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Noether type discrete conserved quantities arising from a finite element approximation of a variational problemJul 31 2012Mar 16 2015In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element discretisation of a model ... More

Second Order $L^\infty$ Variational Problems and the $\infty$-PolylaplacianMay 25 2016Jun 15 2016In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. ... More

A finite element method for fully nonlinear elliptic problemsMar 15 2011Aug 07 2012We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit ... More

Surface permeability, capillary transport and the Laplace-Beltrami problemAug 18 2018We have established previously, in a lead-in study, that the spreading of liquids in particulate porous media at low saturation levels, characteristically less than 10% of the void space, has very distinctive features in comparison to that at higher saturation ... More

Discontinuous Galerkin methods for the $p$--biharmonic equation from a discrete variational perspectiveSep 18 2012Sep 22 2015We study discontinuous Galerkin approximations of the $p$--biharmonic equation from a variational perspective. We propose a discrete variational formulation of the problem based on a appropriate definition of a finite element Hessian and study convergence ... More

An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problemsMay 16 2015In this work we present an a posteriori analysis for classes of inconsistent, nonconforming schemes approximating elliptic problems. We show the estimates coincide with existing ones for interior penalty type discontinuous Galerkin approximations of the ... More

On the finite element approximation of infinity-harmonic functionsNov 02 2015In this note we show that conforming Galerkin approximations for p-harmonic functions tend to infinity-harmonic functions in the limit p \to \infty and h \to 0, where h denotes the Galerkin discretisation parameter.

Applications of nonvariational finite element methods to Monge--Ampère type equationsMar 03 2012The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge--Amp\`ere type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature.

A Symmetry Analysis of the $\infty$-PolylaplacianAug 22 2017Aug 25 2018In this work we use Lie group theoretic methods and the theory of prolonged group actions to study two fully nonlinear partial differential equations (PDEs). First we consider a third order PDE in two spatial dimensions that arises as the analogue of ... More

Second Order $L^\infty$ Variational Problems and the $\infty$-PolylaplacianMay 25 2016Jan 05 2018In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. ... More

Second Order $L^\infty$ Variational Problems and the $\infty$-PolylaplacianMay 25 2016Nov 04 2016In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. ... More

Quasinorms in semilinear elliptic problemsNov 19 2018In this note we examine the a priori and a posteriori analysis of discontinuous Galerkin finite element discretisations of semilinear elliptic PDEs with polynomial nonlinearity. We show that optimal a priori error bounds in the energy norm are only possible ... More

On the Numerical Approximation of $\infty$-Harmonic MappingsNov 04 2015Nov 05 2015Given a map $u : \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} \Delta_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) : \text{D}^2 u\, ... More

A posteriori analysis for dynamic model adaptation in convection dominated problemsJul 28 2016Jul 07 2017In this work we present an a posteriori error indicator for approximation schemes of Runge-Kutta-discontinuous-Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make ... More

A review from the PDE viewpoint of Hamilton-Jacobi-Bellman Equations Arising in Optimal Control with Vectorial CostSep 30 2014Jan 13 2018This paper is a review of results on Optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued ... More

Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamicsMay 19 2014Sep 19 2014We give an a posteriori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component ... More

Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamicsMay 19 2014Sep 19 2014We give an a priori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics which involves an energy density depending not only on the strain but also the strain gradient. A key component ... More

Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow modelJul 31 2013Apr 24 2014We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen-Cahn/Cahn-Hilliard/Navier-Stokes-Korteweg type which allows for phase transitions. We show that the scheme ... More

Discontinuous Galerkin methods for nonvariational problemsApr 08 2013We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the NVFEM ... More

On Hamilton-Jacobi-Bellman Equations Arising in Deterministic and Stochastic Optimal Control with Vectorial CostSep 30 2014We consider the problem of optimally controlling a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation of the cost is considered with respect to a partial ordering generated by a given proper cone $K$. Since in ... More

Quasinorms in semilinear elliptic problemsNov 19 2018Feb 25 2019In this note we examine the a priori and a posteriori analysis of discontinuous Galerkin finite element discretisations of semilinear elliptic PDEs with polynomial nonlinearity. We show that optimal a priori error bounds in the energy norm are only possible ... More

On the numerical approximation of vectorial absolute minimisers in $L^\infty$Dec 28 2018Let $\Omega$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ open}, \] applied to locally ... More

Gradient recovery in adaptive finite element methods for parabolic problemsMay 17 2009Mar 12 2010We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators ... More

A posteriori analysis for dynamic model adaptation in convection dominated problemsJul 28 2016In this work we present an a posteriori error indicator for approximation schemes of Runge-Kutta-discontinuous-Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make ... More

