates – organisms whose genome, during specialisation, has a complex spatial structure that can be measured by the genus; and

(iii) the organisation of the kinetoplast DNA in trypanosomes – whose kinetoplast genome (the `energy power-house’ of the cell) consists of thousands of DNA circles that are chained to each other, forming a non-trivial link that resembles a medieval mailchain armour.

Sketched out in 1992, selected by ESA in 1996, launched in 2009, Planck delivered on March 21st its first full sky maps of the millimetric emission at 9 frequencies, as well as those which follow from them, and in particular Planck map of the anisotropies of the Cosmic Microwave Background (CMB). The later displays minuscule variations as a function of the observing direction of the temperature of the fossile radiation around its mean temperature of 2.725K. I will briefly describe how these high resolution maps with a precision of a few parts in a million have been obtained, from collection to analysis of the first 500 billion samples of our HFI instrument.

CMB anisotropies reveal the imprint of the primordial fluctuations which initiate the growth of the large scale structures of the Universe, as transformed by their evolution, in particular during the first 370 000 years, i.e. till the Universe became transparent and the forming of the image we record today. The statistical characteristics of these anisotropies allow constraining jointly the physics of the creation of the primordial fluctuations and that of their evolution. They teach us the possible value of the parameters of the models which we confront to data. I will describe Planck estimates of the density of the constituents of the Universe (usual matter, cold dark matter or CDM, dark energy…), and their implication in terms of derived quantities like the expansion rate or the spatial curvature. I will review what we learnt on the generation of the fluctuation, and wil discuss extensions of the standard cosmological model, so called « Lambda-CDM », both in term of non minimal physical models — multi-field inflation for instance, or additional constituents – like cosmic strings or a fourth neutrino.

Finally, it will briefly describe other promising results on the matter distribution which is travelled through by the CMB image on its long 13.7 billion years trip towards us. I will mention in particular what we can learn on the dark matter distribution – which is detected through its distorting effet of the CMB image by gravitationnal lensing, or that of hot gaz, which is revealed by the spectral distortion it induces.

I will discuss joint work with Peter Scholze on the pro-étale topology of schemes. The main goal of this work is to give an intuitive perspective on the foundations of l-adic cohomology. After giving the basic definitions, I will explain the local contractibility of the pro-étale site of a scheme (and its homological consequences). Using this, we will see why the category of locally constant sheaves on the pro-étale topology gives rise to a fundamental group that is rich enough to detect all lisse l-adic sheaves through its representation theory (which fails for the groups constructed in SGA1 and SGA3 on non-normal schemes).

Crossovers are formed between homologous chromosomes in meiosis, the sex-specific cell division process during which a diploid cell gives rise to four gametes. In most species, nearby crossovers are rarer than if they were to arise independently. Such a phenomenon, discovered in 1913 by Sturtevant, has been coined « interference ».

In this talk I first review how mathematical modeling has been used to describe interference. Then I provide recent results based on analysing state of the art experimental data, giving novel insights into what new features must be included hereafter in improved modeling approaches. Last but not least, I will consider what happens in « recombinant inbred lines » where meiosis is repeated over many generations. By working within the framework of quantum field theory equations, I will provide a mathematical solution to an open problem going back to 1931, namely how to generalize to any number of loci the 2-locus formula of Haldane and Waddington.

The « very good » property for algebraic stacks was introduced by Beilinson and Drinfeld in their paper « The Quantization of Hitchin’s Integrable System and Hecke Eigensheaves ». They proved that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is « very good » as long as X has genus g > 1. We will introduce the « very good » property in the context of a group action on an algebraic variety, and prove it for a moduli space of parabolic bundles on P1 arising from quiver representations. As a special case, we will consider the « very good » property for the diagonal action of the group PGL(n) on a product of partial flag varieties and its relationship with the space of solutions to the additive Deligne-Simpson problem.