Publications

Quelques aspects de l’élaboration de l’analyse fonctionnelle jusqu’à Banach. Rôle des opérations dans une approche générale

La these se divise en deux parties. Dans la premiere, nous montrons comment les notions de multiplicite et d'ensemble frontiere sont intimement liees aux classes a#m#,#n. Plus precisement nous donnons une condition suffisante en termes de multiplicite pour qu'une contraction appartienne a une classe donnee. Nous conjecturons que ces criteres caracterisent en particulier les classes a#1#,#n. Ces resultats permettent deja de construire avec une tres grande efficacite de multiple exemples d'operateurs dans une classe donnee et de tester certaines conjectures classiques. La deuxieme partie met en place le cadre general permettant d'etudier l'algebre duale engendree par une paire commutative de contractions. En particulier nous caracterisons les paires continues (ie. Possedant un calcul fonctionnel h#(t#2) faible#* continu en termes de mesure spectrale et de bande de mesures. Ensuite, nous donnons des resultats de factorisation fonctionnelle dans l#1() ou est dans une large classe de mesures incluant les mesures spectrales du dessus. Enfin nous mettons en place un processus d'approximation qui est une variante de la technique de s. Brown, basee sur une hypothese de type e#r##,#. On demontre alors qu'une paire de contractions a calcul isometrique avec cette propriete possede un sous-espace invariant non trivial.

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

For a fixed finite group G, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched G-covers of the projective line. The main focus is on determining whether components obtained by "gluing" two other components, both defined over a number field K, are also defined over K. The article presents a list of situations in which a positive answer is obtained. As an application, when G is a semi-direct product of symmetric groups or the Mathieu group M 23 , components defined over Q of small dimension (6 and 4, respectively) are shown to exist.

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

We study the dynamics of random walks hopping on homogeneous hyper-cubic lattices and multiplying at a fertile site. In one and two dimensions, the total number $\mathcal{N}(t)$ of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate $\mu$ at the fertile site. When $d>d_c=2$, the number of walkers may remain finite forever for any $\mu$; it surely remains finite when $\mu\leq \mu_d$. We determine $\mu_d$ and show that $\langle\mathcal{N}(t)\rangle$ grows exponentially if $\mu>\mu_d$. The distribution of the total number of walkers remains broad when $d\leq 2$, and also when $d>2$ and $\mu>\mu_d$. We compute $\langle \mathcal{N}^m\rangle$ explicitly for small $m$, and show how to determine higher moments. In the critical regime, $\langle \mathcal{N}\rangle$ grows as $\sqrt{t}$ for $d=3$, $t/\ln t$ for $d=4$, and $t$ for $d>4$. Higher moments grow anomalously, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{2m-1}$, in the critical regime; the growth is normal, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{m}$, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if $d>2$, there is again a critical multiplication rate, above which $\langle\mathcal{N}(t)\rangle$ grows linearly (not exponentially) in time; when $d\leq d_c=2$, the leading behavior is independent on $\mu$ and $\langle\mathcal{N}(t)\rangle$ exhibits a sub-linear growth.

We study the dynamics of random walks hopping on homogeneous hyper-cubic lattices and multiplying at a fertile site. In one and two dimensions, the total number $\mathcal{N}(t)$ of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate $\mu$ at the fertile site. When $d>d_c=2$, the number of walkers may remain finite forever for any $\mu$; it surely remains finite when $\mu\leq \mu_d$. We determine $\mu_d$ and show that $\langle\mathcal{N}(t)\rangle$ grows exponentially if $\mu>\mu_d$. The distribution of the total number of walkers remains broad when $d\leq 2$, and also when $d>2$ and $\mu>\mu_d$. We compute $\langle \mathcal{N}^m\rangle$ explicitly for small $m$, and show how to determine higher moments. In the critical regime, $\langle \mathcal{N}\rangle$ grows as $\sqrt{t}$ for $d=3$, $t/\ln t$ for $d=4$, and $t$ for $d>4$. Higher moments grow anomalously, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{2m-1}$, in the critical regime; the growth is normal, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{m}$, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if $d>2$, there is again a critical multiplication rate, above which $\langle\mathcal{N}(t)\rangle$ grows linearly (not exponentially) in time; when $d\leq d_c=2$, the leading behavior is independent on $\mu$ and $\langle\mathcal{N}(t)\rangle$ exhibits a sub-linear growth.

Given a diffeomorphism which is homotopic to the identity from the [Formula: see text]-torus to itself, we construct an isotopy whose norm is controlled by that of the diffeomorphism in question.

