Illinois Journal of Mathematics Articles (Project Euclid)
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The latest articles from Illinois Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 09 Mar 2011 09:09 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Hilbertian matrix cross normed spaces arising from normed ideals
http://projecteuclid.org/euclid.ijm/1264170836
<strong>Takahiro Ohta</strong><p><strong>Source: </strong>Illinois J. Math., Volume 53, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space. We study the space of completely bounded mappings between the matrix cross normed spaces obtained in this way and show that the multiplicator norm naturally appears as the completely bounded norm.
</p>projecteuclid.org/euclid.ijm/1264170836_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA half-space theorem for graphs of constant mean curvature $0<H<\frac{1}{2}$ in $\mathbb{H}^{2}\times\mathbb{R}$http://projecteuclid.org/euclid.ijm/1455203158<strong>L. Mazet</strong>, <strong>G. A. Wanderley</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 43--53.</p><p><strong>Abstract:</strong><br/>
We study a half-space problem related to graphs in $\mathbb{H}^{2}\times\mathbb{R}$, where $\mathbb{H}^{2}$ is the hyperbolic plane, having constant mean curvature $H$ defined over unbounded domains in $\mathbb{H}^{2}$.
</p>projecteuclid.org/euclid.ijm/1455203158_20160211100611Thu, 11 Feb 2016 10:06 ESTWall-crossing and invariants of higher rank Joyce–Song stable pairshttp://projecteuclid.org/euclid.ijm/1455203159<strong>Artan Sheshmani</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 55--83.</p><p><strong>Abstract:</strong><br/>
We introduce a higher rank analog of the Joyce–Song theory of stable pairs. Given a nonsingular projective Calabi–Yau threefold $X$, we define the higher rank Joyce–Song pairs given by ${O}^{\oplus r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg0$ is a fixed integer. We equip the higher rank pairs with a Joyce–Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of “ weakly ” semistable objects.
</p>projecteuclid.org/euclid.ijm/1455203159_20160211100611Thu, 11 Feb 2016 10:06 ESTInvariant Basis Number for $C^{*}$-algebrashttp://projecteuclid.org/euclid.ijm/1455203160<strong>Philip M. Gipson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 85--98.</p><p><strong>Abstract:</strong><br/>
We develop the ring-theoretic notion of Invariant Basis Number in the context of unital $C^{*}$-algebras and their Hilbert $C^{*}$-modules. Characterization of $C^{*}$-algebras with Invariant Basis Number is given in $K$-theoretic terms, closure properties of the class of $C^{*}$-algebras with Invariant Basis Number are given, and examples of $C^{*}$-algebras both with and without the property are explored. For $C^{*}$-algebras without Invariant Basis Number, we determine structure in terms of a “Basis Type” and describe a class of $C^{*}$-algebras which are universal in an appropriate sense. We conclude by investigating properties which are strictly stronger than Invariant Basis Number.
</p>projecteuclid.org/euclid.ijm/1455203160_20160211100611Thu, 11 Feb 2016 10:06 ESTThe quadratic complete intersections associated with the action of the symmetric grouphttp://projecteuclid.org/euclid.ijm/1455203161<strong>Tadahito Harima</strong>, <strong>Akihito Wachi</strong>, <strong>Junzo Watanabe</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 99--113.</p><p><strong>Abstract:</strong><br/>
We prove that any quadratic complete intersection with a certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. Furthermore, we discuss under what conditions its ring of invariants by a Young subgroup is a homogeneous complete intersection with a standard grading.
</p>projecteuclid.org/euclid.ijm/1455203161_20160211100611Thu, 11 Feb 2016 10:06 ESTLooking out for Frobenius summands on a blown-up surface of $\mathbb{P}^{2}$http://projecteuclid.org/euclid.ijm/1455203162<strong>Nobuo Hara</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 115--142.</p><p><strong>Abstract:</strong><br/>
For an algebraic variety $X$ in characteristic $p>0$, the push-forward $F^{e}_{*}\mathcal{O}_{X}$ of the structure sheaf by an iterated Frobenius endomorphism $F^{e}$ is closely related to the geometry of $X$. We study the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ into direct summands when $X$ is obtained by blowing up the projective plane $\mathbb{P}^{2}$ at four points in general position. We explicitly describe the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ and show that there appear only finitely many direct summands up to isomorphism, when $e$ runs over all positive integers. We also prove that these summands generate the derived category $D^{b}(X)$. On the other hand, we show that there appear infinitely many distinct indecomposable summands of iterated Frobenius push-forwards on a ten-point blowup of $\mathbb{P}^{2}$.
