Admissible closures of polynomial time computable arithmetic

We propose two admissible closures ${\mathbb{A}({\sf PTCA})}$ and ${\mathbb{A}({\sf PHCA})}$ of Ferreira??s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) ${\mathbb{A}({\sf PTCA})}$ is conservative over PTCA with respect to ${\forall\exists\Sigma^b_1}$ sentences, and (ii) ${\mathbb{A}({\sf PHCA})}$ is conservative over full bounded arithmetic PHCA for ${\forall\exists\Sigma^b_{\infty}}$ sentences. This yields that (i) the ${\Sigma^b_1}$ definable functions of ${\mathbb{A}({\sf PTCA})}$ are the polytime functions, and (ii) the ${\Sigma^b_{\infty}}$ definable functions of ${\mathbb{A}({\sf PHCA})}$ are the functions in the polynomial time hierarchy.     

Crystallographic number systems

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion $$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$ for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.     

Actions infinitésimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible ${\overline{\mathbb Qp}}$ -representation of ${{\rm Gal}(\overline{\mathbb Qp}/\mathbb Qp)}$ and let ${\prod(V)}$ be the ${{\rm GL}_2(\mathbb Qp)}$ Banach representation associated by Colmez??s p-adic Langlands correspondence. We establish a link between the action of the Lie algebra of ${{\rm GL}_2(\mathbb Qp)}$ on the locally analytic vectors ${\prod(V)^{\rm an}}$ of ${\prod(V)}$ , the connection ${\nabla}$ on the ${(\varphi, \Gamma)}$ -module associated to V and the Sen polynomial of V. This answers a question of Harris, concerning the infinitesimal character of ${\prod(V)^{\rm an}}$ . Using this result, we give a new proof of a theorem of Colmez, stating that ${\prod(V)}$ has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge?CTate weights.     

Relative minimizers of energy are relative minimizers of area

Let Γ be a closed, regular Jordan curve in ${{\mathbb R}^3}$ which is of class C ^1,μ , 0 <  μ <  1, and denote by ${{\mathcal C}(\Gamma)}$ the class of the disk-type surfaces ${X : B \to {\mathbb R}^3}$ with continuous, monotonic boundary values, mapping ${\partial B}$ onto Γ. One easily sees that any minimal surface ${X \in {\mathcal C}(\Gamma)}$ is a relative minimizer of energy, i.e. of Dirichlet’s integral D, if it is a relative minimizer of the area functional A. Here we prove conversely: If an immersed ${X \in {\mathcal C}(\Gamma)}$ is a C ¹-relative minimizer of D in ${{\mathcal C}(\Gamma)}$ , then it also is a C ^1,μ -relative minimizer of A in ${{\mathcal C}(\Gamma)}$ .     

Uniqueness of the Group Measure Space Decomposition for Popa’s {\mathcal{HT}} Factors

We prove that if ${\Gamma\curvearrowright (X, \mu)}$ is a free ergodic rigid (in the sense of Popa in Ann Math 163:809–889, 2006) probability measure preserving action of a group Γ with positive first ${\ell^2}$ -Betti number, then the II₁ factor ${L^{\infty}(X)\rtimes\Gamma}$ has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many ${\mathcal{HT}}$ factors, including the II₁ factors associated with the usual actions ${\Gamma\curvearrowright \mathbb{T^2}}$ and ${\Gamma\curvearrowright}$ ${{\rm SL}_2(\mathbb R)/{\rm SL}_2(\mathbb Z)}$ , where Γ is a non-amenable subgroup of ${{\rm SL}_2(\mathbb Z)}$ , have a unique group measure space decomposition.     

