ALBERT

Description: We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of non-autonomous Hamiltonian vector fields.

Description: We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$ . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I $_\infty$ , II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

Description: We study the rate of convergence to zero of the tail entropy of $C^\infty$ maps. We give an upper bound of this rate in terms of the growth in $k$ of the derivative of order $k$ and give examples showing the optimality of the established rate of convergence. We also consider the case of multimodal maps of the interval. Finally, we prove that homoclinic tangencies give rise to $C^r$ $(r\geqslant 2)$ robustly non- $h$ -expansive dynamical systems.

Description: Let ${{\mathscr {C}}}^0_{{{\mathfrak {g}}}}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'({{\mathfrak {g}}})$ and let $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty }$ . In this paper, we investigate the relationship between the categories ${{\mathscr {C}}}^0_{A_{N-1}^{(1)}}$ and ${{\mathscr {C}}}^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur–Weyl duality functors ${\mathcal {F}}^{(t)}$ from $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ to ${{\mathscr {C}}}^0_{A_{N-1}^{(t)}}\ (t=1,2)$ .

Description: We present new constructions of complex and $p$ -adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

Description: We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross–Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed.

Description: Let $k$ and $n$ be positive even integers. For a cuspidal Hecke eigenform $h$ in the Kohnen plus space of weight $k-n/2+1/2$ for $\varGamma _0(4),$ let $I_n(h)$ be the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift of $h$ in the space of cusp forms of weight $k$ for ${\rm Sp}_n({{\bf{Z}}}),$ and $f$ be the primitive form of weight $2k-n$ for ${\rm SL}_2({{\bf{Z}}})$ corresponding to $h$ under the Shimura correspondence. We then express the ratio $\displaystyle {\langle I_n(h), I_n(h) \rangle / \langle h, h \rangle }$ of the period of $I_n(h)$ to that of $h$ in terms of special values of certain $L$ -functions of $f$ . This proves the conjecture proposed by Ikeda concerning the period of the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift.

Description: A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert–Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty $ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko–Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that \[ f(A+B)-f(A)-\left.\frac{d}{dt}(f(A+tB))\right\vert_{t=0}\notin \mathcal S^1. \]

Description: Let $\mu $ be a probability measure on $ \mathbb R^n$ with a bounded density $f$ . We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann–Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $\mu $ , any marginal $\pi _E(\mu )$ , or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

Description: Let $\pi :X\to \mathbb {P}^1_{\mathbb {Q}}$ be a non-singular conic bundle over $\mathbb {Q}$ having $n$ non-split fibres and denote by $N(\pi ,B)$ the cardinality of the fibres of Weil height at most $B$ that possess a rational point. Serre showed in 1990 that a direct application of the large sieve yields \[ N(\pi,B)\ll B^2(\log B)^{-n/2} \] and raised the problem of proving that this is the true order of magnitude of $N(\pi ,B)$ under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most 3. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser–Iwaniec sieve.

Description: We associate a dimer algebra $A$ to a Postnikov diagram $D$ (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian ${\rm Gr}(k,n)$ . We show that $A$ is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module $T$ over the algebra $B$ used to categorify the cluster structure of ${\rm Gr}(k,n)$ by Jensen–King–Su. It follows that $B$ can be realised as the boundary algebra of $A$ , that is, the subalgebra $eAe$ for an idempotent $e$ corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram $D$ , with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

Description: We describe a ring whose category of Cohen–Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$ -planes in $n$ -space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective–injective object is Geiss–Leclerc–Schröer's category Sub $Q_k$ , which categorifies the coordinate ring of the big cell in this Grassmannian.

Description: We develop a way of seeing a complete orientable hyperbolic 4-manifold $ {\mathcal {M}}$ as an orbifold cover of a Coxeter polytope $ {\mathcal {P}} \subset \mathbb {H}^4$ that has a facet colouring. We also develop a way of finding a totally geodesic sub-manifold $ {\mathcal {N}}$ in $ {\mathcal {M}}$ , and describing the result of mutations along $ {\mathcal {N}}$ . As an application of our method, we construct an example of a complete orientable hyperbolic 4-manifold $ {\mathcal {X}}$ with a single non-toric cusp and a complete orientable hyperbolic 4-manifold ${\mathcal {Y}}$ with a single toric cusp. Both $ {\mathcal {X}}$ and $ {\mathcal {Y}}$ have twice the minimal volume among all complete orientable hyperbolic 4-manifolds.

Description: The covariogram $g_{K}$ of a convex body $K$ in $ \mathbb {R}^n$ is the function that associates to each $x\in \mathbb {R}^n$ the volume of the intersection of $K$ with $K+x$ . Determining $K$ from the knowledge of $g_K$ is known as the Covariogram Problem. It is equivalent to determining the characteristic function $1_K$ of $K$ from the modulus of its Fourier transform $\widehat {{1_K}}$ in $ \mathbb {R}^n$ , a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform $\widehat {{1_K}}$ seen as a function in $\mathbb {C}^n$ . The first connection is with the problem of determining $K$ from the knowledge of the zero set of $\widehat {{1_K}}$ in $\mathbb {C}^n$ . To attack this problem T. Kobayashi studied the asymptotic behavior at infinity of this zero set. We obtain this asymptotic behavior assuming less regularity on $K$ and we use this result as an essential ingredient for proving that when $K$ is sufficiently smooth and in any dimension $n$ , $K$ is determined by $g_K$ in the class of sufficiently smooth bodies. The second connection is with the irreducibility of the entire function $\widehat {{1_K}}$ . This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.

Description: Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$ . We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$ -module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $ , then $f\circ \varphi =0$ defines a free divisor on $X$ ; this is generalized in several directions. Among applications we recover a result of Mond–van Straten, generalize a construction of Buchweitz–Conca, and show that a map $\varphi :\mathbb {C}^{n+1}\to \mathbb {C}^n$ with critical set of codimension 2 has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi $ is the algebraic quotient $X\to S=X\!{/\!/} G$ with $X\!{/\!/} G$ smooth, then we describe sufficient conditions for $T^1_{X/S}$ to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on $\mathbb {C}^{n+1}$ lifts under the operation of ‘castling’ to a free divisor on $\mathbb {C}^{n(n+1)}$ , partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.