</p>projecteuclid.org/euclid.ijm/1455203162_20160211100611Thu, 11 Feb 2016 10:06 ESTLong range correlation inequalities for massless Euclidean fieldshttp://projecteuclid.org/euclid.ijm/1455203163<strong>Joseph G. Conlon</strong>, <strong>Arash Fahim</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 143--187.</p><p><strong>Abstract:</strong><br/>
In this paper, new correlation inequalities are obtained for massless Euclidean fields on the $d$ dimensional integer lattice. Some of the inequalities have been obtained previously, in the case where the Lagrangian is a very small perturbation of a quadratic, using the renormalization group method. The results of the present paper apply provided the Lagrangian is uniformly convex. They therefore hold for the Coulomb dipole gas in which particle density can be of order $1$. The approach of the present paper is based on the methodology of Naddaf–Spencer, which relates second moment correlation functions for the Euclidean field to expectations of Green’s functions for parabolic PDE with random coefficients.
</p>projecteuclid.org/euclid.ijm/1455203163_20160211100611Thu, 11 Feb 2016 10:06 ESTFixed curves near fixed pointshttp://projecteuclid.org/euclid.ijm/1455203164<strong>Alastair Fletcher</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 189--217.</p><p><strong>Abstract:</strong><br/>
Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_{0}$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_{0}$. In particular, we find how many curves there are that are fixed by $f$ and that land at $z_{0}$.
</p>projecteuclid.org/euclid.ijm/1455203164_20160211100611Thu, 11 Feb 2016 10:06 ESTUniqueness results for noncommutative spheres and projective spaceshttp://projecteuclid.org/euclid.ijm/1455203165<strong>Teodor Banica</strong>, <strong>Szabolcs Mészáros</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 219--233.</p><p><strong>Abstract:</strong><br/>
It is known that, under strong combinatorial axioms, $O_{N}\subset O_{N}^{*}\subset O_{N}^{+}$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_{\mathbb{R}}\subset S^{N-1}_{\mathbb{R},*}\subset S^{N-1}_{\mathbb{R},+}$, the noncommutative projective spaces $P^{N-1}_{\mathbb{R}}\subset P^{N-1}_{\mathbb{C}}\subset P^{N-1}_{+}$, and the projective orthogonal quantum groups $PO_{N}\subset PO_{N}^{*}\subset PO_{N}^{+}$.
</p>projecteuclid.org/euclid.ijm/1455203165_20160211100611Thu, 11 Feb 2016 10:06 ESTLimit theorems for some critical superprocesseshttp://projecteuclid.org/euclid.ijm/1455203166<strong>Yan-Xia Ren</strong>, <strong>Renming Song</strong>, <strong>Rui Zhang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 1, 235--276.</p><p><strong>Abstract:</strong><br/>
Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.
</p>projecteuclid.org/euclid.ijm/1455203166_20160211100611Thu, 11 Feb 2016 10:06 ESTJulia’s equation and differential transcendencehttp://projecteuclid.org/euclid.ijm/1462450701<strong>Matthias Aschenbrenner</strong>, <strong>Walter Bergweiler</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 277--294.</p><p><strong>Abstract:</strong><br/>
We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially transcendental over the ring of convergent power series. Our results apply, in particular, to the exponential generating function of a sequence arising from work of Shadrin and Zvonkine on Hurwitz numbers.
</p>projecteuclid.org/euclid.ijm/1462450701_20160505081831Thu, 05 May 2016 08:18 EDTAlgebraic properties of small Hankel operators on the harmonic Bergman spacehttp://projecteuclid.org/euclid.ijm/1462450702<strong>Yong Chen</strong>, <strong>Wei He</strong>, <strong>Yunzhong Hu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 295--317.</p><p><strong>Abstract:</strong><br/>
This paper completely characterizes the commuting problem of two small Hankel operators acting on the harmonic Bergman space with the symbols one being bounded and another being quasihomogeneous, or both being harmonic. The characterizations for semi-commuting problem and the product of two small Hankel operators being another small Hankel operator for certain class of symbols are also obtained.
</p>projecteuclid.org/euclid.ijm/1462450702_20160505081831Thu, 05 May 2016 08:18 EDTElliptic curves on Abelian varietieshttp://projecteuclid.org/euclid.ijm/1462450703<strong>Robert Auffarth II</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 319--336.</p><p><strong>Abstract:</strong><br/>
Given a principally polarized Abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its Néron–Severi group.
</p>projecteuclid.org/euclid.ijm/1462450703_20160505081831Thu, 05 May 2016 08:18 EDTA Schauder basis for $L_{1}(0,\infty)$ consisting of non-negative functionshttp://projecteuclid.org/euclid.ijm/1462450704<strong>William B. Johnson</strong>, <strong>Gideon Schechtman</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 337--344.</p><p><strong>Abstract:</strong><br/>
We construct a Schauder basis for $L_{1}$ consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in $L_{p}$, $1\le p<\infty$.