On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functionsJan 25 2017May 14 2018In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in $L^{\infty}$. The associated equation, coined the $\infty$-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by $\Delta^2_\infty u\, := (\Delta u)^3 ... More

Conservative Galerkin methods for dispersive Hamiltonian problemsNov 25 2018In this work we design a conservative discontinuous Galerkin scheme for a generalised third order KdV type equation. The techniques we use allow for the derivation of optimal a priori and a posteriori bounds. We summarise numerical experiments showcasing ... More

A finite element method for second order nonvariational elliptic problemsMar 01 2010We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second ... More

An adaptive finite element method for the infinity LaplacianNov 15 2013We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem may be singular, which has prompted us to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. ... More

Analysis Of Discontinuous Galerkin Methods Using Mesh-Dependent NormsOct 17 2016We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms.

Recovered Finite Element MethodsMay 10 2017We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has a number of ... More

Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms and Applications to Problems with Rough DataOct 17 2016Mar 28 2017We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an ... More

Hydrodynamics of moving contact lines: macroscopic versus microscopicJul 12 2017The fluid-mechanics community is currently divided in assessing the boundaries of applicability of the macroscopic approach to fluid mechanical problems. Can the dynamics of nano-droplets be described by the same macroscopic equations as the ones used ... More

Babuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshesApr 18 2017Oct 24 2018We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general class of meshes, which includes many examples of geometrically graded element ... More

Surface permeability and surface flow tortuosity of particulate porous mediaJun 05 2019The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by super-fast non-linear ... More

The design of conservative finite element discretisations for the vectorial modified KdV equationOct 10 2017We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance ... More

A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation lawsMay 29 2014In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative ... More

A finite element method for the Monge-Ampère equation with transport boundary conditionsJul 10 2018Aug 23 2018We address the numerical solution via Galerkin type methods of the Monge-Amp\`ere equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully nonlinear ... More

A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problemsSep 06 2007Nov 07 2010We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual ... More

Recovered finite element methods on polygonal and polyhedral meshesApr 23 2018Recovered finite element methods (R-FEM) have been recently introduced for meshes consisting of simplicial and/or box-type meshes. Here, utilising the flexibility of R-FEM framework, we extend their definition on polygonal and polyhedral meshes in two ... More

Energy consistent DG methods for the Navier-Stokes-Korteweg systemJul 19 2012We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme ... More

A numerical implementation of the unified transform for evolution problems on a finite intervalOct 14 2016We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary conditions, and ... More

A posteriori error estimates for the virtual element methodMar 18 2016Apr 24 2017An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable ... More

A numerical implementation of the unified Fokas transform for evolution problems on a finite intervalOct 14 2016Oct 18 2016We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary conditions, and ... More

A posteriori error estimates for the virtual element methodMar 18 2016An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable ... More

Residual estimates for post-processors in elliptic problemsJun 11 2019In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient ... More

A Viscosity Method in the Min-max Theory of Minimal SurfacesAug 28 2015Oct 31 2016We present the min-max construction of critical points of the area in arbitrary closed sub-manifold of euclidian spaces by penalization arguments. Precisely, for any immersion of a closed surface ${\Sigma}$, we add to the area functional a term equal ... More

A Viscosity Method in the Min-max Theory of Minimal SurfacesAug 28 2015We present the min-max construction of critical points of the area in arbitrary closed sub-manifold of euclidian spaces by relaxation arguments. Precisely, for any immersion of a closed surface $\Sigma$, we add to the area functional a term equal to the ... More

On Control Of Sobolev Norms For Some Semilinear Wave Equations With Localized DataApr 13 2012Oct 15 2016We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the Hs, s<1, closure of compactly supported data inside a ball of radius R, with R a fixed and positive number. In order to do that we perform an ... More

Global well-posedness of partially periodic KP-I equation in the energy space and applicationJun 21 2017In this article, we address the Cauchy problem for the KP-I equation \[\partial_t u + \partial_x^3 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0\] for functions periodic in $y$. We prove global well-posedness of this problem for any data in the ... More

Continuous Patrolling and Hiding GamesDec 14 2016Mar 29 2018We present two zero-sum games modeling situations where one player attacks (or hides in) a finite dimensional nonempty compact set, and the other tries to prevent the attack (or find him). The first game, called patrolling game, corresponds to a dynamic ... More

A note on the theorem of Maynard and TaoNov 21 2013As a corollary to the recent extraordinary theorem of Maynard and Tao, we re-prove, in a stronger form, a result of Shiu concerning "strings" of consecutive, congruent primes.