We introduce a new functional inspired by Perelman's $\lambda$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $\lambda_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $\lambda_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $\lambda_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluingperturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluingperturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

Lectures grothendieckiennes rassemble les textes qui font suite à un séminaire qui s’est tenu au département de mathématiques de l’École Normale Supérieure de 2017 à 2018. Le livre présente une pensée complexe à l’œuvre, celle de l’un des mathématiciens les plus influents et énigmatiques du 20e siècle : Alexander Grothendieck. Les auteurs, Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Laforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz et Fernando Zalamea, dévoilent à leur façon les conséquences mathématiques ou philosophiques que l’on peut tirer d’une œuvre monumentale qui a transformé le paysage mathématique du 20e siècle et qui a probablement ouvert une nouvelle ère mathématique que nous avons seulement commencé à explorer. Préface de Peter Scholze.

Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\overline{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\overline H}(y,q,p)$ and thus yields an ``effective Hamiltonian''. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\overline H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard.

Soit $X$ une vari\'et\'e projective et lisse, d\'efinie sur un corps de nombres. Sous l'hypoth\`ese $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\`ene et Raskind ont d\'emontr\'e que le sous-groupe de torsion $CH^2(X)_{tors}$ du groupe de Chow en codimension $2$ est fini. Dans cette note, on donne des bornes uniformes pour le groupe fini $CH^2(X)_{tors}$ quand $X$ varie en famille. Let $X$ be a smooth projective variety defined over a number field. Assuming $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\`ene and Raskind proved that the torsion subgroup $CH^2(X)_{tors}$ in the Chow group of cycles of codimension $2$ is finite. In this note, we give uniform bounds for the finite group $CH^2(X)_{tors}$ when $X$ varies in a family.

Let $\mathscr{M}_{\mathbb R}$ be the moduli space of smooth real binary quintics. We show that each connected component of $\mathscr{M}_{\mathbb R}$ is isomorphic to an arithmetic quotient of an open subset of the real hyperbolic plane. Our main result is that the induced metric on $\mathscr{M}_{\mathbb R}$ extends to a complete hyperbolic metric on the moduli space of stable real binary quintics, making it isometric to the hyperbolic triangle of angles $\pi/3$, $\pi/5$ and $\pi/10$.

We construct non-arithmetic lattices in $PO(n,1)$ for each integer $n$ greater than one. To do so, we provide a construction of glueing together complete real hyperbolic orbifolds along totally geodesic subspaces. The hyperbolic orbifolds that we glue are those that arise from anti-unitary involutions on a unitary Shimura variety. This construction generalizes a method of Allcock-Carlson-Toledo, who glued hyperbolic quotient spaces in dimensions two, three and four.

We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds $A$ whose real locus $A(\mathbb R)$ is connected, and for real abelian threefolds $A$ which are a product $A = B \times E$ of an abelian surface $B$ and an elliptic curve $E$ with connected real locus $E(\mathbb R)$. Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the general case to the Jacobian case.

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove the integral Hodge conjecture for one-cycles on complex Jacobian varieties, and the real integral Hodge conjecture modulo torsion for real abelian threefolds. For the theory of real algebraic cycles, and for several other purposes in real algebraic geometry, it is useful to have moduli spaces of real varieties to our disposal. Insight in the topology of a real moduli space provides insight in the geometry of a real variety that defines a point in it, and the other way around. In the moduli space of real abelian varieties, as well as in the Torelli locus contained in it, we prove density of the set of moduli points attached to abelian varieties containing an abelian subvariety of fixed dimension. Moreover, we provide the moduli space of stable real binary quintics with a hyperbolic orbifold structure, compatible with the period map on the locus of smooth quintics. This identifies the moduli space of stable real binary quintics with a non-arithmetic ball quotient.

Abstract For fixed k < g and a family of polarized abelian varieties of dimension g over ${{\mathbb{R}}}$, we give a criterion for the density in the parameter space of those abelian varieties over ${{\mathbb{R}}}$ containing a k-dimensional abelian subvariety over ${{\mathbb{R}}}$. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension g and density of real algebraic curves mapping non-trivially to real k-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the real setting results by Colombo and Pirola [10]. We then consider the real locus of an algebraic stack over $\mathbb{R}$, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for ${{\mathcal{M}}}_g$ and ${{\mathcal{A}}}_g$, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross–Harris [13] and Seppälä–Silhol [23].

We address the problem of deterministic sequential estimation for a nonsmooth dynamics governed by a variational inequality. An example of such dynamics is the Skorokhod problem with a reflective boundary condition. For smooth dynamics, Mortensen introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with Hijab in 1980, several authors established a connection between Mortensen's approach and the vanishing noise limit of the robust form of the so-called Zakai equation. In this paper, we investigate to what extent these methods can be developed for dynamics governed by a variational inequality. On the one hand, we address this problem by relaxing the inequality constraint by penalization: this yields an approximate Mortensen estimator relying on an approximating smooth dynamics. We verify that the equivalence between the deterministic and stochastic approaches holds through a vanishing noise limit. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. In contrast to the case of smooth dynamics, the zero-noise limit of the robust form of the Zakai equation cannot be understood in our case from the Bellman equation on the value function arising in Mortensen's procedure. This unveils a violation of equivalence for dynamics governed by a variational inequality between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.