Description: Let $M^n$ be a compact manifold of dimension $n$ with free $T^k$ -action. We consider collapsings of $M$ on $N=M/T^k$ such that the sectional curvature and diameter of $M$ satisfy $|K(M)|\leq a$ and $ {\rm diam}(M) 〈 d$ , and give examples of collapsings for all $k$ such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of $M$ are bounded above by $c(M)\cdot \hbox {inj}(M)^{2k}$ . Moreover, we prove that the first non-zero eigenvalue of Laplacian acting on 1-forms of all principal $T^k$ -bundle $M$ over $N$ is bounded below by $c(n,a,d,N)\cdot {\rm Vol}(M)^2$ and $c\cdot \hbox {inj}(M)^{2k}$ when $M$ collapses on $N$ .

Description: We give explicit atomic bases of arbitrary coefficient-free cluster algebras of types A and à . This entails showing that the minimal elements of the positive semiring of these cluster algebras form a linear basis over the integers for the cluster algebra.

Description: We prove that strongly F -regular and F -pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F -injective singularities fail to satisfy even the most basic Bertini-type results.

Description: This is the second of a pair of papers on the Delta-group structure on the braid and mapping class groups of a surface. We obtain a description of the homotopy groups of these Delta-groups and generalize to an arbitrary surface the Berrick–Cohen–Wong–Wu exact sequence relating the Brunnian braid groups of the 2-sphere to its homotopy groups. We prove a similar result for Brunnian mapping class groups.

Description: We construct a geometric realization of the Khovanov–Lauda–Rouquier algebra R associated with a symmetric Borcherds–Cartan matrix A = ( a ij ) i , j I via quiver varieties. As an application, if a ii != 0 for any i I , we prove that there exists a one-to-one correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of U A – (g) (respectively, V A ( )) and the set of isomorphism classes of indecomposable projective graded modules over R (respectively, R ).

Description: The purpose of this paper is to study the nature of quasi-invariant measures for finitely generated non-discrete subgroups of Diff ( S 1 ). For this, we apply ideas involving the closure of these groups to find out that the regularity of the measure depends on a ‘measurable version’ of well-known problems concerning stable self-intersection of Cantor sets. As applications, we prove that every d -quasiconformal probability measure for a non-solvable and non-discrete group must be absolutely continuous. Concerning singular quasi-invariant measures, it is also proved that their associated Hausdorff measures must either be zero or of infinite mass, a result contrasting with the case of dynamically defined Cantor sets and also applicable to the examples of singular stationary measures constructed by Kaimanovich and Le Prince. As a further application of our methods, a theorem of rigidity for measurable conjugations between groups as above is obtained.

Description: We study the space of period polynomials associated with modular forms of integral weight for finite-index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite-index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for 1 ( N ): an extension of the Eichler–Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficient theorem of Manin.

Description: We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt (5) and SL 2 (5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite quasisimple groups except for the alternating group Alt (5) are Beauville groups.

Description: Let U R d be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f : U -〉 R can be approximated by real analytic convex functions, uniformly on all of U . We also show that C 0 -fine approximation of convex functions by smooth (or real analytic) convex functions on R d is possible in general if and only if d = 1. Nevertheless, for d ≥ 2, we give a characterization of the class of convex functions on R d which can be approximated by real analytic (or just smoother) convex functions in the C 0 -fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C 1 convex and proper function on U can be approximated by C convex functions in the C 1 -fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Description: Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$ -space. In this paper, we analyze the homotopy type of the homotopy fixed point set $X^{hG}$ , and the natural injection $k\colon X^G\hookrightarrow X^{hG}$ . We show that if $X$ is elliptic, that is, it has finite-dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi _* (k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

Description: We employ the ergodic-theoretic machinery of scenery flows to address classical geometric measure-theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

Description: A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to a classification of such manifolds with thickness 1, 3 or 2 $k$ , for $k\geq 1$ . On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.

Description: Extending a classical result of Widom from 1969, polynomials with small supremum norms are constructed for a large family of compact sets $\Gamma$ : their norm is at most a constant times the theoretical lower limit ${{\rm cap}}(\Gamma )^n$ , where ${{\rm cap}}(\Gamma )$ denotes logarithmic capacity. The construction is based on a discretization of the equilibrium measure, and the polynomials have the additional property that outside the given set $\Gamma$ they increase as fast as possible, namely as ${{\rm cap}}(\Gamma )^n\exp (ng_{ \overline {{{}C}}\setminus \Gamma }(z))$ , with the Green's function with pole at infinity in the exponent. This latter fact allows us to use these polynomials as building blocks in constructing Dirac delta-type polynomials around corners: if a compact set $K$ has a corner at some point $z_0$ , then Dirac delta-type polynomials (fast decreasing polynomials) peaking at $z_0$ are polynomials $P_n(z)$ with $P_n(z_0)=1$ that decrease as $|P_n(z)|\prec \exp (-n^ \beta |z-z_0|^ \gamma )$ on the set $K$ as $z$ moves away from $z_0$ . The possible $(\beta , \gamma )$ pairs are completely described in turn of the angle $\alpha \pi$ at $z_0$ ( $\beta \lt 1$ and $\gamma \ge \beta /(2- \alpha )$ or $\beta =1$ and $\gamma 〉 \beta /(2- \alpha )$ ). As application of these fast decreasing polynomials sharp Nikolskii- and Markov-type inequalities are proved for Jordan domains with corners. The paper uses distortion properties of conformal maps, potential theoretic techniques as well as the theory of weighted logarithmic potentials.

Description: We consider for every $n\in \mathbb {N}$ an algebra $\mathcal {A}_{n}$ of germs at $0\in \mathbb {R}^{n}$ of continuous real-valued functions, such that we can associate to every germ $f\in \mathcal {A}_{n}$ a (divergent) series $\mathcal {T}(f)$ with non-negative real exponents, which can be thought of as an asymptotic expansion of $f$ . We require that the $\mathbb {R}$ -algebra homomorphism $f\mapsto \mathcal {T}(f)$ be injective (quasianalyticity property). In this setting, we prove analogue results to Denef and van den Dries’ quantifier elimination theorem and Hironaka's rectilinearization theorem for subanalytic sets.

