Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

On Plane Waves in Diluted Relativistic Cold Plasmas

We briefly report on some exact results (Fiore in J. Phys. A, Math. Theor. 47:225501, 2014) regarding plane waves in a relativistic cold plasma. If the plasma, initially at rest, is reached by a transverse plane electromagnetic travelling-wave, then its motion has a very simple dependence on this wave in the limit of zero density, otherwise can be determined by an iterative procedure whose accuracy decreases with time or the plasma density. Thus one can describe in particular the impact of a very intense and short laser pulse onto a plasma and determine conditions for the slingshot effect (Fiore et al. in arXiv:1309.1400, 2014) to occur. The motion in vacuum of a charged test particle subject to a wave of the same kind is also determined, for any initial velocity.     

Some remarks on the inverse semigroup associated to the Cuntz-Li algebras

In this article, we prove that the inverse semigroup associated to the Cuntz-Li relations is strongly 0-E unitary and is an F ^?-inverse semigroup. We also identify the universal group of the inverse semigroup. This gives a conceptual explanation for the result obtained in S. Sundar (arXiv:1201.4620v1, 2012).     

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Recently Guth and Katz (arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erd?s??s distinct distances problem, a new method for partitioning finite point sets in ? ^d , based on the Stone?CTukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi?CTrotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach?CSharir bound on the number of incidences with algebraic curves of bounded degree.     

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants,and Normalized Ricci Flow

In this article, we produce infinite families of 4-manifolds with positive first Betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg–Witten invariants of their connected sums. Elementary building blocks used in Ishida and Sasahira (arXiv:0804.3452, 2008) are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov’s simplicial volume is nontrivial, Perelman’s \(\bar{\lambda}\) invariant is negative, and the relevant Gromov–Hitchin–Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. Fang et al. (Math. Ann. 340:647–674, 2008) conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov’s simplicial volume and negative Perelman’s \(\bar{\lambda}\) invariant implies the Gromov–Hitchin–Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds.     

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (mod_Λ), with Λ a finite dimensional algebra.     

Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms

We derive the defocusing cubic Gross–Pitaevskii (GP) hierarchy in dimension d = 3, from an N-body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies (Chen and Pavlovi? in Discr Contin Dyn Syst 27(2):715–739, 2010; http://arxiv.org/abs/0906.2984; Proc Am Math Soc 141:279–293, 2013), which are inspired by the solution spaces based on space-time norms introduced by Klainerman and Machedon (Comm Math Phys 279(1):169–185, 2008). We note that in d = 3, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrödinger equation (NLS) in d = 3.     

On the Brauer-Feit bound for abelian defect groups

We improve the Brauer-Feit bound on the number of irreducible characters in a $p$ -block for abelian defect groups by making use of Halasi and Podoski (Every coprime linear group admits a base of size two. http://arxiv.org/abs/1212.0199v1, [7]) and Kessar and Malle (Ann Math 178(2):321–384, [11]). We also prove Brauer’s $k(B)$ -Conjecture for 2-blocks with abelian defect groups of rank at most 5 and 3-blocks and 5-blocks with abelian defect groups of rank at most 3.     

Effective Dynamics for N-Solitons of the Gross–Pitaevskii Equation

We consider several solitons moving in a slowly varying external field. We present results of numerical computations which indicate that the effective dynamics obtained by restricting the full Hamiltonian to the finite-dimensional manifold of N-solitons (constructed when no external field is present) provides a remarkably good approximation to the actual soliton dynamics. This is quantified as an error of size h ² where h is the parameter describing the slowly varying nature of the potential. This also indicates that previous mathematical results of Holmer and Zworski (Int. Math. Res. Not. 2008: Art. ID runn026, 2008) for one soliton are optimal. For potentials with unstable equilibria, the Ehrenfest time, log(1/h)/h, appears to be the natural limiting time for these effective dynamics. We also show that the results of Holmer et?al. (arXiv:0912.5122, 2009) for two mKdV solitons apply numerically to a larger number of interacting solitons. We illustrate the results by applying the method with the external potentials used in the Bose?CEinstein soliton train experiments of Strecker et?al. (Nature 417:150?C153, 2002).     

Averaging Fluctuations in Resolvents of Random Band Matrices

We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erd?s et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erd?s et al., arXiv:1205.5669, 2013).     

Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space ${\mathbb{H}^n}$ . The graphs are considered as unbounded hypersurfaces of ${\mathbb{H}^{n+1}}$ which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.     

Hall Algebras of Odd Periodic Triangulated Categories

We define the Hall algebra associated to any triangulated category under some finiteness conditions with odd periodic translation functor T. This generalizes the results in Toën (Duke Math J 135(3):587–615, 2006) and Xiao and Xu (Duke Math J 143(2):357–373, 2008).     

A vanishing result for strictly p-convex domains

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for \(q\) -complete domains in \(\mathbb C ^{n}\) , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly \(p\) -convex domains in \(\mathbb R ^n\) in the sense of Harvey and Lawson (The foundations of \(p\) -convexity and \(p\) -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the \({L}^2\) -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

Cell decomposition of some unitary group Rapoport–Zink spaces

In this paper we study the \(p\) -adic analytic geometry of the basic unitary group Rapoport–Zink spaces \(\mathcal {M}_K\) with signature \((1,n-1)\) . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber \(\mathcal {M}_{red}\) defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain \(\mathcal {D}_K\) in \(\mathcal {M}_K\) for the action of \(G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)\) , the product of the associated \(p\) -adic reductive groups, and prove that \(\mathcal {M}_K\) admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Characteristics of Graph Braid Groups

We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.     

Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53?C70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561?C598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric $\mathbb{M}^{d}$ and the hyperbolic metric ? ^d . This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among $\mathbb{E}^{d}$ , cones in $\mathbb{E}^{d+1}$ , $\mathbb{S}^{d}$ , $\mathbb{M}^{d}$ , and ? ^d . We also consider the further extensions associated with the other Cayley?CKlein geometries overlaid on the shared underlying projective geometry.     

The Local Langlands Correspondence for GL n over p-adic fields

We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL _n over p-adic fields as well as the existence of ?-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.