Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces
Marco
Matone
2989-3004
Abstract: We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces correspond to Riemann period matrices satisfying a set of equations which lead to a number theoretical problem. It turns out that these surfaces precisely correspond to branched covering of the torus. This reflects in a Jacobian with a particular kind of complex multiplication.
A new variational characterization of $n$-dimensional space forms
Zejun
Hu;
Haizhong
Li
3005-3023
Abstract: A Riemannian manifold $(M^n,g)$ is associated with a Schouten $(0,2)$-tensor $C_g$ which is a naturally defined Codazzi tensor in case $(M^n,g)$ is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional $\mathcal{F}_k[g]=\int_M\sigma_k(C_g)dvol_g$ defined on $\mathcal{M}_1=\{g\in\mathcal{M}\vert Vol(g)=1\}$, where $\mathcal{M}$ is the space of smooth Riemannian metrics on a compact smooth manifold $M$ and $\{\sigma_k(C_g), 1\leq k\leq n\}$ is the elementary symmetric functions of the eigenvalues of $C_g$ with respect to $g$. We prove that if $n\geq 5$ and a conformally flat metric $g$ is a critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$, then $g$ must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$ characterized the three-dimensional space forms.
The $\forall\exists$-theory of $\mathcal{R}(\leq,\vee,\wedge)$ is undecidable
Russell
G.
Miller;
Andre
O.
Nies;
Richard
A.
Shore
3025-3067
Abstract: The three quantifier theory of $(\mathcal{R},\leq_{T})$, the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of $\mathcal{R}$ that lies between the two and three quantifier theories with $\leq_{T}$ but includes function symbols. Theorem. The two quantifier theory of $(\mathcal{R},\leq ,\vee,\wedge)$, the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on $\mathcal{R}$) is undecidable. The same result holds for various lattices of ideals of $\mathcal{R}$ which are natural extensions of $\mathcal{R}$ preserving join and infimum when it exits.
Szegö kernels and finite group actions
Roberto
Paoletti
3069-3076
Abstract: In the context of almost complex quantization, a natural generalization of algebro-geometric linear series on a compact symplectic manifold has been proposed. Here we suppose given a compatible action of a finite group and consider the linear subseries associated to the irreducible representations of $G$, give conditions under which these are base-point-free and study properties of the associated projective morphisms. The results obtained are new even in the complex projective case.
Homotopy equivalence of two families of complexes
Giandomenico
Boffi;
David
A.
Buchsbaum
3077-3107
Abstract: An explicit homotopy equivalence is established between two families of complexes, both of which generalize the classical Koszul complex.
Trees, parking functions, syzygies, and deformations of monomial ideals
Alexander
Postnikov;
Boris
Shapiro
3109-3142
Abstract: For a graph $G$, we construct two algebras whose dimensions are both equal to the number of spanning trees of $G$. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to $G$-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.
Core versus graded core, and global sections of line bundles
Eero
Hyry;
Karen
E.
Smith
3143-3166
Abstract: We find formulas for the graded core of certain $\mathfrak{m}$-primary ideals in a graded ring. In particular, if $S$ is the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal of $S$ generated by all elements of degrees at least $N$ (for some, equivalently every, large $N$) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.
Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc
Pekka
J.
Nieminen;
Eero
Saksman
3167-3187
Abstract: Let $\phi$ be a holomorphic self-map of the unit disc $\mathbb{D}$. For every $\alpha \in \partial\mathbb{D}$, there is a measure $\tau_\alpha$ on $\partial\mathbb{D}$ (sometimes called Aleksandrov measure) defined by the Poisson representation $\operatorname{Re}(\alpha+\phi(z))/(\alpha-\phi(z)) = \int P(z,\zeta) \,d\tau_\alpha(\zeta)$. Its singular part $\sigma_\alpha$ measures in a natural way the ``affinity'' of $\phi$ for the boundary value $\alpha$. The affinity for values $w$ inside $\mathbb{D}$ is provided by the Nevanlinna counting function $N(w)$ of $\phi$. We introduce a natural measure-valued refinement $M_w$ of $N(w)$ and establish that the measures $\{\sigma_\alpha\}_{\alpha\in\partial\mathbb{D}}$are obtained as boundary values of the refined Nevanlinna counting function $M$. More precisely, we prove that $\sigma_\alpha$ is the weak$^*$ limit of $M_w$ whenever $w$ converges to $\alpha$non-tangentially outside a small exceptional set $E$. We obtain a sharp estimate for the size of $E$ in the sense of capacity.
On Diophantine definability and decidability in some infinite totally real extensions of $\mathbb Q$
Alexandra
Shlapentokh
3189-3207
Abstract: Let $M$ be a number field, and $W_M$ a set of its non-Archimedean primes. Then let $O_{M,W_M} = \{x \in M\vert \operatorname{ord}_{\mathfrak{t}}x \geq 0, \, \forall \mathfrak{t} \, \not \in W_M\}$. Let $\{p_1,\ldots,p_r\}$ be a finite set of prime numbers. Let $F_{inf}$ be the field generated by all the $p_i^{j}$-th roots of unity as $j \rightarrow \infty$ and $i=1,\ldots,r$. Let $K_{inf}$ be the largest totally real subfield of $F_{inf}$. Then for any $\varepsilon > 0$, there exist a number field $M \subset K_{inf}$, and a set $W_M$ of non-Archimedean primes of $M$ such that $W_M$ has density greater than $1-\varepsilon$, and $\mathbb{Z}$ has a Diophantine definition over the integral closure of $O_{M,W_M}$ in $K_{inf}$.