A Viscosity Method in the Min-max Theory of Minimal SurfacesAug 28 2015Oct 27 2017We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $\Sigma$ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^q$ norm of the ... More

The Regularity of Conformal Target Harmonic MapsOct 31 2016May 26 2017In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal ... More

*-Ring OrderingsJan 10 2016Feb 26 2016We examine a number of *-ring orderings, generalizing classical properties of *-positive elements to *-accretives. We also examine *-rings satisfying versions of Blackadar's property (SP), generalizing some basic properties of Rickart *-rings to Blackadar ... More

Conformally Invariant Variational ProblemsJun 11 2012Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl ... More

Sequences of Smooth Global Isothermic ImmersionsFeb 06 2012In the present work we study the behavior of sequences of smooth global isothermic immersions of a given closed surface and having a uniformly bounded total curvature. We prove that, if the conformal class of this sequence is bounded in the Moduli space ... More

Intersection homology D-Modules and Bernstein polynomials associated with a complete intersectionSep 11 2007May 25 2008Let X be a complex analytic manifold. Given a closed subspace $Y\subset X$ of pure codimension p>0, we consider the sheaf of local algebraic cohomology $H^p_{[Y]}({\cal O}_X)$, and ${\cal L}(Y,X)\subset H^p_{[Y]}({\cal O}_X)$ the intersection homology ... More

Bubbling and regularity issues in geometric non-linear analysisApr 24 2003Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which ... More

On the Cauchy problem for the periodic fifth-order KP-I equationDec 04 2017The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_t u - \partial_x^5 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0,~(t,x,y)\in\mathbb{R}\times\mathbb{T}^2\] We prove global well-posedness ... More

*-Annihilators in Proper *-SemigroupsApr 06 2014We generalize some basic C*-algebra and von Neumann algebra theory on hereditary C*-subalgebras and projections. In particular, we extend Murray-von Neumann equivalence from projections to *-annihilators and show that several of its important properties ... More

Communicating Zero-Sum Product Stochastic GamesSep 04 2017Aug 06 2018We study two classes of zero-sum stochastic games with compact action sets and a finite product state space. These two classes assume a communication property on the state spaces of the players. For strongly communicating on one side games, we prove the ... More

Review of AdS/CFT Integrability, Chapter II.2: Quantum Strings in AdS5xS5Dec 17 2010Dec 20 2010We review the semiclassical analysis of strings in AdS5xS5 with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in ... More

On the half-space theorem for minimal surfaces in Heisenberg spaceJun 18 2012Nov 02 2015We propose a simple proof of the vertical half-space theorem for Heisenberg space.

Conservation laws for conformal invariant variational problemsMar 15 2006We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries ... More

Analysis aspects of Willmore surfacesDec 18 2006We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally ... More

Addendum to "Quivers with loops and perverse sheaves"Sep 22 2015We prove a character formula for the Hopf algebra defined in arXiv:1401.5302 that generalizes quantum groups, as well as for the simple modules associated to dominant integral weights defined in arXiv:1403.0846.

A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrodinger equationsAug 02 2015Aug 31 2016We prove a weak form of the soliton resolution conjecture of bounded solutions of high-dimensional fourth-order Schrodinger equations. The result relies upon two properties to be proved: the asymptotic frequency localization and the asymptotic spatial ... More

Scattering above energy norm of solutions of a loglog energy-supercritical Schrodinger equation with radial dataNov 01 2009We prove scattering of $\tilde{H}^{k} $ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})} $, $0 < c < c_{n}$, $n=\{3,4}$, with radial data $u(0)=u_{0} ... More

Disk-sphere field duality theoremDec 04 2018This paper presents a new reformulated theorem for fields embedded on a sphere or a disk. We focus in particular on the associated sphere of a disk when closing its only one boundary. We call this the disk-sphere duality theorem for the study of fields ... More

Irreducible components of the global nilpotent coneDec 20 2017Jun 27 2018This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus $\ge2$. The first main result of this paper states that the set of irreducible components of the global nilpotent ... More

Products of shifted primes simultaneously taking perfect power valuesAug 11 2010Aug 12 2010Let $r \ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products $\prod_{p \mid n} (p+a)$ (over primes $p$) are perfect $r$th ... More

Semicontinuity in Ordered Banach SpacesApr 11 2016We extend the C*-algebra semicontinuity theory of Akemann, Brown and Pedersen to (pre)ordered Banach spaces.

