Asymptotic ergodicity of the eigenvalues of random operators in the localized phase

We prove that, for a general class of random operators, the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of Minami (Spectra of random operators and related topics, 2011). Minami conjectured this to be the case for discrete Anderson model in the localized regime. We also provide a local analogue of this result. From the asymptotics ergodicity, one can recover the statistics of the level spacings as well as a number of other spectral statistics. Our proofs rely on the analysis developed in Germinet and Klopp (Spectral statistics for random Schrödinger operators in the localized regime, 2010).     

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 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.     

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.     

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.     

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).     

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes (Sel. Math. 1(3):411–457, 1995). In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Consani and Marcolli (Adv. Math. 214(2):761–831, 2007), and using a classification result of Borger and de Smit (arXiv:1105.4662) for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic (arXiv:1010.4766).     

Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces

In the previous work (Zhang and Zhu in J Differ Geom, http://arxiv.org/pdf/1012.4233v3, 2012), the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1–14, 1994), Chen and Wang (Sci Sin (A) 40:384–394, 1997) and Bakry–Qian (Adv Math 155:98–153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li–Yau’s estimate for positve solutions of heat equations on Alexandrov 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.     

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.     

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).     

Growth,entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat005 2013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).     

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.     

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).     

Finite number of fibre products of Kummer covers and curves with many points over finite fields

We study fibre products of a finite number of Kummer covers of the projective line over finite fields. We determine the number of rational points of the fibre product over a rational point of the projective line, which improves the results of Özbudak and Temür (Appl Algebra Eng Commun Comput 18:433–443, 2007) substantially. We also construct explicit examples of fibre products of Kummer covers with many rational points, including a record and two new entries for the current table (http://www.manypoints.org, 2011).     

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.     

Large time behavior and Lyapunov functionals for a nonlocal differential equation

A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016). Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016).     

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.     

Phase Transition for Accessibility Percolation on Hypercubes

In this paper, we consider accessibility percolation on hypercubes, i.e., we place i.i.d. uniform [0, 1] random variables on vertices of a hypercube, and study whether there is a path connecting two vertices such that the values of these random variables increase along the path. We establish a sharp phase transition depending on the difference of the values at the two endpoints and determine the critical window of the phase transition. Our result completely resolves a conjecture of Berestycki et al. (Accessibility percolation with backsteps. Preprint, available at http://arxiv.org/abs/1401.6894).     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.