Showing papers in "Proceedings of The London Mathematical Society in 2017"

Local uniqueness and periodicity induced by concentration

Abstract: We consider poly-harmonic equations with critical exponents. Under some conditions on the coefficient K(y) in the equations near its critical points, we prove the existence and local uniqueness of solutions with infinitely many bubbles. The local uniqueness result implies that some bubbling solutions preserve the symmetry of the scalar curvature K(y). Moreover, we also show that the conditions imposed are optimal to obtain such results.

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

Abstract: We prove that all hierarchically hyperbolic groups have finite asymptotic dimension. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic groups and spaces. We also prove a small-cancellation result namely: if G is a hierarchically hyperbolic group, H⩽G is a suitable hyperbolically embedded subgroup, and N◃H is ‘sufficiently deep’ in H, then G/N is a relatively hierarchically hyperbolic group. This new class provides many new examples to which our asymptotic dimension bounds apply. Along the way, we prove new results about the structure of HHSs, for example: the associated hyperbolic spaces are always obtained, up to quasi-isometry, by coning off canonical coarse product regions in the original space (generalizing a relation established by Masur and Minsky between the complex of curves of a surface and Teichmuller space).

The twist for positroid varieties

Abstract: The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Abstract: We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed.

Manolescu invariants of connected sums

Abstract: We give inequalities for the Manolescu invariants α,β,γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a Z∞ subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y, for large n.

Laplacian flow of homogeneous G2-structures and its solitons

Abstract: We use the bracket flow/algebraic soliton approach to study the Laplacian flow of G2-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (that is, a G-invariant G2-structure on a homogeneous space G/K that flows by pull-back of automorphisms of G up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a ‘diagonal’ evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long-time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.

The principal series of p‐adic groups with disconnected center

Abstract: Let G be a split connected reductive group over a local non-Archimedean field. We classify all irreducible complex G-representations in the principal series, irrespective of the (dis)connectedness of the center of G. This leads to a local Langlands correspondence for principal series representations of G. It satisfies all expected properties, in particular it is functorial with respect to homomorphisms of reductive groups. At the same time, we show that every Bernstein component s in the principal series has the structure of an extended quotient of Bernstein's torus by Bernstein's finite group (both attached to s).

Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

Abstract: We prove that the maximal operator associated with variable homogeneous planar curves (t,utα)t∈R, α≠1 positive, is bounded on Lp(R2) for each p>1, under the assumption that u:R2→R is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t,utα)t∈R, α≠1 positive, is bounded on Lp(R2) for each p>1, under the assumption that u:R2→R is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, TT∗ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.

Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions

Abstract: Over the past fifteen years, the theory of Wasserstein gradient flows of convex (or, more generally, semiconvex) energies has led to advances in several areas of partial differential equations and analysis. In this work, we extend the well-posedness theory for Wasserstein gradient flows to energies satisfying a more general criterion for uniqueness, motivated by the Osgood criterion for ordinary differential equations. We also prove the first quantitative estimates on convergence of the discrete gradient flow or "JKO scheme" outside of the semiconvex case. We conclude by applying these results to study the well-posedness of constrained nonlocal interaction energies, which have arisen in recent work on biological chemotaxis and congested aggregation.

Méthode de mahler : relations linéaires, transcendance et applications aux nombres automatiques

Abstract: — Cet article est consacre a la methode de Mahler. Nous decrivons en detail la structure des relations de dependance lineaire entre les valeurs aux points algebriques de fonctions mahleriennes. Etant donnes un corps de nombres k, une fonction mahlerienne f (z) ∈ k{z} et α un nombre algebrique, 0 < |α| < 1, qui n'est pas un pole de f , nous montrons notamment que l'on peut toujours determiner si le nombre f (α) est transcendant ou non. Dans ce dernier cas, nous obtenons que f (α) appartient necessairement a l'extension k(α). Nous considerons egalement les consequences remarquables de cette theorie concernant un probleme arithmetique classique : l'etude de la suite des chiffres des nombres algebriques dans une base entiere ou, plus generalement, algebrique. Nos resultats sont obtenus a partir d'un theoreme recent de Philippon [31] que nous raffinons et dont nous simplifions la demonstration.