Minmax Hierarchies and Minimal Surfaces in ManifoldsMay 27 2017Jun 05 2017We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in Banach manifolds equipped with Finsler structures. We call the resulting tree ... More

Introduction to scattering for radial 3D NLKG below energy normSep 23 2008Aug 22 2016We prove scattering for the radial nonlinear Klein-Gordon equation $ \partial_{tt} u - \Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) \in H^{s} \times H^{s-1} $, $ 1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4 \geq p > 3 ... More

Nearness PosetsFeb 21 2019We extend nearness frames to posets representing bases and even subbases of $T_1$ spaces. This allows us to put a classic duality due to Wallman, between compact $T_1$ spaces and abstract simplicial complexes, into a general nearness framework. Within ... More

The Regularity of Conformal Target Harmonic MapsOct 31 2016In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal ... More

Piecewise Strong Convexity of Neural NetworksOct 30 2018We study the loss surface of a fully connected neural network with ReLU non-linearities, regularized with weight decay. We start by expressing the output of the network as a matrix determinant, which allows us to establish that the loss function is piecewise ... More

Type Decomposition in PosetsMar 17 2014Motivated by the classical type decomposition of von Neumann algebras, and various more recent extensions to other structures, we develop a type decomposition theory for general posets.

Strings of congruent primes in short intervals IIOct 30 2011Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $\epsilon$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes $p_r$, up to $X$, ... More

Lower Semi-Continuity of the Index in the Visosity Method for Minimal SurfacesJul 30 2018The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of an euclidian space $R^Q$. Second, ... More

Critical weak immersed Surfaces within Sub-manifolds of the Teichmüller SpaceJul 20 2013We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichm\"uller space satisfy the corresponding ... More

Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental FormJul 18 2010In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed ... More

Scattering above energy norm of solutions of a loglog energy-supercritical Schrodinger equation with radial dataNov 01 2009Feb 14 2018We prove scattering of $\tilde{H}^{k} $ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})}$, $0 < c < c_{n}$, $n={3,4}$, with radial data $u(0):=u_{0} ... More

The Variations of Yang-Mills LagrangianJun 15 2015Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One ... More

Tori in $S^3$ minimizing locally the conformal volumeMay 11 2014We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally ... More

Global well-posedness for the radial defocusing cubic wave equation on $\mathbb{R}^{3}$ and for rough dataAug 17 2007Feb 10 2008We prove global well-posedness for the radial defocusing wave equation and for data in $H^{s} \times H^{s-1}$, $1>s>{7/10}$.

Remark on the semilinear ill-posedness for a periodic higher order KP-I equationMay 05 2018We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.

3D quadratic NLS equation with electromagnetic perturbationsMar 23 2019In this paper we study the asymptotic behavior of a quadratic Schr\"{o}dinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. ... More

MAD Families of Projections on l^2 and Real-Valued Functions on OmegaOct 07 2012Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality ... More

The Projection CalculusMar 17 2012Jan 22 2013We develop some tools for manipulating and constructing projections in C*-algebras. These are then applied to give short proofs of some standard projection homotopy results, as well as strengthen some fundamental classical results for C*-algebras of real ... More

Yoneda CompletenessMar 11 2015Mar 15 2016We characterize Yoneda completeness for non-symmetric distances by combinations of metric and directed completeness. One of these generalizes the Kostanek-Waszkiewicz theorem on formal balls.

Logarithmic comparison theorem and D-modules: an overviewOct 20 2005Let D be a divisor in a complex analytic manifold X. A natural problem is to determine when the de Rham complex of meromorphic forms on X with poles along D is quasi-isomorphic to its subcomplex of logarithmic forms. In this mostly expository note, we ... More

Bourbaki Seminar 1081 : Min-max methods and the Willmore conjecture, after Fernando Codá Marques and André Arroja NevesFeb 06 2014Two years ago, F.C. Marques and A.A. Neves implemented, in the framework of closed rectifiable 2-dimensional currents of the 3-dimensional sphere, a min-max method in geometric measure theory due to F. Almgren and J. Pitts. Using this approach they succeeded ... More

Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)May 17 2019A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more ... More

Short intervals with a given number of primesAug 01 2015A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is asymptotically ... More

Strings of congruent primes in short intervalsMay 25 2010Aug 26 2010Fix \epsilon > 0, and let p_1 = 2, p_2 = 3,... be the sequence of all primes. We prove that if (q,a) = 1 then there are infinitely many pairs p_r, p_{r+1} such that p_r \equiv p_{r+1} \equiv a \mod q and p_{r+1} - p_r < \epsilon\log p_r. The proof combines ... More

Willmore Minmax Surfaces and the Cost of the Sphere EversionDec 30 2015We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere ... More

Sub-criticality of Schroedinger Systems with Antisymmetric PotentialsNov 05 2009Let $m$ be an integer larger or equal to 3. We prove that Schroedinger systems on $B^m$ with $L^{m/2}-$antisymmetric potential $\Omega$ of the form $$ -\Delta v=\Omega v $$ can be written in divergence form and we deduce that solutions $v$ in $L^{m/(m-2)}$ ... More

A Generalized Construction of Calabi-Yau Models and Mirror SymmetryNov 30 2016We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive ... More

Concave elliptic equations and generalized Khovanskii-Teissier inequalitiesMar 26 2019We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.