Localizing solutions of the Einstein constraint equations

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of $N$-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time $t\in(0,T)$, in striking contrast with the Newtonian gravity scenario.Comment: Final version, to appear on Inventiones Mathematica

Finer estimates on the 22-dimensional matching problem

We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semi-discrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.Comment: 25 pages, Revised versio

Estimates and rigidity for stable solutions to some nonlinear elliptic problems

My thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabré, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincaré sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.Mi tesis se encaja en el estudio de las EDPs elípticas. Está dividida en dos partes: la primera trata una ecuación no-lineal con el p-Laplaciano, la segunda de un problema no-local. En la primera parte, estudiamos la regularidad de las soluciones estables de una ecuación no lineal con el p-Laplaciano en un dominio acotado. Esta ecuacion es la versión no-lineal de la ámpliamente estudiada ecuacion semilineal con el Laplaciano. Cabré, Figalli, Ros-Oton, y Serra han demostrado recientemente que las soluciones estables de las ecuaciones semilineales son acotadas, y por tanto regulares, hasta la dimensión 9. Este resultado es optimal. En el caso del p-Laplaciano, la regularidad de las soluciones estables se conjetura de ser cierta hasta una dimension critica y, de hecho, se conocen ejemplos de soluciones no acotadas cuando la dimension llega al valor critico. Además, se ha demostrado que en el caso radial o assumiendo hipótesis fuertes sobre la no-linealidad las soluciones estables son acotadas hasta la dimension critica. En el primer capítulo, demostramos que las soluciones estables son acotadas, bajo una nueva condición en n y p, que es optimal en el caso radial, y más restrictiva en el caso general. Esta investigación mejora conocidos resultados del tema y es el primer ejemplo, para el p-Laplaciano, de un método que produce un resultado para el caso general y un resultado optimal en el caso radial. En la primera parte, nos ocupamos también de las desigualdades funcionales del tipo Hardy y Sobolev sobre hipersuperfícies del espacio Euclideo, todas conteniendo un término de curvatura media. Nuestra motivación proviene de varias apliaciones que tienen estas desigualdades en el estudio de estimaciones para las soluciones estables. En detalle, damos una demostración simple de la conocida desigualdad de Michael-Simon y Allard, obtenemos dos formas nuevas de la desigualdad de Hardy sobre hipersuperfícies, y otra desigualdad de Hardy-Poincaré. En la segunda parte, nos ocupamos de un problema de Dirichlet-Neumann que emerge de un modelo para las ondas en el agua. El sistema se describe con una ecuación de difusión en una tira de altura fija, que contiene un parámetro a en (-1,1). La parte superior de la tira es dotada de una condicion 0 de Neumann, mientras en la parte inferior tenemos un dato de Dirichlet y una ecuación con una nonlinearidad regular. Este problema puede ser reformulado como una ecuación no-local sobre la componente dotada del dato de Dirichlet, definiendo un operador de Dirichlet-Neumann apropiado. Primero, demostramos un teorema del tipo Liouville, que garantiza la simetría unidimensional de las soluciones monótonas, asumiendo un control sobre el crecimiento de la energía asociada. Como consecuencia, obtenemos la simetría 1D de las soluciones estables en dimension 2. Para n=3, obtenemos estimaciónes optimales de la energía para las soluciones que minimizan la energía y para las soluciones monótonas. Estas estimaciones nos conducen a la simetría 1D de estas clases de soluciones, aplicando