The problem reads: Is there a general principle(s) that underlies the processes of human learning the following?

1. Native languages.

2. Mathematics.

3. Bipedal locomotion.

We shall bring (speculative) evidence for the existence of such principle(s) and indicate a possible direction of (quasi)mathematical modeling of (conjectural) universal learning.

Starting from the known unfaithful spinorial representations of the compact subalgebra KE(10) of the hyperbolic Kac Moody algebra E(10), new fermionic `higher spin’ representations are constructed in a second quantized framework.

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.

Alternative Splicing is an important source of variety in the process of protein expression. It is deregulated in human diseases including cancer. The precise mechanism through which a splicing variant is selected for expression is not known at present. Here, we demonstrate that alternative splicing involves the RNA interference protein machinery, previously discovered as important for gene regulation at a distinct step, i. e. at the translational level.

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.

We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

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The Hodge index theorem of Faltings and Hriljac asserts that the Neron-Tate height pairing on a projective curve over a number field is equal to a certain intersection pairing in the setting of Arakelov geometry. In the talk, I will present an extension of this result to adelic line bundles on higher dimensional varieties over finitely generated fields. Then I will talk about its relation to the non-archimedean Calabi-Yau theorem and its application to algebraic dynamics. This is a joint work with Shou-Wu Zhang.

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