Description: Suppose that $F(x)\in \mathbb {Z}[\![x]\!]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$ for an integer $b\geq 2$ . In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If, in addition, $F(x)$ is regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$ -number or a $T$ -number in Mahler's classification of real numbers.

Description: We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$ -adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

Description: We give a bordism-theoretic characterization of those closed almost contact $(2q{+ }1)$ -manifolds (with $q\geq 2$ ) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's $h$ -principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

Description: Casson-type invariants emerging from Donaldson theory over certain negative-definite four-manifolds were recently suggested by Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism programme of proving Witten's conjecture, we use a moduli space of ${\rm PU}(2)$ Seiberg–Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

Description: The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems and includes a positive voltage parameter $\lambda$ . It is shown that there is a threshold value $\lambda _* 〉 0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda 〉 \lambda _* $ , while at least two such solutions exist for $\lambda \in (0,\lambda _* )$ . Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda 〉 \lambda _* $ .

Description: Assuming the generalized Riemann hypothesis, we prove a quantitative estimate for the number of simple zeros on the critical line for $L$ -functions attached to classical holomorphic newforms.

Description: In this paper, we consider a $\mathbb {Q}$ -Fano $3$ -fold weighted complete intersection of codimension $2$ in the $85$ families listed in Iano-Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to another explicitly described Mori fiber space. As a consequence, nineteen families are proved to be birationally rigid and the remaining $66$ families are proved to be birationally non-rigid.

Description: Suppose that a sequence of numbers $x_n$ (a ‘signal’) is transmitted through a noisy channel. The receiver observes a noisy version of the signal with additive random fluctuations, $x_n + \xi _n$ , where $\xi _n$ is a sequence of independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space ${\mathscr {X}}$ of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted? In this paper, we consider the case in which signals are infinite sequences and the recovery or detection are required to hold with probability 1. We provide conditions on the space ${\mathscr {X}}$ for checking whether detection or recovery are possible. We also analyze in detail several examples including spaces of Fourier transforms of measures, spaces with fixed amplitudes and the space of almost periodic functions. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible.

Description: In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in $\mathbb {Q}^n$ to any acyclic quiver $Q$ , and this simplicial fan allows one to completely determine the canonical presentation of any element in $\mathbb {Z}^n$ . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-1 subsets of $\mathbb {Q}^n$ , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

Description: The $j$ -multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the $\epsilon$ -multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the $j$ -multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$ -multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$ - and $\epsilon$ -multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

Description: Let $\pi : X \to Y$ be a morphism of projective varieties and suppose that $\alpha $ is a pseudo-effective numerical cycle class satisfying $\pi _{*}\alpha =0$ . A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha $ is a limit of classes of effective cycles contracted by $\pi $ . We establish new cases of the conjecture for higher codimension cycles. In particular, we prove a strong version when $X$ is a fourfold and $\pi $ has relative dimension 1.

Description: Let $R$ be a group of prime order $r$ that acts on the $r'$ -group $G$ , let $RG$ be the semidirect product of $G$ with $R$ , let ${\mathbb {F}}$ be a field and $V$ be a faithful completely reducible $\mathbb {F}[{RG}]$ -module. Trivially, $C_{G}({R})$ acts on $C_{V}({R})$ . Let $K$ be the kernel of this action. What can be said about $K$ ? This question is considered when $G$ is soluble. It turns out that $K$ is subnormal in $G$ or $r$ is a Fermat or half-Fermat prime. In the latter cases, the subnormal closure of $K$ in $G$ is described. Several applications to the theory of automorphisms of soluble groups are given.

Description: The mono-epi (ME) exact structure on the morphisms of an exact category $(\mathcal {A}; \mathcal {E})$ is introduced and used to prove ideal versions of Salce's Lemma, Christensen's (Ghost) Lemma, and Wakamatsu's Lemma for an exact category. Salce's Lemma establishes a bijective correspondence $\mathcal {I} \mapsto \mathcal {I}^{\perp }$ between the class of special precovering ideals of $(\mathcal {A}; \mathcal {E})$ and that of its special preenveloping ideals. ME-extensions of morphisms are used to define an extension $\mathcal {I} \diamond \mathcal {J}$ of ideals. Christensen's Lemma asserts that the class of special precovering (respectively, special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma satisfies $(\mathcal {I} \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \diamond \mathcal {I}^{\perp }$ and $(\mathcal {I} \diamond \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \mathcal {I}^{\perp }.$ Wakamatsu's Lemma asserts that if a covering ideal $\mathcal {I}$ is closed under ME-extensions, then it is a special precovering ideal. As an application, it is proved that if $G$ is a finite group and $\Phi $ is the ideal of phantom morphisms in the category $k[G]$ - $\rm Mod,$ then $\Phi ^{n-1}$ is the object ideal generated by projective modules, where $n$ is the nilpotency index of the Jacobson radical $J.$ If $R$ is a semiprimary ring, with $J^n =0,$ then $\Phi ^n$ is generated by projective modules. For a right coherent ring $R$ over which every cotorsion left $R$ -module has a coresolution of length $n$ by pure injective modules, $\Phi ^{n+1}$ is generated by flat modules.

Description: We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.

Description: Let $P_{n}(x)= \sum _{i=0}^n \xi _i x^i$ be a Kac random polynomial where the coefficients $\xi _i$ are i.i.d. copies of a given random variable $\xi $ . Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root. As an application, we consider the problem of estimating the number of real roots of $P_n$ , which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi $ , that the expected number of real roots of $P_n(x)$ is exactly $({2}/{\pi }) \log n +C +o(1)$ , where $C$ is an absolute constant depending on the atom variable $\xi $ . Prior to this paper, such a result was known only for the case when $\xi $ is Gaussian.

Description: The goal of this article was to study the Iwasawa theory of an abelian variety $A$ that has complex multiplication by a complex multiplication (CM) field $F$ that contains the reflex field of $A$ , which has supersingular reduction at every prime above $p$ . To do so, we make use of the signed Coleman maps constructed in our companion article [Kâzım Büyükboduk and Antonio Lei, ‘Integral Iwasawa theory of motives for non-ordinary primes’, 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed $p$ -adic $L$ -function via a reciprocity conjecture that we formulate for the (conjectural) Rubin–Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank- $g$ Euler–Kolyvagin systems to be applied with Rubin–Stark elements and deduce the main conjecture for the maximal $\mathbb {Z}_p$ -power extension of $F$ for the primes failing the ordinary hypothesis of Katz.