nuestro teorema del tipo Liouville. Relativo a este problema, estudiamos también la naturaleza del operador de Dirichlet-Neumann. Primero, deducimos su expresión como operador de Fourier, que anteriormente solo se conocía para a=0. Este resultado evidencia la naturaleza del operador, que es no-local pero no puramente fraccionaria. Estudiamos en profundidad este comportamiento mixto del operador a través del estudio de la G-convergencia de un funcional energía asociado al operador. Demostramos la G-convergencia de nuestro funcional a un límite que corresponde a una energía de interacción pura cuando a en (0,1) y al perímetro clásico cuando a en (-1,0]. El límite a=0, así como el G-límite para el régimen a en (-1,0], es común a otros problemas no-locales tratados en la literatura. Al contrario, el funcional límite en el régimen puramente no-local es nuevo y diferente a otros funcionaQuesta tesi si occupa di equazioni differenziali alle derivate parziali di tipo ellittico. È divisa in due parti: la prima riguarda un’equazione nonlineare per il p-Laplaciano, mentre la seconda è incentrata su un problema nonlocale, che può essere formulato per mezzo di un operatore di Dirichlet-Neumann collegato con il Laplaciano frazionario. Nella prima parte, studiamo la regolarità delle soluzioni stabili dell’equazione nonlineare per il p-Laplaciano dove W è un dominio limitato, p 2 (1,+¥) e f è una nonlinearità C1. Questa equazione è la versione nonlineare dell’equazione semilineare ������������Du = f (u) in un dominio limitato W Rn, che è stata ampiamente studiata in letteratura. Molto recentemente, Cabré, Figalli, Ros-Oton, e Serra [38] hanno dimostrato che le soluzioni stabili delle equazioni semilineari sono limitate, e quindi regolari, in dimensione n 9. Questo risultato è ottimale, dato che esempi di soluzioni illimitate e stabili sono noti in dimensione n 10. Inoltre, i risultati in [38] forniscono una risposta completa ad un annoso problema aperto, proposto da Brezis e Vázquez [25], sulla regolarità delle soluzioni estremali dell’equazione ������������Du = l f (u). Queste ultime sono infatti esempi non banali di soluzioni stabili di equazioni semilineari, che possono essere limitate o illimitate in dipendenza della dimensione n, del dominio W, e della nonlinearità f . In questa tesi studiamo la limitatezza delle soluzioni stabili di (0.4), che si congettura essere vera fino alla dimensione n < p + 4p/(p ������������ 1). Sono infatti noti esempi di soluzioni stabili e illimitate quando n p + 4p/(p ������������ 1), anche quando il dominio è la palla unitaria. Inoltre, nel caso radiale o assumendo ipotesi forti sulla nonlinearità, è stato dimostrato che le soluzioni stabili di (0.4) sono limitate quando n < p + 4p/(p ������������ 1). Nel Capitolo 1 della tesi dimostriamo una nuova stima L¥ a priori per le soluzioni stabili di (0.4), assumendo una nuova condizione su n e p, che è ottimale nel caso radiale e più restrittiva nel caso generale. Il nostro risultato migliora ciò che è noto in letteratura e ed è il primo esempio di tecnica che produce sia un risultato nel caso non radiale sia il risultato ottimale nel caso radiale. Per ottenere questo risultato estendiamo al caso del p-Laplaciano una tecnica sviluppata da Cabré [30] per il caso classico del problema, con p = 2. La strategia si basa su una disuguaglianza di Hardy sugli insiemi di livello della soluzione, combinata con una disuguaglianza di tipo geometrico per le soluzioni stabili di (0.4). Nella prima parte della tesi ci occupiamo anche di disuguaglianze funzionali di tipo Hardy e Sobolev, su ipersuperfici dello spazio euclideo. Nel fare ciò siamo motivati dalle varie applicazioni di questo tipo di risultati allo studio di stime a priori per le soluzioni stabili, sia nel caso semilineare che nel caso nonlineare ..

