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We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class. Rigidity for these coordinated frameworks is a generic property, and we characterize the rigid graphs in terms of redundant rigidity in the standard $d$-dimensional rigidity matroid. We also interpret our main results in terms of matroid unions., 19 pages, 5 figures

For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak--McCarthy., 15 pages, 9 figures. Minor revisions. To appear in Algebraic and Geometric Topology

We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)& \mbox{ for }xc, \end{array}\right. $$ where we assume that $\beta>1$. We show that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $2^\infty$ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum-Coullet-Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results., Comment: Accepted for publication in Communications in Mathematical Physics

We consider a $\varphi$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $\varphi(t)$ is a non-negative convex function vanishing only at $t=0$. We show that this property is always satisfied in dimension $n=2$, while in higher dimension it requires some further restriction on $\varphi$. In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e., when $\varphi(t) = ct^2$) in dimension $n\ge 4$. The validity of the quadratic rigidity, which we prove in dimension $n=2$, implies the existence of the trace of a divergence-measure vector field $\xi$ on a $\mathcal{H}^{1}$-rectifiable set $S$, as soon as its weak normal trace $[\xi\cdot \nu_S]$ is maximal on $S$. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense., Comment: 19 pages, 3 figures

For every finitely generated free group ▫$F$▫, we construct an irreducible open 3-manifold ▫$M_F$▫ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of ▫$M_F$▫ isomorphic to ▫$F$▫. The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group ▫$G$▫, an irreducible open 3-manifold ▫$M_G$▫ with end homogeneity group ▫$G$▫. The method used in the proof of our main result also shows that if ▫$G$▫ is a group with a Cayley graph in ▫$mathbb{R}^3$▫ such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold ▫$M_G$▫ with end homogeneity group ▫$G$▫.

International audience; Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional.We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy.Our methods provide alternative proofs in the classical Aviles-Giga context.

A classical result by K.B. Lee states that every group morphism between almost crystallographic groups is induced by an affine map on the nilpotent Lie group whereon these groups by definition act. It is the main technique for studying morphisms between virtually nilpotent groups, having important applications in fixed point theory, for example as a tool to compute Nielsen numbers for self-maps on infra-nilmanifolds. In this paper we generalize this result to morphisms between virtually polycyclic groups, which are known to act crystallographically by affine transformations on a nilpotent Lie group. The main method is to study the special class of translation-like actions, which behave well under taking subgroups and restricting the action to invariant right cosets., Comment: 19 pages

We prove that bounded Beltrami fields must be symmetric if a proportionality factor depends on 2 variables in the cylindrical coordinate and admits a regular level set diffeomorphic to a cylinder or a torus., 12 pages

For every Banach space (Y, parallel to . parallel to(Y)) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y) is an element of(0, infinity) with the following property. Suppose that n is an element of N and that X is an n-dimensional normed space with unit ball B-X. Then for every 1-Lipschitz function f : B-X -> Y and for every epsilon is an element of(0, 1/2] there exists a radius r >= exp (1/epsilon(cn)), a point x is an element of B-X with x + r B-X subset of B-X, and an affine mapping Lambda : X -> Y such that parallel to f (y) - Lambda (y)parallel to(Y)

In this paper, we consider Serrin’s overdetermined problems in warped product manifolds and we prove Serrin’s type rigidity results by using the [Formula: see text]-function approach introduced by Weinberger.

We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies., Comment: 87 pages, a computational mistake in the proof of Prop. 4.5 corrected. Changes compared to the published version in Annals of Probability are highlighted in color

An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove W*-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type $II_1$ equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction., Comment: v2: The examples in propositions 4.1 and 4.2 have been corrected, taking into account the exceptional isomorphisms for SO(n,m) when n+m=4. There were no other changes

We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated subgroup $\Gamma$ of $\mathcal{F}(\overline{\mathbb{Q}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X(\mathbb{C}_p)$ satisfying $nP\in\Gamma$ for some $n\in\mathbb{N}$, the minimal such orders $n$ are uniformly bounded whenever $X$ does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full $p$-adic formal Mordell--Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in $p$-adic deformations. Specifically, we do so in the context of the nearly ordinary $p$-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida., Comment: 19 pages, section 2.3. and consequences have been revised to correct an erroneous result

We investigate the zero set of a stationary Gaussian process on the real line, and in particular give lower bounds for the variance of the number of points on a large interval, in all generality. We prove that this point process is never hyperuniform, i.e. the variance is at least linear, and give necessary conditions to have linear variance, which are close to be sharp. We study the class of symmetric Bernoulli convolutions and give an example where the zero set is super rigid, weakly mixing, and not hyperuniform.; On {\'e}tudie l'ensemble des z{\'e}ros d'un processus gaussien stationnaire sur la ligne r{\'e}{\'e}lle, on donne en particulier des bornes inf{\'e}rieures pour la variance du nombre de points et de statistiques lin{\'e}aires sur un grand intervalle, en tout g{\'e}n{\'e}ralit{\'e}. On prouve que ce processus ponctuel n'est pas hyper-uniforme, i.e. la variance est toujours au moins lin{\'e}aire, et on donne une condition n{\'e}cessaire et suffisante pour avoir une variance lin{\'e}aire, qui est proche d'\^etre suffisante. On {\'e}tudie la classe des convolutions sym{\'e}triques de Bernoulli et on exhibe un exemple o\`u l'ensemble des z{\'e}ros est max-rigide, faiblement m{\'e}langeant, et non-hyper-uniforme.\\