Description: We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way, we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold-valued rough paths. Finally, this framework is used to show that there is a one-to-one correspondence between rough paths on a $d$ -dimensional manifold and rough paths on $d$ -dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.

Description: We show that several important normal subgroups $\Gamma $ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure-preserving action $\Gamma \curvearrowright X$ is stably $OE$ -superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from Chifan–Ioana–Kida [‘ $W^*$ -superrigidity for arbitrary actions of central quotients of braid groups’, Math. Ann. 361 (2015) 925–959], we deduce that any free, ergodic, probability measure-preserving action of almost any surface braid group is stably $W^*$ -superrigid, that is, it can be completely reconstructed from its von Neumann algebra.

Description: Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting contractions on a finite-dimensional Hilbert space and studying their joint generalized eigenspaces. Non-tangential regularity of rational functions on the polydisk is also studied. One result states that rational inner functions on the polydisk have non-tangential limits at every point of the $n$ -torus. An algebraic characterization of higher non-tangential regularity is given. We also make some connections with the earlier material and prove that rational functions on the bidisk which are square integrable on the two-torus are non-tangentially bounded at every point. Several examples are provided.

Description: We propose a construction of a tensor exact category $\mathcal {F}_X^m$ of Artin–Tate motivic sheaves with finite coefficients $\mathbb {Z}/m$ over an algebraic variety $X$ (over a field $K$ of characteristic prime to $m$ ) in terms of étale sheaves of $\mathbb {Z}/m$ -modules over $X$ . Among the objects of $\mathcal {F}_X^m$ , in addition to the Tate motives $\mathbb {Z}/m(j)$ , there are the cohomological relative motives with compact support $\mathcal {M}_{cc}^m(Y/X)$ of varieties $Y$ quasi-finite over $X$ . Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties $Y\longrightarrow X$ act on the exact categories $\mathcal {F}_X^m$ . Assuming the existence of triangulated categories of motivic sheaves $\mathcal {D}\mathcal {M}(X,\mathbb {Z}/m)$ over algebraic varieties $X$ over $K$ and a weak version of the ‘six operations’ in these categories, we identify $\mathcal {F}_X^m$ with the exact subcategory in $\mathcal {D}\mathcal {M}(X,\mathbb {Z}/m)$ consisting of all the iterated extensions of the Tate twists $\mathcal {M}_{cc}^m(Y/X)(j)$ of the motives $\mathcal {M}_{cc}^m(Y/X)$ . An isomorphism of the $\mathbb {Z}/m$ -modules ${\rm Ext}$ between the Tate motives $\mathbb {Z}/m(j)$ in the exact category $\mathcal {F}_X^m$ with the motivic cohomology modules predicted by the Beilinson–Lichtenbaum étale descent conjecture (recently proved by Voevodsky, Rost et al. ) holds for smooth varieties $X$ over $K$ if and only if the similar isomorphism holds for Artin–Tate motives over fields containing $K$ . When $K$ contains a primitive $m$ -root of unity, the latter condition is equivalent to a certain Koszulity hypothesis, as shown in our previous paper [Positselski, ‘Mixed Artin–Tate motives with finite coefficients’, Mosc. Math. J. 11 (2011) 317–402].

Description: We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let $X$ be an arithmetic scheme (scheme of finite type over $\textbf {Z}$ ), and for a prime $p$ let $\zeta _{X_p}(s)$ be the local factor of its zeta function. We present an algorithm that computes $\zeta _{X_p}(s)$ for a single prime $p$ in time $p^{1/2+o(1)}$ , and another algorithm that computes $\zeta _{X_p}(s)$ for all primes $p 〈 N$ in time $N \log ^{3+o(1)} N$ . These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.

Description: We undertake a systematic study of asymptotically hereditarily aspherical (AHA) groups, the class of groups introduced by Tadeusz Januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups. We provide many new examples of AHA groups, also in high dimensions. We relate the AHA property with the topology at infinity of a group, and deduce in this way some new properties of (weakly) systolic groups. We also exhibit an interesting property of boundaries at infinity for a few classes of AHA groups.

Description: We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space ${\overline {\mathcal M}}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In particular, we prove that codimension 2 boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in ${\overline {\mathcal M}}_{3,1}$ and the locus of hyperelliptic curves in ${\overline {\mathcal M}}_4$ are extremal cycles. In addition, we exhibit infinitely many extremal codimension 2 cycles in ${\overline {\mathcal M}}_{1,n}$ for $n\geq 5$ and in ${\overline {\mathcal M}}_{2,n}$ for $n\geq 2$ .

Description: In this paper, we consider the so-called Toda System in planar domains under Dirichlet boundary condition. We show the existence of continua of solutions for which one component is blowing-up at a certain number of points. The proofs use singular perturbation methods.

Description: According to the celebrated Jaworski theorem, a finite-dimensional aperiodic dynamical system $(X,T)$ embeds in the one-dimensional cubical shift $([0,1]{}^{\mathbb {Z}},\hbox {shift})$ . If $X$ admits periodic points (still assuming $\dim (X) 〈 \infty $ ), then we show in this paper that periodic dimension $\hbox {perdim}(X,T) 〈 {d}/{2}$ implies that $(X,T)$ embeds in the $d$ -dimensional cubical shift $(([0,1]^{d})^{\mathbb {Z}},\hbox {shift})$ . This verifies a conjecture by Lindenstrauss and Tsukamoto for finite-dimensional systems. Moreover, for an infinite-dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb {Z}},\hbox {shift})$ . Furthermore, we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti–Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $\hbox {mdim}(X,T) 〈 {d}/{16}$ embeds in the $(d+1)$ - cubical shift $(([0,1]^{d+1})^{\mathbb {Z}},\hbox {shift})$ .