L^{\infty } estimation of tensor truncations

Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $\ell ^{2}$ or $L^{2}$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require $\ell ^{\infty }$ or $L^{\infty }$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\| \cdot \right\| _{\infty }$ by $\left\| \cdot \right\| _{2}$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to $\left\| \cdot \right\| _{\infty }$ can be derived from the Gagliardo–Nirenberg inequality because of the special nature of the singular value decomposition truncation.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Heegner cycles and higher weight specializations of big Heegner points

Let ${\mathbf{{f}}}$ be a $p$ -ordinary Hida family of tame level $N$ , and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis relative to $N$ . By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level $\Gamma _0(N)\cap \Gamma _1(p^s)$ , Howard has constructed a canonical class $\mathfrak{Z }$ in the cohomology of a self-dual twist of the big Galois representation associated to ${\mathbf{{f}}}$ . If a $p$ -ordinary eigenform $f$ on $\Gamma _0(N)$ of weight $k>2$ is the specialization of ${\mathbf{{f}}}$ at $\nu $ , one thus obtains from $\mathfrak{Z }_{\nu }$ a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes $\mathfrak{Z }_{\nu }$ to the étale Abel-Jacobi images of Heegner cycles when $p$ splits in $K$ .     

Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem

We give a new characterization of the strict $\forall {\Sigma^b_j}$ sentences provable using ${\Sigma^b_k}$ induction, for 1 ?? j ?? k. As a small application we show that, in a certain sense, Buss??s witnessing theorem for strict ${\Sigma^b_k}$ formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of $\forall {\Sigma^b_1}$ sentences.     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Bochner–Riesz Means with Respect to a Generalized Cylinder

We study the L ^p boundedness of the generalized Bochner–Riesz means S ^λ which are defined as $$S^{\lambda}f(x) = \mathcal{F}^{-1} \left[\left(1 - \rho \right)_{+}^{\lambda} \widehat{f} \right](x)$$ where ${\rho(\xi) = {\rm max}\{|\xi_{1}|, \ldots, |\xi_{\ell}|\}}$ for ${\xi = (\xi_{1},\ldots, \xi_{\ell}) \in \mathbb{R}^{{d}_{1}} \times \cdots \times \mathbb{R}^{{d}_{\ell}}}$ and ${\mathcal{F}^{-1}}$ is the inverse Fourier transform.     

An explicit growth condition for middle-degree cuspidal cohomology of arithmetically defined quaternionic hyperbolic n-manifolds

Let ${G/\mathbb Q}$ be the simple algebraic group Sp(n, 1) and ${\Gamma=\Gamma(N)}$ a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group ${G(\mathbb R)}$ . Then a double quotient ${\Gamma\backslash G(\mathbb R)/K}$ is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology ${H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)}$ in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkhäuser, Boston), pp. 1–48, 1984).     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra ${\mathcal{A}}$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that ${\mathcal{A}}$ separates points in the sense that for each distinct pair ${x, y \in X}$ , there exists an ${f \in \mathcal{A}}$ such that ${f(x) \neq f(y)}$ . If ${\mathcal{A}}$ does not separate points, it is known that there exists an algebra ${\widehat{\mathcal{A}}}$ on a compact Hausdorff space ${(\widehat{X}, \widehat{\tau})}$ that does separate points such that the map ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume ${\mathcal{A}}$ separates points. The construction of ${{\widehat{\mathcal{A}}}}$ and ${(\widehat{X}, \widehat{\tau})}$ does not require that ${\mathcal{A}}$ has any algebraic structure nor that ${(X, \tau)}$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family ${\mathcal{A}}$ of bounded, complex-valued, continuous functions on any topological space ${(X, \tau)}$ . We also demonstrate that further structures may be preserved by the mapping ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Sur la densité des représentations cristallines de \text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p)

Let ${\mathfrak X }_d$ be the $p$ -adic analytic space classifying the semisimple continuous representations $\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p) \rightarrow \mathrm GL _d(\overline{\mathbb Q }_p)$ . We show that the crystalline representations are Zarski-dense in many irreducible components of ${\mathfrak X }_d$ , including the components made of residually irreducible representations. This extends to any dimension $d$ previous results of Colmez and Kisin for $d = 2$ . For this we construct an analogue of the infinite fern of Gouvêa–Mazur in this context, based on a study of analytic families of trianguline $(\varphi ,\Gamma )$ -modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline $(\varphi ,\Gamma )$ -modules, as well as the density of the crystalline $(\varphi ,\Gamma )$ -modules in this family. These results may be viewed as a local analogue of the theory of $p$ -adic families of finite slope automorphic forms and they are new already in dimension $2$ . The technical heart of the paper is a collection of results about the Fontaine–Herr cohomology of families of trianguline $(\varphi ,\Gamma )$ -modules.