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq \Omega \longrightarrow \mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot \Delta u = 0 \ \, \text{ in }\Omega. \] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^\infty$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $\mathrm{D} u$ is equal to one or $n=2$ and $u$ has the additively separated form $u(x,y)=f(x)+g(y)$. As a consequence we obtain corresponding flatness results for the images of $p$-Harmonic maps, $p\in [2,\infty]$., Comment: 15 pages, 2 figures

The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that $\langle u_v, u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent vertices in $G$. The study of the orthogonality dimension of graphs is motivated by various applications in information theory and in theoretical computer science. The contribution of the present work is two-fold. First, we prove that there exists a constant $c$ such that for every sufficiently large integer $t$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of an input graph over $\mathbb{R}$ is at most $t$ or at least $3t/2-c$. At the heart of the proof lies a geometric result, which might be of independent interest, on a generalization of the orthogonality dimension parameter for the family of Kneser graphs, analogously to a long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976). Second, we study the smallest possible orthogonality dimension over finite fields of the complement of graphs that do not contain certain fixed subgraphs. In particular, we provide an explicit construction of triangle-free $n$-vertex graphs whose complement has orthogonality dimension over the binary field at most $n^{1-\delta}$ for some constant $\delta >0$. Our results involve constructions from the family of generalized Kneser graphs and they are motivated by the rigidity approach to circuit lower bounds. We use them to answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field., Comment: 19 pages

In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem., Comment: 48 pages, minor corrections after refereeing process

International audience; We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type G2), but doubled in the certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.

Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $\pi_1(X,x)\to SL (3,{\mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions., Comment: v3, 49 pages; final version, to appear in Compositio Math

It is known that if the special automorphism group $\text{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we address the question whether this is the only possibility for the automorphism group $\text{Aut}(X)$ to act infinitely transitively on $X$. We show that this is the case provided $X$ admits a nontrivial $\mathbb{G}_a$- or $\mathbb{G}_m$-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity., 10 pages

We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1407.4763

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras., 14 pages. Comments welcome !

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of a parabolic subgroup. We show that the class of all right-angled Coxeter groups is not smooth and establish some general combinatorial criteria for such classes to be abstract elementary classes (AECs), for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translates into a machinery to build concrete examples of AECs satisfying given model-theoretic properties. We exhibit the power of our method by constructing three concrete examples of finitary classes. We show that the first and third classes are nonhomogeneous and that the last two are tame, uncountably categorical, and axiomatizable by a single L-omega 1,L- omega-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is kappa-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

The sine process is a rigid point process on the real line, which means that for almost all configurations $X$, the number of points in an interval $I = [-R,R]$ is determined by the points of $X$ outside of $I$. In addition, the points in $I$ are an orthogonal polynomial ensemble on $I$ with a weight function that is determined by the points in $X \setminus I$. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length $|I|=2R$ tends to infinity, thereby answering a question posed by A.I. Bufetov., Comment: 26 pages, no figures, revised version with Appendix B

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $\max_{jk} {\mathbb E} |X_{jk}|^{4+\delta} < \infty, \delta > 0$, it was proved in [G\"otze, Naumov and Tikhomirov, Bernoulli, 2018] that with high probability the typical distance between the Stieltjes transforms $m_n(z), z = u + i v$, of the empirical spectral distribution (ESD) and the Stieltjes transforms $m_{sc}(z)$ of the semicircle law is of order $(nv)^{-1} \log n$. The aim of this paper is to remove $\delta>0$ and show that this result still holds if we assume that $\max_{jk} {\mathbb E} |X_{jk}|^{4} < \infty$. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors., Comment: to appear in Journal of Theoretical Probability; 28 pages

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^*$-algebras with additional $C^*$-algebraic structure. Our approach naturally applies to algebras arising from $C^*$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs., Comment: 1 figure, 28 pages. To appear in Compositio Mathematica

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this article, we show that if $R$ tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs., Comment: 26 pages

If [math] is a henselian pair with an action of a finite group [math] and [math] is an integer coprime to [math] such that [math] , then the reduction map of mod- [math] equivariant [math] -theory spectra ¶ K G ( R ) ∕ n → ≃ K G ( R ∕ I ) ∕ n ¶ is an equivalence. We prove this by revisiting the recent proof of nonequivariant rigidity by Clausen, Mathew, and Morrow.