Description: Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type defined over number fields. We prove that, for any $\sigma \in \mathrm {Gal}(\bar {\mathbb Q}/{\mathbb Q})$ different from the identity and the complex conjugation, there is a Beauville surface $S$ such that $S$ and its Galois conjugate $S^{\sigma }$ have non-isomorphic fundamental groups. This in turn easily implies that the action of $\mathrm {Gal}(\bar {\mathbb Q}/{\mathbb Q})$ on the set of isomorphism classes of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that $\mathrm {Gal}(\bar {\mathbb Q}/{\mathbb Q})$ acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above-mentioned authors. These results on Beauville surfaces heavily depend on the fact that the absolute Galois group acts faithfully on regular dessins, a result that we prove in this paper. Moreover, we show the stronger result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

Description: It is elementary to observe that functions interpolating an extremal two-point Pick problem on the polydisc are left inverses to complex geodesics. In the present article, we show that the same property holds for a three-point Pick problem on polydiscs, that is, the solution may be expressed in terms of 3-complex geodesics. Using this idea, we are able to obtain formulas and a uniqueness theorem for solutions of extremal problems. In particular, we determine a class of rational inner functions that solve the interpolation problem. Possible extensions and further investigations are also discussed.

Description: We prove that connected higher-rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces ${\mathcal E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that ${\rm SL}(3,\mathbb {R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second-named author asserting that ${\rm SL}(3,\mathbb {R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher-rank groups, it is sufficient to prove strong property (T) for ${\rm Sp}(2,\mathbb {R})$ and its universal covering group. As consequences of our main result, it follows that for $X \in {\mathcal E}_{10}$ , connected higher-rank simple Lie groups and their lattices have property ( $\hbox {F}_X$ ) of Bader, Furman, Gelander and Monod, and that the expanders constructed from a lattice in a connected higher-rank simple Lie group do not admit a coarse embedding into $X$ .

Description: We study the action of ${\rm Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ on the category of Belyĭ functions (finite étale covers of $\mathbb {P}^1_{{\overline {\mathbb {Q}}}}\setminus \{0,1,\infty \}$ ). We describe a new combinatorial ${\rm Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ -invariant for Belyĭ functions whose monodromy cycle types above 0 and $\infty $ are the same. We use a version of our invariant to prove that ${\rm Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ acts faithfully on the set of Belyĭ functions whose monodromy cycle types above 0 and $\infty $ are the same; the proof of this result involves a version of Belyĭ's Theorem for meromorphic functions of odd degree. Using our invariant, we obtain that for all $k 〈 2^{\sqrt {\frac {2}{3}}}$ and all positive integers $N$ , there exists a positive integer $n \le N$ such that the set of degree $n$ Belyĭ functions of a particular rational Nielsen class must split into at least $\Omega (k^{\sqrt {N}})$ Galois orbits.

Description: In this paper, we find approximate solutions of certain Riemann–Hilbert boundary value problems for minimal surfaces in $\mathbb {R}^n$ and null holomorphic curves in $\mathbb {C}^n$ for any $n\ge 3$ . With this tool in hand, we construct complete conformally immersed minimal surfaces in $\mathbb {R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb {R}^n$ which extend continuously up to the boundary; for $n\ge 5,$ we find embeddings with these properties.

Description: We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\mathbb {Z}_p^d$ -extensions of function fields ramifying at a finite set of places.

Description: The observation that the zero-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$ of Bieri–Neumann–Strebel–Renz can be interpreted as a horospherical limit set opens a direct trail from Poincaré's limit set $\Lambda (\Gamma )$ of a discrete group $\Gamma $ of Möbius transformations (which contains the horospherical limit set of $\Gamma $ ) to the roots of tropical geometry (closely related to $\Sigma ^{0}(G;A)$ when $G$ is abelian). We explore this trail by introducing the horospherical limit set, $\Sigma (M;A)$ , of a $G$ -module $A$ when $G$ acts by isometries on a proper $\textit {CAT}(0)$ metric space $M$ . This is a subset of the boundary at infinity $\partial M$ . On the way, we meet instances where $\Sigma (M;A)$ is the set of all conical limit points ( $G$ geometrically finite and $M$ hyperbolic), the complement of the radial projection of a tropical variety ( $G$ abelian and $M$ Euclidean) or the complement of a spherical building ( $G$ arithmetic and $M$ symmetric).

Description: In this paper, we give a geometric construction of the quantum group $U_t({\mathcal G})$ using Nakajima categories, which were developed in Scherotzke (‘Quiver varieties and self-injective algebras’, Preprint, 2014, arxiv.org/abs/1405.4729). Our methods allow us to establish a direct connection between the algebraic realization of the quantum group as Hall algebra by Bridgeland (‘Quantum groups via Hall algebras of complexes’, Annals of Maths. 177 (2013) 739–759) and its geometric counterpart by Qin (‘Quantum groups via cyclic quiver varieties I’, Compos. Math. , Preprint, 2013, arxiv.org/abs/1312.1101).

Description: Let $T$ be a tile in $\mathbb {Z}^n$ , meaning a finite subset of $\mathbb {Z}^n$ . It may or may not tile $\mathbb {Z}^n$ , in the sense of $\mathbb {Z}^n$ having a partition into copies of $T$ . However, we prove that $T$ does tile $\mathbb {Z}^d$ for some $d$ . This resolves a conjecture of Chalcraft.

Description: In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [Z. Z. Tang and W. J. Yan, ‘Isoparametric foliation and Yau conjecture on the first eigenvalue’, J. Differential. Geom. 94 (2013) 521–540; Z. Z. Tang, Y. Q. Xie and W. J. Yan, ‘Isoparametric foliation and Yau conjecture on the first eigenvalue, II’, J. Funct. Anal. 266 (2014) 6174–6199]. Eells and Lemaire [ Selected topics in harmonic maps , C.B.M.S. Regional Conference Series in Mathematics 50 (American Mathematical Society, Providence, RI, 1983)] posed a problem to characterize the compact Riemannian manifold $M$ for which there is an eigenmap from $M$ to $S^n$ . As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [‘Minimal submanifolds in a sphere II’, Bull. London Math. Soc. 23 (1991) 387–390] (posed in 1991) on minimal submanifolds in unit spheres. Note that these conjectures of Leung have been proved in the case that the normal connection is flat [T. Hasanis and T. Vlachos, ‘Ricci curvatures and minimal submanifolds’, Pacific J. Math. 197 (2001) 13–24].