Estimates and rigidity for stable solutions to some nonlinear elliptic problems

Tesi en modalitat de cotutela: Universitat Politècnica de Catalunya i Università degli Studi di MilanoMy thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabré, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincaré sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.Mi tesis se encaja en el estudio de las EDPs elípticas. Está dividida en dos partes: la primera trata una ecuación no-lineal con el p-Laplaciano, la segunda de un problema no-local. En la primera parte, estudiamos la regularidad de las soluciones estables de una ecuación no lineal con el p-Laplaciano en un dominio acotado. Esta ecuacion es la versión no-lineal de la ámpliamente estudiada ecuacion semilineal con el Laplaciano. Cabré, Figalli, Ros-Oton, y Serra han demostrado recientemente que las soluciones estables de las ecuaciones semilineales son acotadas, y por tanto regulares, hasta la dimensión 9. Este resultado es optimal. En el caso del p-Laplaciano, la regularidad de las soluciones estables se conjetura de ser cierta hasta una dimension critica y, de hecho, se conocen ejemplos de soluciones no acotadas cuando la dimension llega al valor critico. Además, se ha demostrado que en el caso radial o assumiendo hipótesis fuertes sobre la no-linealidad las soluciones estables son acotadas hasta la dimension critica. En el primer capítulo, demostramos que las soluciones estables son acotadas, bajo una nueva condición en n y p, que es optimal en el caso radial, y más restrictiva en el caso general. Esta investigación mejora conocidos resultados del tema y es el primer ejemplo, para el p-Laplaciano, de un método que produce un resultado para el caso general y un resultado optimal en el caso radial. En la primera parte, nos ocupamos también de las desigualdades funcionales del tipo Hardy y Sobolev sobre hipersuperfícies del espacio Euclideo, todas conteniendo un término de curvatura media. Nuestra motivación proviene de varias apliaciones que tienen estas desigualdades en el estudio de estimaciones para las soluciones estables. En detalle, damos una demostración simple de la conocida desigualdad de Michael-Simon y Allard, obtenemos dos formas nuevas de la desigualdad de Hardy sobre hipersuperfícies, y otra desigualdad de Hardy-Poincaré. En la segunda parte, nos ocupamos de un problema de Dirichlet-Neumann que emerge de un modelo para las ondas en el agua. El sistema se describe con una ecuación de difusión en una tira de altura fija, que contiene un parámetro a en (-1,1). La parte superior de la tira es dotada de una condicion 0 de Neumann, mientras en la parte inferior tenemos un dato de Dirichlet y una ecuación con una nonlinearidad regular. Este problema puede ser reformulado como una ecuación no-local sobre la componente dotada del dato de Dirichlet, definiendo un operador de Dirichlet-Neumann apropiado. Primero, demostramos un teorema del tipo Liouville, que garantiza la simetría unidimensional de las soluciones monótonas, asumiendo un control sobre el crecimiento de la energía asociada. Como consecuencia, obtenemos la simetría 1D de las soluciones estables en dimension 2. Para n=3, obtenemos estimaciónes optimales de la energía para las soluciones que minimizan la energía y para las soluciones monótonas. Estas estimaciones nos conducen a la simetría 1D de estas clases de soluciones, aplicando nuestro teorema del tipo Liouville. Relativo a este problema, estudiamos también la naturaleza del operador de Dirichlet-Neumann. Primero, deducimos su expresión como operador de Fourier, que anteriormente solo se conocía