Kudla and Millson constructed a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space using Howe's differential operators. It is a crucial ingredient in their theory of theta lifting. This form can be seen as a Thom form of a real oriented vector bundle. In \cite{mq} Mathai and Quillen constructed a {\em canonical} Thom form and we show how to recover the Kudla-Millson form via their construction. A similar result was obtained by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to an arbitrary signature.

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as \[ \delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0, \] where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as \[ \delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0, \] where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as \[ \delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0, \] where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in R P 2 \mathbb {RP}^2 in the spirit of Hilbert’s 16th problem.

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic Del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.

Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.

Soit k un corps ultramétrique complet. Nous démontrons un substitut au théorème de la fibre réduite (de Bosch, Lütkebohmert et Raynaud) valable pour tout morphisme Y Ñ X plat et à fibres géométriquement réduites entre espaces k-affinoïdes au sens de Berkovich, sans supposer que X et Y sont stricts ni que la dimension relative de Y sur X est constante. Nous ne faisons pas appel au théorème de la fibre réduite original, ni aux techniques ou au langage de la géométrie formelle. Notre énoncé est formulé en termes de réduction graduée à la Temkin ; notre preuve repose sur un théorème de finitude de Grauert et Remmert et sur la théorie de la réduction (graduée) des germes d'espaces analytiques, due à Temkin. Abstract (Reduction of affinoid spaces in family).-Let k be a nonarchimedean complete field. We prove a substitute for the reduced fiber theorem (of Bosch, Lütkebohmert and Raynaud) that holds for every morphism Y Ñ X flat and with geometrically reduced fibers between k-affinoid spaces in the sense of Berkovich, without assuming that X and Y are strict, nor that the relative dimension of Y over X is constant. We do not use the original reduced fiber theorem, nor the language or the techniques of formal geometry. Our statement is formulated in terms of Temkin's graded reduction; our proof rests on a finiteness result of Grauert and Remmert and on Temkin's theory of (graded) reduction of germs of analytic spaces. Lors de la rédaction de cet article, l'auteur a bénéficié du soutien de l'ANR à travers les projets Valuations, combinatoire et théorie des modèles (ANR-13-BS01-0006), et Définissabilité en géométrie non archimédienne (ANR-15-CE40-0008), ainsi que de celui de l'IUF dont il était membre junior d'octobre 2012 à octobre 2017. Il a aussi profité en mars 2019 de l'hospitalité de l'université hébraïque de Jérusalem, avec le soutien du projet ERC Consolidator 770922 (BirNonArchGeom) de Michael Temkin.

Nous développons dans cet article des techniques d'aplatissement des faisceaux cohérents en géométrie de Berkovich, en nous inspirant de la stratégie générale que Raynaud et Gruson ont mise en oeuvre pour traiter le problème analogue en théorie des schémas. Nous donnons ensuite quelques applications à l'étude des morphismes entre espaces analytiques compacts, et obtenons notamment une description de l'image d'un tel morphisme. Abstract (Blow-ups flattening in non-archimedean analytic geometry) We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. We then give some applications to the study of morphisms between compact analytic spaces; among other things, we get a description of the image of such a morphism.

We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration.

We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.

We prove that the semigroup generated by a reversible Mealy automaton contains a free subsemigroup of rank two if and only if it contains an element of infinite order.

Let A ∼ = k〈X 〉/I be an associative algebra. A finite word over alphabet X is I-reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal Ireducible word. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth. Résumé. Soit A ∼ = k〈X 〉/I une algèbre associative. Un mot fini sur l'alphabet X est I-réductible si son image dans A est une combinaison linéaire k de mots de longueur lexicographiquement moindre. Une obstruction dans un mot minimal I-réductible. Si le nombre d'obstructions est fini, alors I a une base finie Gröbner, et le mot problème pour l'algèbre est décidable. Une fonction co-croissance est le nombre d'obstructions de longueur ≤ n. Nous montrons que la fonction de co-croissance d'une algèbre finement présentée est soit bornée, soit au moins logarithmique. Nous montrons également qu'un mot uniformément récurrent a au moins une co-croissance logarithmique.

We prove that for a compact metric space the property of having finite covering dimension is equivalent to the existence of a total order with finite snake number.

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler-Fourier and Euler-Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated to (sub)level-sets persistence of random Gaussian filtrations.

Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.

Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_\sigma$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.

Given a countable group $G$ and a $G$-flow $X$, a measure $\mu\in P(X)$ is called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal $G$-flow, for any group $G$, which does not admit a characteristic measure. We construct for every countable group $G$ such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group $G$ and a collection of infinite subgroups $\{\Delta_i: i\in I\}$, when is there a faithful $G$-flow for which every $\Delta_i$ acts minimally?