Description: For $K$ belonging to the class of convex bodies in $ \mathbb {R}^n$ , we consider the $\lambda _1$ -product functional, defined by $\lambda _ 1 (K) \lambda _ 1 (K^o)$ , where $K^o$ is the polar body of $K$ , and $\lambda _1 (\cdot )$ is the first Dirichlet eigenvalue of the Dirichlet Laplacian. As a counterpart of the classical Blaschke–Santaló inequality for the volume product, we prove that the $\lambda _1$ -product is minimized by balls. Much more challenging is the problem of maximizing the $\lambda _1$ -product modulo invertible linear transformations, which is the analog of the famous Mahler conjecture for the volume product in Convex Geometry. We solve the problem in dimension $n=2$ for axisymmetric convex bodies, by proving that the solution is the square. To that aim we first reduce our problem to a reverse Faber–Krahn inequality for axisymmetric convex octagons, and then we identify an optimal octagon with the one that degenerates into a square. For this latter challenge, we employ a hybrid method inspired by the Polymath blog by Tao, which is based on the joint use of theoretical arguments to settle octagons lying in computable ‘neighborhoods’ of the square, and of a numerical argument (rigorously working thanks to the monotonicity by inclusions of the involved functionals) to settle octagons lying outside the confidence zones.

Description: In this paper, we prove existence and uniqueness of matings of a large class of renormalizable cubic polynomials with one fixed critical point and the other cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part toward a conjecture by L. Tan, stating that all (cubic) Newton maps can be described as matings or captures.

Description: We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of $H^1$ -curves around a critical point or a critical $ {\mathbb R}^1$ orbit of a Finsler isometry-invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll–Meyer's splitting lemmas on Riemannian manifolds [Gromoll and Meyer, ‘On differentiable functions with isolated critical points’, Topology 8 (1969) 361–369; Gromoll and Meyer ‘Periodic geodesics on compact Riemannian manifolds’, J. Differential Geom. 3 (1969) 493–510]. As an application, we extend to Finsler manifolds a result by Grove and Tanaka [‘On the number of invariant closed geodesics’, Acta Math. 140 (1978) 33–48; Tanaka, ‘On the existence of infinitely many isometry-invariant geodesics’, J. Differential Geom. 17 (1982) 171–184] about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.

Description: We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of Rémond. This connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties.

Description: Koenig and Xi introduced affine cellular algebras . Kleshchev and Loubert (and Miemietz) showed that an important class of infinite-dimensional algebras, the Khovanov–Lauda–Rouquier algebras $R(\Gamma )$ of finite Lie type $\Gamma$ , are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of quasihereditary algebras of Cline–Parshall–Scott in a finite-dimensional situation. A fundamental result of Cline–Parshall–Scott says that a finite-dimensional algebra $A$ is quasihereditary if and only if the category of finite-dimensional $A$ -modules is a highest weight category . On the other hand, S. Kato and Brundan–Kleshchev–McNamara proved that the category of finitely generated graded $R(\Gamma )$ -modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an affine quasihereditary algebra and an affine highest weight category . In particular, we prove an affine analog of the Cline–Parshall–Scott Theorem. We also develop stratified versions of these notions.

Description: Let $k$ be a complete non-Archimedean non-trivially real-valued algebraically closed field. Let $\phi$ be a finite endomorphism of $\mathbb {P}^1_k$ . Given a closed point $x \in \mathbb {P}^1_k$ , we are interested in the radius $f(x)$ of the largest Berkovich open ball centred at $x$ over which the morphism $\phi ^{\mathrm {an}}$ is a topological fibration. Interestingly, the function $f: \mathbb {P}_k^1(k) \to \mathbb {R}_{\geq 0}$ admits a strong tameness property in that it is controlled by a non-empty finite graph contained in $\mathbb {P}^{1,\mathrm {an}}_k$ . We show that this result can be generalized to the case of finite morphisms $\phi : V' \to V$ between integral projective $k$ -varieties where $V$ is normal. These results are applications of Hrushovski and Loeser's work on the homotopy type of non-Archimedean analytic spaces [E. Hrushovski and F. Loeser, ‘Non-Archimedean tame topology and stably dominated types’, Preprint, 2010, arXiv:1009.0252v3].

Description: Recently, the second author has associated a finite ${{\mathbf {F}}}_q[T]$ -module $H$ to the Carlitz module over a finite extension of ${{\mathbf {F}}}_q(T)$ . This module is an analogue of the ideal class group of a number field. In this paper, we study the Galois module structure of this module $H$ for ‘cyclotomic’ extensions of ${{\mathbf {F}}}_q(T)$ . We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$ -adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular units, and give a negative answer to Anderson's Kummer–Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second author's class number formula for the Carlitz module.

Description: Hofer asked (1989): if $n\geq 3$ can the cylinder ${B}^2(1) \times \mathbb {R}^{2(n-1)}$ be symplectically embedded into ${B}^{2(n-1)}(R) \times \mathbb {R}^2$ for some $R 〉 0$ ? We show that this is the case if $R \geq \sqrt {2^{n-1}+2^{n-2}-2}$ . It follows that there are no intermediate symplectic capacities, between $1$ -capacities, first constructed by Gromov in 1985, and $n$ -capacities like the volume. In 2008, Guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property .

Description: We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras ${{\mathcal {R}}^\Lambda _{{n}}}$ of type $A$ when $e=0$ (the linear quiver) or $e 〉 n$ . We prove that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0,$ we show that the Khovanov–Lauda–Rouquier grading on the quiver Hecke algebras is compatible with the Koszul grading on the blocks of parabolic category ${{\mathcal {O}}^\Lambda _{{}}}$ given by Backelin, building on the work of Beilinson, Ginzburg and Soergel. As a consequence, $e=0$ our cyclotomic quiver Schur algebras are Koszul over fields of characteristic zero. Finally, we give an Lascoux–Leclerc–Thibon-like algorithm for computing the graded decomposition numbers of the cyclotomic quiver Schur algebras in characteristic zero.

Description: Over a global field $K$ (number field, or function field of a curve over a finite field $F$ ), arithmetic duality theorems for the Galois cohomology of tori and finite Galois modules have long been known. More recent work investigates the case where $K$ is the function field of a curve over a $p$ -adic field. For $K$ the function field of a curve over the formal series field $F={{\mathbb C}}((t))$ , we establish analogous duality theorems. We thus control the obstruction to the local–global principle and to weak approximation for homogeneous spaces of tori. There are differences with the afore described cases. For example, the Hasse principle need not hold for principal homogeneous spaces of a $K$ -rational torus.

