Meromorphic groups
Anand
Pillay;
Thomas
Scanlon
3843-3859
Abstract: We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results of Fujiki. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus, answering a question of Hrushovski. As a consequence, we show that a simple compact complex manifold has algebraic and Kummer dimension zero if and only if its generic type is trivial.
Poset block equivalence of integral matrices
Mike
Boyle;
Danrun
Huang
3861-3886
Abstract: Given square matrices $B$ and $B'$ with a poset-indexed block structure (for which an $ij$ block is zero unless $i\preceq j$), when are there invertible matrices $U$ and $V$ with this required-zero-block structure such that $UBV = B'$? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain $\mathcal R$. As one application, when $\mathcal R$ is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of $U$ and $V$ have determinant $1$. The invariants involve an associated diagram (the ``$K$-web'') of $\mathcal R$-module homomorphisms. The study is motivated by applications to symbolic dynamics and $C^*$-algebras.
Multiple orthogonal polynomials for classical weights
A.
I.
Aptekarev;
A.
Branquinho;
W.
Van Assche
3887-3914
Abstract: A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to $p > 1$ weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order $p+1$. We also obtain explicit formulas and recurrence relations for these polynomials.
Maximal singular loci of Schubert varieties in $SL(n)/B$
Sara
C.
Billey;
Gregory
S.
Warrington
3915-3945
Abstract: Schubert varieties in the flag manifold $SL(n)/B$ play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety $X_w$ is nonsingular if and only if $w$ avoids the patterns $4231$ and $3412$. They also gave a conjectural description of the singular locus of $X_w$. In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety $X_w$ for any element $w\in \mathfrak{S}_n$. In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from $w$ by a cycle depending naturally on a $4231$ or $3412$ pattern in $w$. Our description of the irreducible components is computationally more efficient ($O(n^6)$) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.
Quandle cohomology and state-sum invariants of knotted curves and surfaces
J.
Scott
Carter;
Daniel
Jelsovsky;
Seiichi
Kamada;
Laurel
Langford;
Masahico
Saito
3947-3989
Abstract: The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in $3$-space and knotted surfaces in $4$-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.
Spin Borromean surgeries
Gwénaël
Massuyeau
3991-4017
Abstract: In 1986, Matveev defined the notion of Borromean surgery for closed oriented $3$-manifolds and showed that the equivalence relation generated by this move is characterized by the pair (first Betti number, linking form up to isomorphism). We explain how this extends for $3$-manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure. We then show that the equivalence relation among closed spin $3$-manifolds generated by spin Borromean surgeries is characterized by the triple (first Betti number, linking form up to isomorphism, Rochlin invariant modulo $8$).
Fractafolds based on the Sierpinski gasket and their spectra
Robert
S.
Strichartz
4019-4043
Abstract: We introduce the notion of ``fractafold'', which is to a fractal what a manifold is to a Euclidean half-space. We specialize to the case when the fractal is the Sierpinski gasket SG. We show that each such compact fractafold can be given by a cellular construction based on a finite cell graph $G$, which is $3$-regular in the case that the fractafold has no boundary. We show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph, using the spectral decimation method of Fukushima and Shima. This enables us to obtain isospectral pairs of nonhomeomorphic fractafolds. We also show that although SG is topologically rigid, there are fractafolds based on SG that are not topologically rigid.
Generalized hyperelliptic surfaces
Francesco
Zucconi
4045-4059
Abstract: This article presents some results on the surfaces of general type whose Albanese morphism is a holomorphic fibre bundle.
Tribasic integrals and identities of Rogers-Ramanujan type
M.
E. H.
Ismail;
D.
Stanton
4061-4091
Abstract: Some integrals involving three bases are evaluated as infinite products using complex analysis. Many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products. The resulting identities are identities of Rogers-Ramanujan type. Some integer partition interpretations of these identities are given. Generalizations of the Rogers-Ramanujan type identities involving polynomials are given again as corollaries of integral evaluations.
Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion
Yoonweon
Lee
4093-4110
Abstract: The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.
