Character varieties of abelian groups

We prove that for every reductive group $G$ with a maximal torus ${\mathbb {T}}$ and the Weyl group $W,\, {\mathbb {T}}^N/W$ is the normalization of the irreducible component, $X_G^0({\mathbb {Z}}^N)$ , of the $G$ -character variety $X_G({\mathbb {Z}}^N)$ of ${\mathbb {Z}}^N$ containing the trivial representation. We also prove that $X_G^0({\mathbb {Z}}^N)={\mathbb {T}}^N/W$ for all classical groups. Additionally, we prove that even though there are no irreducible representations in $X_G^0({\mathbb {Z}}^N)$ for non-abelian $G$ , the tangent spaces to $X_G^0({\mathbb {Z}}^N)$ coincide with $H^1({\mathbb {Z}}^N, Ad\, \rho )$ . Consequently, $X_G^0({\mathbb {Z}}^2)$ , has the “Goldman” symplectic form for which the combinatorial formulas for Goldman bracket hold.     

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$ , and let $\mu :\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$ , $O(V)$ , or $Sp(V)$ . For these classes of examples, $\mu ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $\mu ^{-1}(0)/\!/G$ .     

On the Union of Increasing Chains of Torsion-Free Modules Over Integral Domains

Motivated by Hill’s criterion of freeness for abelian groups, we establish a generalization of that result to categories ${\mathcal{C}}$ of torsion-free modules over integral domains, which are closed with respect to the formation of direct sums, and in which every object can be decomposed into direct sums of objects of ${\mathcal{C}}$ of rank at most a fixed limit cardinal number κ. Our main result states that a module belongs to ${\mathcal{C}}$ if it is the union of a continuous, well-ordered, ascending chain of length κ, consisting of pure submodules which are objects of ${\mathcal{C}}$ . As corollaries, we derive versions of Hill’s theorem for some classes of torsion-free modules over domains, and a generalization of a well-known result by Kaplansky.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

The distribution of the zeros of the Goss zeta-function for A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)

Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

A converse of Baer’s theorem

Schur’s classical theorem states that for a group $G$ , if $G/Z(G)$ is finite, then $G'$ is finite. Baer extended this theorem for the factor group $G/Z_n(G)$ , in which $Z_n(G)$ is the $n$ -th term of the upper central series of $G$ . Hekster proved a converse of Baer’s theorem as follows: If $G$ is a finitely generated group such that $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ denotes the $(n+1)$ st term of the lower central series of $G$ . In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$ . Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ .     

Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta _p u=f(u)$ in the case $2<p< 3$ and $f(\cdot )$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot )$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta _p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.     

Symbolic extensions and dominated splittings for generic C^{1}-diffeomorphisms

Let $\mathrm{Diff }^1(M)$ be the set of all $C^1$ -diffeomorphisms $f:M\rightarrow M$ , where $M$ is a compact boundaryless d-dimensional manifold, $d\ge 2$ . We prove that there is a residual subset $\mathfrak R $ of $\mathrm{Diff }^1(M)$ such that if $f\in \mathfrak R $ and if $H(p)$ is the homoclinic class associated with a hyperbolic periodic point $p$ , then either $H(p)$ admits a dominated splitting of the form $E\oplus F_1\oplus \dots \oplus F_k\oplus G$ , where $F_i$ is not hyperbolic and one-dimensional, or $f|_{H(p)}$ has no symbolic extensions.     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$      

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Strongly embedded subspaces of p-convex Banach function spaces

Let $X(\mu )$ be a p-convex ( $1\le p<\infty $ ) order continuous Banach function space over a positive finite measure  $\mu $ . We characterize the subspaces of  $X(\mu )$ which can be found simultaneously in  $X(\mu )$ and a suitable $L^1(\eta )$ space, where $\eta $ is a positive finite measure related to the representation of  $X(\mu )$ as an $L^p(m)$ space of a vector measure  $m$ . We provide in this way new tools to analyze the strict singularity of the inclusion of  $X(\mu )$ in such an $L^1$ space. No rearrangement invariant type restrictions on  $X(\mu )$ are required.     

Open Gromov–Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X , \omega )$ . We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$ . Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$ , every relative homology class $d \in H_2 (X , L ; \mathbb{Z })$ and adequate number of incidence conditions in $L$ or $X$ , the weighted number of $J$ -holomorphic discs with boundary on $L$ , homologous to $d$ , and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$ , provided that at least one incidence condition lies in $L$ . These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring $A$ .     

Singular gradient flow of the distance function and homotopy equivalence

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ .     

Normal generation and \ell ^2-Betti numbers of groups

The normal rank of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell ^2$ -Betti number and conjecture the inequality $\beta _1^{(2)}(G) \le \mathrm{nrk}(G)-1$ for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $n\ge 2$ and every $\varepsilon >0$ , we give an example of a simple group $Q$ (with torsion) such that $\beta _1^{(2)}(Q) \ge n-1-\varepsilon $ . These groups also provide examples of simple groups of rank exactly $n$ for every $n\ge 2$ ; existence of such examples for $n> 3$ was unknown until now.     

The Saito–Kurokawa lifting and Darmon points

Let $E_{/_\mathbb{Q }}$ be an elliptic curve of conductor $Np$ with $p\not \mid N$ and let $f$ be its associated newform of weight $2$ . Denote by $f_\infty $ the $p$ -adic Hida family passing though $f$ , and by $F_\infty $ its $\varLambda $ -adic Saito–Kurokawa lift. The $p$ -adic family $F_\infty $ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde{A}_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$ . We relate explicitly certain global points on $E$ (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their $p$ -adic derivatives, evaluated at weight $k=2$ .     

Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

According to Mukai and Iliev, a smooth prime Fano threefold $X$ of genus $9$ is associated with a surface $\mathbb{P }(\mathcal{V })$ , ruled over a smooth plane quartic $\varGamma $ , and the derived category of $\varGamma $ embeds into that of $X$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $2$ stable sheaves on $X$ with odd determinant. For each $c_2 \ge 7$ , we prove that a component of their moduli space $\mathsf{M}_X(2,1,c_2)$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $\varGamma $ , having enough sections when twisted by $\mathcal{V }$ . For $c_2=7$ , we prove that $\mathsf{M}_X(2,1,7)$ is isomorphic to the blow-up of the Picard variety $\text{ Pic}^{2}({\varGamma })$ along the curve parametrizing lines contained in $X$ .