para a=0. Este resultado evidencia la naturaleza del operador, que es no-local pero no puramente fraccionaria. Estudiamos en profundidad este comportamiento mixto del operador a través del estudio de la G-convergencia de un funcional energía asociado al operador. Demostramos la G-convergencia de nuestro funcional a un límite que corresponde a una energía de interacción pura cuando a en (0,1) y al perímetro clásico cuando a en (-1,0]. El límite a=0, así como el G-límite para el régimen a en (-1,0], es común a otros problemas no-locales tratados en la literatura. Al contrario, el funcional límite en el régimen puramente no-local es nuevo y diferente a otros funcionaQuesta tesi si occupa di equazioni differenziali alle derivate parziali di tipo ellittico. È divisa in due parti: la prima riguarda un’equazione nonlineare per il p-Laplaciano, mentre la seconda è incentrata su un problema nonlocale, che può essere formulato per mezzo di un operatore di Dirichlet-Neumann collegato con il Laplaciano frazionario. Nella prima parte, studiamo la regolarità delle soluzioni stabili dell’equazione nonlineare per il p-Laplaciano dove W è un dominio limitato, p 2 (1,+¥) e f è una nonlinearità C1. Questa equazione è la versione nonlineare dell’equazione semilineare ������������Du = f (u) in un dominio limitato W Rn, che è stata ampiamente studiata in letteratura. Molto recentemente, Cabré, Figalli, Ros-Oton, e Serra [38] hanno dimostrato che le soluzioni stabili delle equazioni semilineari sono limitate, e quindi regolari, in dimensione n 9. Questo risultato è ottimale, dato che esempi di soluzioni illimitate e stabili sono noti in dimensione n 10. Inoltre, i risultati in [38] forniscono una risposta completa ad un annoso problema aperto, proposto da Brezis e Vázquez [25], sulla regolarità delle soluzioni estremali dell’equazione ������������Du = l f (u). Queste ultime sono infatti esempi non banali di soluzioni stabili di equazioni semilineari, che possono essere limitate o illimitate in dipendenza della dimensione n, del dominio W, e della nonlinearità f . In questa tesi studiamo la limitatezza delle soluzioni stabili di (0.4), che si congettura essere vera fino alla dimensione n < p + 4p/(p ������������ 1). Sono infatti noti esempi di soluzioni stabili e illimitate quando n p + 4p/(p ������������ 1), anche quando il dominio è la palla unitaria. Inoltre, nel caso radiale o assumendo ipotesi forti sulla nonlinearità, è stato dimostrato che le soluzioni stabili di (0.4) sono limitate quando n < p + 4p/(p ������������ 1). Nel Capitolo 1 della tesi dimostriamo una nuova stima L¥ a priori per le soluzioni stabili di (0.4), assumendo una nuova condizione su n e p, che è ottimale nel caso radiale e più restrittiva nel caso generale. Il nostro risultato migliora ciò che è noto in letteratura e ed è il primo esempio di tecnica che produce sia un risultato nel caso non radiale sia il risultato ottimale nel caso radiale. Per ottenere questo risultato estendiamo al caso del p-Laplaciano una tecnica sviluppata da Cabré [30] per il caso classico del problema, con p = 2. La strategia si basa su una disuguaglianza di Hardy sugli insiemi di livello della soluzione, combinata con una disuguaglianza di tipo geometrico per le soluzioni stabili di (0.4). Nella prima parte della tesi ci occupiamo anche di disuguaglianze funzionali di tipo Hardy e Sobolev, su ipersuperfici dello spazio euclideo. Nel fare ciò siamo motivati dalle varie applicazioni di questo tipo di risultati allo studio di stime a priori per le soluzioni stabili, sia nel caso semilineare che nel caso nonlineare ...Postprint (published version

