Two smoothly embedded surfaces in a 4-manifold are called exotic if they are C^0 ambient isotopic without being smoothly equivalent. In the talk, I will briefly introduce this peculiar phenomenon and show how it is possible to construct an infinite family of pairwise exotic 2-knots (embedded spheres) in a smooth 4-manifold. Furthermore we will see that these examples bound a topologically embedded 3-ball but not a smoothly embedded one. Finally, the disks obtained by removing a ball from the ambient manifold are, in some sense, exotic. The talk is based on a joint work with Benyahia and Torres (see https://arxiv.org/abs/2206.09659).

Since Antiquity, mathematicians have been trying to find rational solutions to systems of polynomials equations (sometimes in a less precise language). Such questions are called Diophantine equations. During the 19th century, it became clear that the key to solving these equations lay in the study of fields and especially in a central tool: the Galois group, which measures how "symmetric" a field is. A natural yet unsolved question is the *inverse Galois problem*: can every finite group be realized as the Galois group of a finite extension of the field of rational numbers? In this talk, we will tell the story of how our viewpoint on this problem has shifted in the last century. We will first see that the problem is actually related to geometry: are there certain covers of the projective line which are defined by equations with rational coefficients? Then, we will see that covers may themselves be seen as points on a bigger geometrical space: the Hurwitz space. The inverse Galois problem is then asking about the existence of rational points on this space: this is itself a Diophantine equation! We will mention some of the achievements obtained using this paradigm, for example the realization of the Monster group as a Galois group.

Let S be a closed, orientable surface without boundary of genus at least two. In the 1950s Fenchel and Nielsen showed how to give geometrical coordinates to the Teichmueller space of S. We can reinterpret these coordinates algebraically by viewing Teichmueller space as the space of conjugacy classes of discrete, faithful, totally loxodromic, marked representations of the fundamental group of S to PSL(2,R). These coordinates have been generalised to representations to PSL(2,C), SL(3,R) and SU(2,1) by Kourouniotis, Tan, Goldman and Parker-Platis. In this talk, I will generalise to the case of SL(3,C), and show how these coordinates specialise in each of the cases listed above. The talk will be joint work with my PhD student Rodrigo Davila.

Consider the level-sets of smooth Gaussian fields on the d-dimensional Euclidean space, which defines a percolation model undergoing a non-trivial phase transition. Assuming that the field has a slow decay of correlations, we compute the exact exponential rate of decay for the size of subcritical connected components, which is proportional to the inverse correlation times the square of the distance to criticality. This differs drastically from fields with fast decay of correlations, for which the cluster size always decays exponentially, and the precise rate constant is not well understood. Our result is an evidence in support of physicists' predictions of the correlation length exponent for fields with slow algebraic decay of correlations, and also opens to way to the study of other large deviations questions in Gaussian percolation. In this talk, I aim at explaining how the existence of strong correlations leads to a (perhaps surprisingly) better understanding of these large deviation questions. Based on a joint work with Stephen Muirhead.

Après avoir rappelé la loi de Dirichlet et deux de ses fonctions de vraisemblance (en tant que la loi à priori conjuguée), nous donnons une généralisation de cette loi sur un graphe décomposable, sa structure Markovian/réseau bayésien et les contreparties (au cas classique) de ses fonctions de vraisemblance. Si le temps permet, nous parlons un peu de ses liens avec le monoïde partiellement commutatif de Cartier--Foata et d’arithmétique sur les randonnées. Travail en collaboration avec Kołodziejek et Wesolowski.

L'équation de fragmentation avec diffusion est une équation parabolique linéaire non locale qui décrit l'évolution de la fonction de distribution des volumes d'une population de particles dont le volume varie sous l'effet combiné d'un mécanisme de diffusion et d'un mécanisme de fragmentation spontanée. La compétition entre ces deux effets conduit à prédire la convergence vers un état stationnaire et c'est ce que nous étudions, après avoir établi le caractère bien posé de l'équation en utilisant la théorie des semi-groupes (collaboration avec Christoph Walker, Hannover).

Un modèle critique est, presque par définition, censé concilier une décroissance lente des corrélations entre observables microscopiques avec des propriétés de mélange pour des observables macroscopiques. Une formalisation relativement forte de cette notion de mélange est de dire que changer les conditions au bord du modèle ne peux pas avoir un effet singulier (au sens de la théorie de la mesure) sur le modèle loin du bord. Je présenterais une preuve de ce résultat pour l'arbre couvrant uniforme et le modèle de dimères avec condition au bord de Temperley.

Two-dimensional electron gases under a strong magnetic field have tremendously expanded our understanding of many-body physics, with the discovery of integer and fractional quantum Hall effects, together with chiral edge states, fractional excitations, anyons. Another striking effect is the strong coupling between charge and spin/valley degrees of freedom, which takes place near integer filling M of the magnetic Landau levels. More precisely, because of the large energy gap associated to cyclotron motion, any slow spatial variation of the spin background induces a variation of the electronic density proportional to the topological density of the spin background. Minimizing Coulomb energy leads to an exotic class of two-dimensional crystals, which exhibit a periodic non-collinear spin texture called a Skyrmion lattice. I will focus on the limit where a perfect SU(N) symmetry is assumed to hold in the space of internal electronic states. In this case, minimal energy Skyrmion lattices may be described in terms of holomorphic maps from a torus to the Grassmannian manifold Gr(M,N), such that the associated topological charge density is as uniform as possible. Such maps can be constructed by choosing N linearly independent sections of a rank M holomorphic vector bundle on a torus. The main outcome of such analysis is the existence of two regimes depending on whether the topological charge on the torus is smaller (unfrustrated case) or larger (frustrated case) than the number of internal states N accessible to electrons. In the former case, minimal energy Skyrmion lattices for M = 1 are obtained by choosing an orthonormal basis of sections of the associated line bundle with respect to the usual geometric quantization hermitian product. In the general M > 1 case, I will show that we can, to a large extent, identify minimal energy Skyrmion lattices by combining the solution of the M = 1 case with Atiyah’s explicit description of rank M vector bundles on a torus.

Le but de l'exposé sera de montrer que les revêtements ramifiés cycliques sont de très bons invariants pour les noeuds alternés. Plus précisément on a le théorème suivant : Soit K un noeud premier alterné et soit n>2. Si un noeud K' a le même revêtement ramifié cyclique à n feuillets que K, alors K et K' sont équivalents. Je commencerai avec des rappels sur les résultats connus portant sur la détermination des noeuds par leurs revêtements ramifiés cycliques. Ces résultats montrent en particulier que les hypothèses du théorème sont nécessaires. On verra aussi quelles particularités des noeuds alternés sont exploitées dans la preuve, dont la stratégie sera présentée à la fin de l'exposé.