Uncorrelatedness and orthogonality for vector-valued processes
Peter
A.
Loeb;
Horst
Osswald;
Yeneng
Sun;
Zhixiang
Zhang
3209-3225
Abstract: For a square integrable vector-valued process $f$ on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for $f$ are presented. The process $f$ is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for $f$ involving the biorthogonal representation for the conditional expectation of $f$ with respect to the usual product $\sigma$-algebra is presented.
Existence of local sufficiently smooth solutions to the complex Monge-Ampère equation
Saoussen
Kallel-Jallouli
3227-3242
Abstract: We prove the $C^{\infty }$ local solvability of the $n$-dimensional complex Monge-Ampère equation $\det \left( u_{i\overline{j}}\right) =f\left( z,u,\nabla u\right)$, $f\geq 0$, in a neighborhood of any point $z_{0}$where $f\left( z_{0}\right) =0$.
Construction and recognition of hyperbolic 3-manifolds with geodesic boundary
Roberto
Frigerio;
Carlo
Petronio
3243-3282
Abstract: We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition. Our theory has been exploited to classify all the orientable finite-volume hyperbolic $3$-manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.
Infinitely many solutions to fourth order superlinear periodic problems
Monica
Conti;
Susanna
Terracini;
Gianmaria
Verzini
3283-3300
Abstract: We present a new min-max approach to the search of multiple $T$-periodic solutions to a class of fourth order equations \begin{displaymath}u^{iv}(t)-c u''(t)=f(t,u(t)),\hspace{5mm}t\in[0,T],\end{displaymath} where $f(t,u)$ is continuous, $T$-periodic in $t$ and satisfies a superlinearity assumption when $\vert u\vert\to\infty$. For every $n\in\mathbb{N}$, we prove the existence of a $T$-periodic solution having exactly $2n$ zeroes in $(0,T]$.
The cohomology of certain Hopf algebras associated with $p$-groups
Justin
M.
Mauger
3301-3323
Abstract: We study the cohomology $H^*(A)=\operatorname{Ext}_A^*(k,k)$ of a locally finite, connected, cocommutative Hopf algebra $A$ over $k=\mathbb{F} _p$. Specifically, we are interested in those algebras $A$ for which $H^*(A)$ is generated as an algebra by $H^1(A)$ and $H^2(A)$. We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras $F\rightarrow A\rightarrow B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be semi-Koszul. Special attention is given to the case in which $A$ is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central series of a $p$-group. We show that the algebras arising in this way from extensions by $\mathbb{Z} /(p)$ of an abelian $p$-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 $p$-groups, and it is shown that these are all semi-Koszul for $p\geq 5$.
On the divergence of the Rogers-Ramanujan continued fraction on the unit circle
Douglas
Bowman;
James
Mc Laughlin
3325-3347
Abstract: This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number $t \in (0,1)$be denoted by $[0,e_{1}(t),e_{2}(t),\cdots]$ and let the $i$-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \begin{displaymath}S=\{t \in (0,1): e_{i+1}(t) \geq \phi^{d_{i}(t)} \text{ infinitely often}\}, \end{displaymath} where $\phi = (\sqrt{5}+1)/2$. Let $Y_{S} =\{\exp(2 \pi i t): t \in S \}$. It is shown that if $y \in Y_{S}$, then the Rogers-Ramanujan continued fraction $R(y)$ diverges at $y$. $S$ is an uncountable set of measure zero. It is also shown that there is an uncountable set of points $G \subset Y_{S}$such that if $y \in G$, then $R(y)$ does not converge generally. It is further shown that $R(y)$ does not converge generally for $\vert y\vert > 1$. However we show that $R(y)$ does converge generally if $y$ is a primitive $5m$-th root of unity, for some $m \in \mathbb{N}$. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.
Integrals, partitions, and cellular automata
Alexander
E.
Holroyd;
Thomas
M.
Liggett;
Dan
Romik
3349-3368
Abstract: We prove that \begin{displaymath}\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab},\end{displaymath} where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.
The flat model structure on $\mathbf{Ch}(R)$
James
Gillespie
3369-3390
Abstract: Given a cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ with enough $\mathcal{A}$ objects and enough $\mathcal{B}$ objects, we define two cotorsion pairs in the category $\mathbf{Ch(\mathcal{C})}$ of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when $(\mathcal{A},\mathcal{B})$ is hereditary. We then show that both of these induced cotorsion pairs are complete when $(\mathcal{A},\mathcal{B})$ is the ``flat'' cotorsion pair of $R$-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat'' model category structure on $\mathbf{Ch}(R)$. In the last section we use the theory of model categories to show that we can define $\operatorname{Ext}^n_R(M,N)$using a flat resolution of $M$ and a cotorsion coresolution of $N$.
Rationality, regularity, and $C_2$-cofiniteness
Toshiyuki
Abe;
Geoffrey
Buhl;
Chongying
Dong
3391-3402
Abstract: We demonstrate that, for vertex operator algebras of CFT type, $C_2$-cofiniteness and rationality is equivalent to regularity. For $C_2$-cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and $C_2$-cofinite, $V_L^+$ is $C_2$-cofinite, and the fusion rules are finite.
Errata to ``On the structure of weight modules"
Ivan
Dimitrov;
Olivier
Mathieu;
Ivan
Penkov
3403-3404