Models for the Speiser class

Abstract: The Eremenko-Lyubich class B consists of transcendental entire functions with bounded singular set and the Speiser class S ⊂ B is made up of functions with a finite singular set. In [4] I gave a method for constructing Eremenko-Lyubich functions that approximate certain simpler functions called models. In this paper, I show that all models can be approximated in a weaker sense by Speiser class functions, and that the stronger approximation of [4] can fail for the Speiser class. In particular, I give geometric restrictions on the geometry of a Speiser class function that need not be satisfied by general Eremenko-Lyubich functions. Date: January 2017. 1991 Mathematics Subject Classification. Primary: 30D15 Secondary: 30C62, 37F10 .

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

Abstract: We prove the existence of a global fundamental solution Γ(x;y) (with pole x) for any Hormander operator L=∑i=1mXi2 on Rn which is δλ-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δλ of the form δλ(x)=(λσ1x1,…,λσnxn), with 1=σ1⩽⋯⩽σn. Due to a global lifting method for homogeneous operators proved by Folland [Comm. Partial Differential Equations 2 (1977) 161–207], there exists a Carnot group G and a polynomial surjective map π:G→Rn such that L is π-related to a sub-Laplacian LG on G. We show that it is always possible to perform a (global) change of variable on G such that the lifting map π becomes the projection of G≡Rn×Rp onto Rn. We prove that an integration argument over the (non-compact) fibers of π provides a fundamental solution for L. Indeed, if ΓG(x,x′;y,y′) (x,y∈Rn; x′,y′∈Rp) is the fundamental solution of LG, we show that ΓG(x,0;y,y′) is always integrable with respect to y′∈Rp, and its y′-integral is a fundamental solution for L.

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Abstract: The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.

Vinogradov's theorem with almost equal summands

Abstract: Let θ>11/20. We prove that every sufficiently large odd integer n can be written as a sum of three primes n=p1+p2+p3 with |pi−n/3|⩽nθ for i∈{1,2,3}.

The topology of Kuranishi atlases

Abstract: Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Starting from the same core idea (patching local finite-dimensional reductions), we develop a theory of topological Kuranishi atlases and cobordisms that transparently resolves algebraic and topological challenges in this virtual regularization approach. It applies to any Kuranishi-type setting, for example, atlases with isotropy, boundary and corners, or lack of differentiable structure.

The Assouad dimensions of projections of planar sets

Abstract: The first named author is supported by a Leverhulme Trust Research Fellowship and the second named author is supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number 274512.

Hopf measuring comonoids and enrichment

Abstract: We study the existence of universal measuring comonoids P(A,B) for a pair of monoids A, B in a braided monoidal closed category, and the associated enrichment of the category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if A is a bimonoid and B is a commutative monoid, then P(A,B) is a bimonoid; in addition, if A is a cocommutative Hopf monoid then P(A,B) always is Hopf. If A is a Hopf monoid, not necessarily cocommutative, then P(A,B) is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable P(A,B)-comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.

P-adic L-functions of Bianchi modular forms

Abstract: The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the p-adic L-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular symbol attached to a Bianchi modular form and critical values of its L-function, which then allows us to construct p-adic L-functions of Bianchi modular forms.

Whitney extension theorems for convex functions of the classes C1 and C1,ω

Abstract: Let C be a subset of ℝn (not necessarily convex), f : C → R be a function and G : C → ℝn be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f, G for the existence of a convex function F ∈ CC1ω(ℝn) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C1 convex functions on ℝn, with a good control of the Lipschitz constants of the extensions (namely, Lip(F) ≲ ∥G∥∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of ℝn by boundaries of C1 or C1,1 convex bodies with prescribed outer normals on K.

Stark units in positive characteristic

Abstract: We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant A-harmonic series. We apply this to obtain a class formula a la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.