Inverse functions of polynomials and orthogonal polynomials as operator monotone functions
Mitsuru
Uchiyama
4111-4123
Abstract: We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let $\{p_n\}_{n=0}^{\infty}$ be a sequence of orthonormal polynomials and $p_{n+}$ the restriction of $p_n$ to $[a_n, \infty)$, where $a_n$ is the maximum zero of $p_n$. Then $p_{n+}^{-1}$ and the composite $p_{n-1}\circ p_{n+}^{-1}$ are operator monotone on $[0, \infty)$. Furthermore, for every polynomial $p$ with a positive leading coefficient there is a real number $a$ so that the inverse function of $p(t+a)-p(a)$ defined on $[0,\infty)$is semi-operator monotone, that is, for matrices $A,B \geq 0$, $(p(A+a)-p(a))^2 \leq ((p(B+a)-p(a))^{2}$ implies $A^2\leq B^2.$
The structure of equicontinuous maps
Jie-Hua
Mai
4125-4136
Abstract: Let $(X,d)$ be a metric space, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $R(f)$ is compact, and $\omega (x,f)\not =\emptyset$ for all $x\in X$, then $f$ is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism $h$ and a non-expanding map $g$ that is pointwise convergent to a fixed point $v_{0}$ such that $f$ is uniformly conjugate to a subsystem $(h\times g)\vert S$ of the product map $h\times g$. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.
Heegner zeros of theta functions
Jorge
Jimenez-Urroz;
Tonghai
Yang
4137-4149
Abstract: Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant $-7$. This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.
Geometry of graph varieties
Jeremy
L.
Martin
4151-4169
Abstract: A picture $\mathbf{P}$ of a graph $G=(V,E)$ consists of a point $\mathbf{P}(v)$ for each vertex $v \in V$ and a line $\mathbf{P}(e)$ for each edge $e \in E$, all lying in the projective plane over a field $\mathbf k$ and subject to containment conditions corresponding to incidence in $G$. A graph variety is an algebraic set whose points parametrize pictures of $G$. We consider three kinds of graph varieties: the picture space $\mathcal{X}(G)$ of all pictures; the picture variety $\mathcal{V}(G)$, an irreducible component of $\mathcal{X}(G)$ of dimension $2\vert V\vert$, defined as the closure of the set of pictures on which all the $\mathbf{P}(v)$ are distinct; and the slope variety $\mathcal{S}(G)$, obtained by forgetting all data except the slopes of the lines $\mathbf{P}(e)$. We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties: (1) a description and combinatorial interpretation of equations defining each variety set-theoretically; (2) a description of the irreducible components of $\mathcal{X}(G)$; (3) a proof that $\mathcal{V}(G)$ and $\mathcal{S}(G)$ are Cohen-Macaulay when $G$ satisfies a sparsity condition, rigidity independence. In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.
Generalized associahedra via quiver representations
Robert
Marsh;
Markus
Reineke;
Andrei
Zelevinsky
4171-4186
Abstract: We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.
Fibred knots and twisted Alexander invariants
Jae
Choon
Cha
4187-4200
Abstract: We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.
Sub-bundles of the complexified tangent bundle
Howard
Jacobowitz;
Gerardo
Mendoza
4201-4222
Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.
The autohomeomorphism group of the Cech-Stone compactification of the integers
Juris
Steprans
4223-4240
Abstract: It is shown to be consistent that there is a nontrivial autohomeomorphism of $\beta{\mathbb N} \setminus {\mathbb N}$, yet all such autohomeomorphisms are trivial on a dense $P$-ideal. Furthermore, the cardinality of the autohomeomorphism group of $\beta{\mathbb N} \setminus {\mathbb N}$ can be any regular cardinal between $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.
The geometry of 1-based minimal types
Tristram
de Piro;
Byunghan
Kim
4241-4263
Abstract: In this paper, we study the geometry of a (nontrivial) 1-based $SU$ rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any $\omega$-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.
Errata to ``Metric character of Hamilton--Jacobi equations''
Antonio
Siconolfi
4265-4265