Computing the conformal barycenter

The conformal barycenter of a point cloud on the sphere at infinity of the Poincar\'e ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edgelengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton-Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff-Ye iteration.Comment: 29 pages, 8 figure

Quantum space-time: theory and phenomenology

Modern physics is based on two fundamental pillars: quantum mechanics and Einstein's general relativity. Even if, when taken separately, they can claim success in describing satisfactorily a plenty of physical phenomena, so far any attempt to make them compatible with each other failed. It is a central goal of theoretical physics to find a common approach to coherently merge quantum theory and general relativity. Such an effort is not only motivated by the conceptual necessity of completeness that imposes us to look for a unified theoretical framework that gives a consistent picture at all scales, but also by the existence of physical regimes we can not fully describe without a quantum theory of gravity, e.g. the first instants of early universe cosmology. This problem has remained open for more than eighty years now and keeps challenging physicists that, in the struggle to find a solution, have proposed a myriad of models, none of whom can claim full success. In fact, mainly due to the lack of experimental hints, the landscape of quantum gravity currently looks like a variegated compound of approaches that start from different conceptual premises and use different mathematical formalisms. In the majority of cases, it is not clear whether different models reach compatible predictions or even if they produce observable outcomes at all. Given the impossibility to achieve a unique acknowledged theory, it is of pivotal importance to seek insightful connections between different approaches. Such a strategy may help identifying few promising hot spots that may catalyze forthcoming efforts in the quantum gravity research community. In particular, more synergy between top-down and bottom-up models is certainly needed. Besides shedding light on some formal aspects of the models and eventually giving further support to specific ideas, reducing the gap between full-fledged quantum gravity proposals and simpler models that try to capture at least some expected features might produce tangible progress in the field of quantum gravity phenomenology. Indeed, it is now well-established that some effects introduced genuinely at the Planck scale by heuristic models can be efficiently investigated in ongoing and forthcoming experiments. Moreover, the exciting era of multi-messenger and multi-wavelength has started where several satellites, telescopes, and new generation detectors are furnishing us with an incomparable amount of data to probe the structure of gravity on cosmological scales and in new regimes which had remained inaccessible. Finding ways to rigorously derive Planckian testable effects from quantum gravity theories is then needed to enter another phase of maturity of quantum gravity phenomenology, i.e. the passage from the search for Planck-scale signals to the falsification of actual theories. This thesis represents a small step in this direction. To put this plan into action, we start recognizing that, despite the aforementioned heterogeneity, there is the common expectation that near the Planck scale our description of the spacetime as a smooth continuum, a picture shared by both general relativity and quantum mechanics, should break down and be replaced by some ``fuzzy" structure we generically refer to as \textit{quantum spacetime}. Again there are different ways to implement such an idea in different models, however we feel that the most relevant feature that characterizes spacetime quantization from a physical point of view is the associated departure from classical spacetime symmetries that most significantly encode spacetime's properties. In this regard, in the literature of the last three decades there has been much interest in the development of deformations of the Poincar\'e symmetries of special relativity, which most notably took the form of \textit{quantum groups} or \textit{Hopf algebras}, with the aim of modeling Planck scale physics. However, almost the totality of these studies is confined to the limit where gravitational effects are negligible, i.e. a sort of ``quantum Minkowski regime". With the objective to bridge quantum gravity and, in general, beyond general relativity theories with quantum or non-standard Minkowski spacetime models we here devote our attention to the symmetry content of general relativity, synthesized in the \textit{hypersurface deformation algebra}, and explore possible deformations caused by non-classical spacetime effects. Candidate modifications of the algebra of diffeomorphisms have been already obtained in some recent analyses, others will be derived in this thesis for the first time. We then translate modifications of the hypersurface deformation algebra into corresponding deviations from special relativistic symmetry with the main objective of looking for phenomenological opportunities. In particular, studying the Minkowski limit of deformed diffeomorphism algebras, we shall infer two much studied Planckian phenomena, namely \textit{modified dispersion relation} and \textit{the running of spacetime dimensions} with the probed scale. In this thesis we focus in particular on four different paths toward the characterization of non-classical (to be meant in a general sense as non-standard) spacetime properties: \textit{noncommutative geometry}, \textit{loop quantum gravity}, \textit{multifractional geometry}, and \textit{non-Riemannian geometry}; only the second being widely recognized as a candidate full-fledged quantum gravity theory. We first motivate why these two phenomenological Planck scale effects, i.e. dimensional reduction and modifications of particles' dispersion in vacuum, can be ascribed to spacetime fuzziness or quantization intended as an intrinsic obstruction to the measurability of spacetime distances below the Planck scale, an effect which can be deduced from the heuristic combination of general relativistc and quantum mechanical principles. Modified dispersion relation is derived rigorously in the framework of noncommutative geometries and we discuss two different noncommutative models which are of interest for this thesis: $\theta$-Minkowski and $\kappa$-Minkowski. The phenomenon of dimensional flow is instead presented from the perspective of