In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including sytems with degenerate singularities which are invariant under a torus action., Comment: 17 pages, 1 figure

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained., Bruno Pini Mathematical Analysis Seminar, Seminars 2018

We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of $n$ disks in the plane with generic radii cannot have more than $2n-3$ pairs of disks in contact. The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly $2n-3$ contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly, et al. on the number of contacts in a jammed packing of disks with generic radii., v2, edits for typographical errors and clarity of exposition. final version

We prove that the norm of the Euler class E for flat vector bundles is $2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev is sharp. We construct a new cocycle representing E and taking only the two values $\pm 2^{-n}$; a null-set obstruction prevents any cocycle from existing on the projective space. We establish the uniqueness of an antisymmetric representative for E in bounded cohomology., Comment: 19 pages

We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $p \neq 0$. This gives the same conclusion for Totaro's examples of Fano varieties violating Kodaira vanishing. We also provide an alternative purely geometric proof of non-liftability mod $p^2$ and to characteristic zero of the Frobenius homomorphism of a reductive group of non-exceptional type., Comment: 20 pages; comments welcome!

In this note we generalize the Ball-James rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint. In particular, we prove the rigidity for approximate solutions to the two-state inclusion with incompatible states for merely L 1 -bounded sequences. In this way, our theorem can be seen as a result of compensated compactness in the linear-growth setting.

International audience; We introduce an asymmetric distance function, which we call the ``left Hausdorff distance function", on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $\pi_1 \Sigma_g$, or factors through a finite cyclic group. For $g \geq 3$, all finite actions are trivial. This answers a question of Farb.

We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups. Further results include a Tits alternative and a description of the topological dynamics. We briefly discuss to what extent our results hold for more general topological curves., minor additions to v2

We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group $\Gamma$ in this class we determine the general structure of its possible lattice embeddings, i.e. of all compactly generated, locally compact groups that contain $\Gamma$ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature., Comment: incorporated suggestions and corrections from referee report; fixed an issue in proof of thm B and generalized Thm 5.11

We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by $C^1$ diffeomorphisms of the interval., Comment: A more detailed proof of Proposition 4.15 added

Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of $C^1$ diffeomorphisms. Furthermore, we show that the group of Lodha-Moore has no nonabelian $C^1$ action on the interval. We also show that many Monod's groups $H(A)$, for instance when $A$ is such that $\mathsf{PSL}(2,A)$ contains a rational homothety $x\mapsto \tfrac{p}{q}x$, do not admit a $C^1$ action on the interval. The obstruction comes from the existence of hyperbolic fixed points for $C^1$ actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable., 26 pages, 1 figure. Arithmetic conditions in the main theorems have been weakened

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)

Let $F$ be a non-discrete non-Archimedean local field. For any subset $S\subset F$ with finite Haar measure, there is a stationary determinantal point process on $F$ with correlation kernel $\widehat{\mathbb{1}}_S(x-y)$, where $\widehat{\mathbb{1}}_S$ is the Fourier transform of the indicator function $\mathbb{1}_S$. In this note, we give a geometrical condition on the subset $S$, such that the associated determinantal point process is rigid in the sense of Ghosh and Peres. Our geometrical condition is very different from the Euclidean case., 14 pages

International audience; This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

We study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singular. In the second case, we show that all reduced Palm measures, $\textit{of all orders}$, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary., Accepted version, accepted in Communications in Mathematical Physics

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u \i\ curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. Andr\'e and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results., Comment: Accepted for the EMS Surveys in Mathematical Sciences

A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset $S$ of $X$, and fix an endomorphism $A(x) \in \text{End}(E_x)$ for every $x \in S$. It is natural to ask when there is a logarithmic connection on $E$ singular over $S$ with residue $A(x)$ at every $x \in S$. We give a necessary and sufficient condition for it under the assumption that the residues $A(x)$ are rigid., Final version

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere., 9 pages

We present a new proof of a recent ✏-regularity of G. Tian and J.Viaclovsky. Our idea also provides a new proof of a classical result of M. Anderson about volume rigidity of Einstein manifolds. Eventually, we also obtain new rigidity results for critical metrics.

This paper is devoted to rigidity results for some elliptic PDEs and to optimal constants in related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. The largest value of this parameter provides an estimate for the optimal constant in the corresponding interpolation inequality. Our approach relies on a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works on rigidity. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constants in the interpolation inequalities.

International audience; Consider two $k$-gons $P$ and $Q$. We say that the billiard flows in $P$ and $Q$ are homotopically equivalent if the set of conjugacy classes in the fundamental group of $P$ which contain a periodic billiard orbit agrees with the analogous set for $Q$. We study this equivalence relationship and compare it to the equivalence relations, order equivalence and code equivalence, introduced in \cite{BT1,BT2}. In particular we show if $P$ is a rational polygon, and $Q$ is homotopically equivalent to $P$, then $P$ and $Q$ are similar, or affinely similar if all sides of $P$ are vertical and horizontal.