Description: We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c. groups are elementary and $[A]$ -regular t.d.l.c.s.c. groups are elementary.

Description: We study the decomposition matrices of the unipotent $\ell$ -blocks of finite special unitary groups ${{\operatorname {SU}}}_n(q)$ for unitary primes $\ell$ larger than $n$ . Up to few unknown entries, we give a complete solution for $n=2,\ldots ,10$ . We also prove a general result for two-column partitions when $\ell$ divides $q+1$ . This is achieved using projective modules coming from the $\ell$ -adic cohomology of Deligne–Lusztig varieties.

Description: We show that the $\mathscr {B}$ -free subshift $(S,X_{\mathscr {B}})$ associated to a $\mathscr {B}$ -free system is intrinsically ergodic, that is, it has exactly one measure of maximal entropy. Moreover, we study invariant measures for such systems. It is proved that each ergodic invariant measure is of joining type, determined by a joining of the Mirsky measure of a $\mathscr {B}'$ -free subshift contained in $(S,X_{\mathscr {B}})$ and an ergodic invariant measure of the full shift on $\{0,1\}^{{{\mathbb {Z}}}}$ . Moreover, each ergodic joining type measure yields a measure-theoretic dynamical system with infinite rational part of the spectrum corresponding to the above Mirsky measure. Finally, we show that, in general, hereditary systems may not be intrinsically ergodic.

Description: Let $X$ be a smooth curve defined over ${\bar {{\mathbb Q}}}$ , let $a,b\in {{\mathbb P}}^1({\bar {{\mathbb Q}}})$ and let $f_{\lambda }(x)\in {\bar {{\mathbb Q}}}(x)$ be an algebraic family of rational maps indexed by all $\lambda \in X({{\mathbb C}})$ . We study whether there exist infinitely many $\lambda \in X({{\mathbb C}})$ such that both $a$ and $b$ are preperiodic for $f_{\lambda }$ . In particular, we show that if $P,Q\in {\bar {{\mathbb Q}}}[x]$ such that $\deg (P)\ge 2+ \deg (Q)$ , and if $a,b\in {\bar {{\mathbb Q}}}$ such that $a$ is periodic for ${P(x)}/{Q(x)}$ , but $b$ is not preperiodic for ${P(x)}/{Q(x)}$ , then there exist at most finitely many $\lambda \in {{\mathbb C}}$ such that both $a$ and $b$ are preperiodic for $P(x)/Q(x)+ \lambda$ . We also prove a similar result for certain two-dimensional families of endomorphisms of ${{\mathbb P}}^2$ . As a by-product of our method, we extend a recent result of Ingram [‘Variation of the canonical height for a family of polynomials’, J. reine. angew. Math. 685 (2013), 73–97] for the variation of the canonical height in a family of polynomials to a similar result for families of rational maps.

Description: We obtain an essentially optimal estimate for the moment of order $32/3$ of the exponential sum having argument ${{\alpha }}x^3+ {{\beta }}x^2$ . Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds $10$ .

Description: We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in {{\mathbb N}}$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$ -periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$ . Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above, we show that rays which are invariant under $f^p$ , together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p$ -periodic point or one parabolic immediate basin invariant under $f^p$ . This result generalizes the Goldberg–Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel disks; that ‘hidden components’ of a bounded Siegel disk have to be either wandering domains or preperiodic to the Siegel disk itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

Description: This paper studies the local geometry of compactified Jacobians. The main result is a presentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve. The authors have investigated the geometric and combinatorial properties of these rings in previous work, and consequences for compactified Jacobians are presented in this paper. Similar results are given for the local structure of the universal compactified Jacobian over the moduli space of stable curves.

Description: We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over ${\mathbb {Q}}$ . Our main technique consists in arithmetic deformations of Delsarte surfaces, but we also use K3 surfaces and wild automorphisms.

Description: The lower spectral radius, or joint spectral subradius, of a set of real $d \times d$ matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set. The lower spectral radius arises naturally in connection with a number of topics including combinatorics on words, the stability of linear inclusions in control theory, and the study of random Cantor sets. In this article, we apply some ideas originating in the study of dominated splittings of linear cocycles over a dynamical system to characterize the points of continuity of the lower spectral radius on the set of all compact sets of invertible $d \times d$ matrices. As an application, we exhibit open sets of pairs of $2 \times 2$ matrices within which the analogue of the Lagarias–Wang finiteness property for the lower spectral radius fails on a residual set, and discuss some implications of this result for the computation of the lower spectral radius.

Description: The paper generalizes, for a wide class of elliptic curves defined over ${{\mathbb {Q}}}$ , the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curve $X_0(49)$ , including showing that there is a large class of explicit quadratic twists whose complex $L$ -series does not vanish at $s=1$ , and for which the full Birch–Swinnerton-Dyer conjecture is valid.

Description: Mixed $f$ -divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log-concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov–Fenchel-type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log-concave functions. Special cases of $f$ -divergences are mixed $L_\lambda$ -affine surface areas for log-concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santaló-type inequality.

Description: Let $P^+ (n)$ denote the largest prime factor of the integer $n$ and $\Phi _{12} (n)=n^4-n^2+1$ . We prove that for $X$ large enough we have: \[P^+ \left ( \prod _{X\lt n\le 2X}\Phi _{12}(n)\right ) \ge X^{1+c}\ {\rm with}\ c= 10^{-26\,531}.\] Résumé On note $P^+ (n)$ le plus grand facteur premier de l’entier $n$ et $\Phi _{12} (n)=n^4-n^2+1$ . Nous montrons pour $X$ assez grand la minoration : \[P^+ \left ( \prod _{X\lt n\le 2X}\Phi _{12}(n)\right ) \ge X^{1+c}\ {\rm avec}\ c= 10^{-26\,531}.\]

Description: By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo–Nirenberg (GN) inequalities on the half-space ${\mathbb R}^{n-1}\times {\mathbb R}_+ $ , $n\geq 1$ equipped with the weight $\omega (x) = x_n^a$ , $a\geq 0$ . It amounts to work with the fractional dimension $n_a = n+a$ . The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domain ${\mathbb R}^{n-m}\times {\mathbb R}^m_+ $ and the weights are $\omega (x,t) = t_1^{a_1}\cdots t_m^{a_m}$ , where $n\geq m$ , $m\geq 0$ and $a_1,\ldots ,a_m\geq 0$ . A weighted $L^p$ -logarithmic Sobolev inequality is derived from these inequalities.