multifractional geometry. Within this framework we show how dimensional flow and spacetime fuzziness are deeply connected. We illustrate how the assumption of an anomalous scaling of the spacetime dimension in the ultraviolet and a slow change of the dimension in the infrared is enough to produce a scale-dependent deformation of the integration measure with also a fuzzy spacetime structure. We also compare the multifractional correction to lengths with the types of Planckian uncertainty for distance and time measurements. This may offer an explanation why dimensional flow is encountered in almost the totality of quantum gravity models. We then introduce the (classical) hypersurface deformation algebra and constructively present two different ways of deriving it which we designate as representations of the algebra: the gravitational constraint representation, where the brackets are reproduced by the time and spatial diffeomorphism generators, and the Gaussian vector field representation, in which the algebra can be read off from the Lie bracket involving the components of a certain class of vector fields. Using this second realization, we study possible Drinfeld twists of space-time diffeomorphisms with Hopf-algebra techniques. We consider both deformed and twisted diffeomorphisms and compute the associated hypersurface deformation algebra. We then turn our attention to recent loop-qauntum-gravity-inspired studies that have motivated a restricted class of modifications of the algebra of gravitational constraints. We discuss these new results in the light of the possibility to identify an effective quantum-spacetime picture of loop quantum gravity, applicable in the Minkowski regime, where the large-scale (coarse-grained) spacetime metric is flat. We show that these symmetry-algebra results are consistent with a description of spacetime given in terms of the $\kappa$-Minkowski noncommutative spacetime, whose relevance for the study of the quantum-gravity problem has already been proposed for independent reasons. We exploit this unexpected link to extract viable testable predictions out of loop quantum gravity models. These loop-qauntum-gravity-inspired corrections to spacetime symmetries are used to analyze both the consequences on particle propagation and on dimensional running. Adopting a different strategy, we also construct a set of three operators suitable for identifying coordinate-like quantities on a spin-network configuration on the kinematical Hilbert space. Computing their action on coherent coarse-grained states, we are able to study some relevant properties such us the spectra, which are discrete. After that we scrutinize the symmetry structure of multifractional theories with either weighted or $q$- derivatives. These theories have the property that the spacetime dimensions are anomalous since they change with the scale of observation. Despite their different mathematical formalisms, both noncommutative and multifractional geometries allow for the spacetime dimension to vary with the probed scale. For this reason, we compare their symmetries and prove that, despite the presence of many contact points claimed by precedent studies, they are are physically inequivalent, yet one can describe certain aspects of $\kappa$-Minkowski noncommutative geometry as a multifractional theory and vice versa. Turning gravity on, we calculate the algebras of gravitational first-class constraints in the multifractional theories with $q$- and weighted derivatives and discuss their differences with respect to the deformed algebras of $\kappa$-Minkowski spacetime and of loop quantum gravity. Finally, with the aim of traducing multiscale formal properties into physical effects, we derive black hole solution in multifractional gravity theories and highlight new properties in the horizon structure as well as in the thermodynamical properties. Potential phenomenological signatures are underlined. The fourth non-standard spacetime approach we consider is given by non-Riemannian geometries with non-metricity. Among other reasons to modify classical general relativity, one motivation is that modified Einstein-Hilbert action could provide either a better behaved theory in the ultraviolet, while Einstein's theory is not renormalizable, or encode effective corrections to classical gravity, which could be remnants of quantum effects at low energy scales. In this context it is often claimed that a relaxation of the Riemannian condition to incorporate geometric quantities such as torsion and non-metricity may allow to explore new physics associated with defects in a sort of ``spacetime microstructure". We show that non-metricity modifies particles' equations of motion. In particular, we find that it produces observable effects in quantum fields in the form of 4-fermion contact interactions. The analysis we present is carried out in the framework of a wide class of theories of gravity in the metric-affine approach having a modified Lagrangian of the form $\mathcal{L}(g_{\mu\nu}, g^{\mu\rho}R_{(\rho \nu)})$. Finally, we compute the non-metric deformations of the hypersurface deformation algebra by using the Gaussian-vector-field method and make a qualitative comparison with loop quantum gravity results. The final part of this thesis is dedicated to the search for quantum spacetime effects on the propagation of very-high energy particles in the form of in-vacuo dispersion, i.e. a linear correlation between time of observation and energy of particles. Motivated by some recent studies that exposed rather strong statistical evidence of in-vacuo-dispersion-like spectral lags for gamma-ray bursts in the energy range above 40 GeV, we analyze 7 gamma-ray burst events detected by Fermi-LAT in the period 2008-2016 by extending the window of the statistical analysis down to 5 GeV. Intriguingly, we find results that are consistent with what had been previously noticed at higher energies and, thus, could be of quantum-spacetime origin. Reduced samples of the data set based on different energy cuts are also considered with the objective to strengthen the results of the study. Besides the obvious interest of the feature we find, the main importance of our study stands in the fact that it represents one of the first analyses done over a collection of gamma-ray burst events. This paves the way to statistical analyses needed to produce more robust and reliable results despite huge uncertainties on the astrophysical mechanisms behind the formation, the emission and the propagation of photons produced in gamma-ray explosions