Arithmetic of singular character varieties and their E-polynomials

Abstract: We calculate the E-polynomials of the SL3(C)- and GL3(C)-character varieties of compact oriented surfaces of any genus and the E-polynomials of the SL2(C)- and GL2(C)-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the E-polynomials of the SL2(C)-character varieties of compact orientable surfaces, which were computed by Logares, Munoz and Newstead for genus g=1,2 and by Martinez and Munoz for g⩾3. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.

Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Abstract: Let G be a connected bipartite graph with colour classes E and V and root polytope Q. Regarding the hypergraph H=(V,E) induced by G, we prove that the interior polynomial of H is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of H and its transpose H¯=(E,V) agree. When G is a complete bipartite graph, our result recovers a well-known hypergeometric identity due to Saalschutz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group.

The ideal structure of algebraic partial crossed products

Abstract: Given a partial action of a discrete group G on a Hausdorff, locally compact, totally disconnected topological space X, we consider the correponding partial action of G on the algebra Lc(X) consisting of all locally constant, compactly supported functions on X, taking values in a given field K. We then study the ideal structure of the algebraic partial crossed product Lc(X)⋊G. After developing a theory of induced ideals, we show that every ideal in Lc(X)⋊G may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic version of the Effros–Hahn conjecture.

Universal L2‐torsion, polytopes and applications to 3‐manifolds

Abstract: Given an L2-acyclic connected finite CW-complex, we define its universal L2-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group Whw(G). We study its main properties such as homotopy invariance, sum formula, product formula and Poincare duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group Whw(G) to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal L2-torsion can be identified with many invariants such as the L2-torsion, the L2-torsion function, twisted L2-Euler characteristics and, in the case of a 3-manifold, the dual Thurston norm polytope.

On regularity and the word problem for free idempotent generated semigroups

Abstract: The category of all idempotent generated semigroups with a prescribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E) Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups In this paper we take these investigations in a new direction by considering the word problem for IG(E) We prove two principal results, one positive and one negative We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗represents a regular element; if in addition one assumes that all maximal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups

Highest rank of a polytope for An

Abstract: This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, and by the Portuguese Foundation for Science and Technology (FCT-Fundacao para a Ciencia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Abstract: A central question in arrangement theory is to determine whether the characteristic polynomial $\Delta_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $\Delta_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $\beta_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $\mathcal{A}$ defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of $\mathcal{A}$ to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over $\mathbb{C}$) for matroids, and we estimate the number of essential components in the first complex resonance variety of $\mathcal{A}$. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of ${\rm SL}_2(\mathbb{C})$-representation varieties, which are governed by the Maurer--Cartan equation.

Projectively related metrics, Weyl nullity, and metric projectively invariant equations

Abstract: A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries which is defined on a rank $(n+1)$-bundle. We show this connection is linked to the metrisability equations that govern the existence of metrics compatible with the structure. The fundamental 2-tensor also leads to a new class of invariant linear differential operators that are canonically associated to these geometries; included is a third equation studied by Gallot et al. We apply the results to study the metrisability equation, in the nullity setting described. We obtain strong local and global results on the nature of solutions and also on the nature of the geometries admitting such solutions, obtaining classification results in some cases. We show that closed Sasakian and Kahler manifold do not admit nontrivial solutions. We also prove that, on a closed manifold, two nontrivially projectively equivalent metrics cannot have the same tracefree Ricci tensor. We show that on a closed manifold a metric having a nontrivial solution of the metrisablity equation cannot have two-dimensional nullity space at every point. In these statements the meaning of trivial solution is dependent on the context. There is a function $B$ naturally appearing if a metric projective structure has nullity. We analyse in detail the case when this is not a constant, and describe all nontrivially projectively equivalent Riemannian metrics on closed manifolds with nonconstant $B$.

The heat kernel of a Schrödinger operator with inverse square potential

Abstract: We consider the Schrodinger operator H=−Δ+V(|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x,y,t) of the type 00, where U is a positive harmonic function of H. Third, if U2 is an A2 weight on RN, then we prove a lower bound of a similar type.

Elliptic curves over a finite field and the trace formula

Abstract: We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2(Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2(Z) which include as special cases the groups Γ1(N) and Γ(N). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z)2 and congruence subgroups Γ0(2),Γ0(4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vladuţ, Gekeler, Howe and others.