Description: We study positive solutions of equation (E1) $-\Delta u +u^p|\nabla u|^q= 0$ ( $0\leq p$ , $0\leq q\leq 2$ , $p+q 〉 1$ ) and (E2) $-\Delta u +u^p + |\nabla u|^q =0$ ( $p 〉 1$ , $1\lt q\leq 2$ ) in a smooth bounded domain $\Omega \subset \mathbb {R}^N$ . We obtain a sharp condition on $p$ and $q$ under which, for every positive, finite Borel measure $\mu$ on $\partial \Omega$ , there exists a solution such that $u=\mu$ on $\partial \Omega$ . Furthermore, if the condition mentioned above fails, then any isolated point singularity on $\partial \Omega$ is removable, namely, there is no positive solution that vanishes on $\partial \Omega$ everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of $\partial \Omega$ . In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when $p=0$ but not the general case.

Description: We develop the general theory of Jack–Laurent symmetric functions, which are certain generalizations of the Jack symmetric functions, depending on an additional parameter $p_0$ .

Description: We study critical orbits and bifurcations within the moduli space ${\mathrm {M}}_2$ of quadratic rational maps, $f: {{\mathbb P}}^1\to {{\mathbb P}}^1$ . We focus on the family of curves, ${\mathrm {Per}}_1(\lambda ) \subset {\mathrm {M}}_2$ for $\lambda \in {{\mathbb C}}$ , defined by the condition that each $f\in {\mathrm {Per}}_1(\lambda )$ has a fixed point of multiplier $\lambda$ . We prove that the curve ${\mathrm {Per}}_1(\lambda )$ contains infinitely many postcritically finite maps if and only if $\lambda =0$ , addressing a special case of Baker and DeMarco (‘Special curves and postcritically finite polynomials’, Forum of Math. Pi 1 (2013), doi: 10.1017/fmp.2013.2; Conjecture 1.4). We also show that the two critical points of $f$ define distinct bifurcation measures along ${\mathrm {Per}}_1(\lambda )$ .

Description: We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base field. As an application, we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field is, in a suitable sense, the branch of an algebraic curve.

Description: We define a simple property on an infinite directed graph G and show that it is necessary and sufficient for the existence of a transient potential on the associated countable Markov shift.

Description: We consider groups G interpretable in a supersimple finite rank theory T such that T eq eliminates . It is shown that G has a definable soluble radical. If G has rank 2, then if G is pseudofinite, it is soluble-by-finite, and partial results are obtained under weaker hypotheses, such as ‘functional unimodularity’ of the theory. A classification is obtained when T is pseudofinite and G has a definable and definably primitive action on a rank 1 set.

Description: We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group and use this equivalence to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of topologically distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces.

Description: In this paper, we generalize the construction of the Bloch–Kato exponential map to complete discrete valuation fields of mixed characteristic (0, p ) whose residue fields have a finite p -basis. As an application, we prove an explicit reciprocity law, generalizing Théorème IV.2.1 in [F. Cherbonnier and P. Colmez, ‘Théorie d’Iwasawa des représentations p -adiques d'un corps local', J. Amer. Math. Soc. 12 (1999) 241–268]. This result relies on the calculation of the Galois cohomology of a p -adic representation V in terms of its ( , G )-module.

Description: Given a set C R d , let p (C) be the probability that a random d -dimensional unimodular lattice, chosen according to Haar measure on SL( d , Z)\ SL( d , R), is disjoint from C \ { 0 }. For special convex sets C we prove bounds on p (C) that are sharp up to a scaling of C by a constant. We also prove bounds on a variant of p (C) where the probability is conditioned on the random lattice containing a fixed given point p != 0 . Our bounds have applications, among other things, to the asymptotic properties of the collision kernel of the periodic Lorentz gas in the Boltzmann–Grad limit, in arbitrary dimension d .

Description: Let K be a convex body in R n . We introduce a new affine invariant, which we call K , that can be found in three different ways: as a limit of normalized L p -affine surface areas; as the relative entropy of the cone measure of K and the cone measure of K °; as the limit of the volume difference of K and L p -centroid bodies. We investigate properties of K and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.

Description: We establish the precise bounds for the amount of determinacy provable in second-order arithmetic. We show that, for every natural number n , second-order arithmetic can prove that determinacy holds for Boolean combinations of n many classes, but it cannot prove that all finite Boolean combinations of classes are determined. More specifically, we prove that , but that , where is the n th level in the difference hierarchy of classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that, for every true 1 4 sentence T (as, for instance, -DET) and every and .

Description: We describe a method for computing equations of hyperelliptic Shimura curves attached to indefinite quaternion algebras over Q and Atkin–Lehner quotients of them. It exploits Cerednik–Drinfeld 's non-archimedean uniformization of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points, as introduced in Molina [‘Ribet bimodules and specialization of Heegner points’, Israel Journal of Mathematics ]. We provide a list of equations of Shimura curves and quotients of them obtained by our method that had been conjectured by Kurihara.

Description: We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W R 3 . The Cartesian coordinates ( x , y , z ) are chosen so that the mean free surface of the liquid F lies in the ( x , z )-plane and the y -axis is directed upwards. We study the case when W is an axially symmetric, convex, bounded domain such that W F x (–, 0). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction , there is a change of x , z -coordinates by a rotation around the y -axis so that is odd in x -variable. The second result of the paper gives the following monotonicity property of the fundamental eigenfunction . If is odd in x -variable, then it is strictly monotonic in x -variable. This property has the following hydrodynamical meaning. If the liquid oscillates freely with the fundamental frequency according to , then the free surface elevation of the liquid is increasing along each line parallel to the x -axis during one half-period of time and decreasing during the other half-period. The proof of the second result is based on the method developed by Jerison and Nadirashvili for the hot-spots problem for the Neumann–Laplacian.