</p>projecteuclid.org/euclid.ijm/1462450704_20160505081831Thu, 05 May 2016 08:18 EDTOn solvable subgroups of the Cremona grouphttp://projecteuclid.org/euclid.ijm/1462450705<strong>Julie Déserti</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 345--358.</p><p><strong>Abstract:</strong><br/>
The Cremona group $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is the group of birational self-maps of $\mathbb{P}^{2}_{\mathbb{C}}$. Using the action of $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ on the Picard-Manin space of $\mathbb{P}^{2}_{\mathbb{C}}$, we characterize its solvable subgroups. If $\mathrm{G}\subset\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is solvable, nonvirtually Abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $\mathrm{G}$ is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.
We also give some corollaries.
</p>projecteuclid.org/euclid.ijm/1462450705_20160505081831Thu, 05 May 2016 08:18 EDTAn alternate description of the Szlenk index with applicationshttp://projecteuclid.org/euclid.ijm/1462450706<strong>R. M. Causey</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 359--390.</p><p><strong>Abstract:</strong><br/>
We discuss an alternate method for computing the Szlenk index of an arbitrary $w^{*}$ compact subset of the dual of a Banach space. We discuss consequences of this method as well as offer simple, alternative proofs of a number of results already found in the literature.
</p>projecteuclid.org/euclid.ijm/1462450706_20160505081831Thu, 05 May 2016 08:18 EDTNonsimple polyominoes and prime idealshttp://projecteuclid.org/euclid.ijm/1462450707<strong>Takayuki Hibi</strong>, <strong>Ayesha Asloob Qureshi</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 391--398.</p><p><strong>Abstract:</strong><br/>
It is known that the polyomino ideal arising from a simple polyomino comes from a finite bipartite graph and, in particular, it is a prime ideal. A class of nonsimple polyominoes $\mathcal{P}$ for which the polyomino ideal $I_{\mathcal{P}}$ is a prime ideal and for which $I_{\mathcal{P}}$ cannot come from a finite simple graph will be presented.
</p>projecteuclid.org/euclid.ijm/1462450707_20160505081831Thu, 05 May 2016 08:18 EDTMinimal genus of links and fibering of canonical surfaceshttp://projecteuclid.org/euclid.ijm/1462450708<strong>A. Stoimenow</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 399--448.</p><p><strong>Abstract:</strong><br/>
This paper contains some further applications of the study of knot diagrams by genus. Introducing a procedure of regularization for knot generators, and using invariants derived from the Jones polynomial (degrees, congruences, and the Fiedler–Polyak–Viro Gauß diagram formulas for its Vassiliev invariants), we examine the existence of genus-minimizing diagrams for almost alternating and almost positive knots. In particular, we examine the existence of such knots such that either all or none of their almost alternating/positive diagrams have the minimal genus property. We prove that the genus of almost positive non-split links is determined by the Alexander polynomial.
</p>projecteuclid.org/euclid.ijm/1462450708_20160505081831Thu, 05 May 2016 08:18 EDTUnique pseudo-expectations for $C^{*}$-inclusionshttp://projecteuclid.org/euclid.ijm/1462450709<strong>David R. Pitts</strong>, <strong>Vrej Zarikian</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 449--483.</p><p><strong>Abstract:</strong><br/>
Given an inclusion $\mathcal{D}\subseteq\mathcal{C}$ of unital $C^{*}$-algebras (with common unit), a unital completely positive linear map $\Phi$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ which extends the inclusion of $\mathcal{D}$ into $I(\mathcal{D})$ is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ of all pseudo-expectations is a convex set, and when $\mathcal{D}$ is Abelian, we prove a Krein–Milman type theorem showing that $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ can be recovered from its set of extreme points. In general, $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ is not a singleton. However, there are large and natural classes of inclusions (e.g., when $\mathcal{D}$ is a regular MASA in $\mathcal{C}$) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For general inclusions of $C^{*}$-algebras with $\mathcal{D}$ Abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when $\mathcal{C}$ is Abelian, we are able to give a topological description of the unique pseudo-expectation property.
As applications, we show that if an inclusion $\mathcal{D}\subseteq\mathcal{C}$ has a unique pseudo-expectation $\Phi$ which is also faithful, then the$C^{*}$-envelope of any operator space $\mathcal{X}$ with $\mathcal{D}\subseteq\mathcal{X}\subseteq\mathcal{C}$ is the $C^{*}$-subalgebra of $\mathcal{C}$ generated by $\mathcal{X}$; we also show that for many interesting classes of $C^{*}$-inclusions, having a faithful unique pseudo-expectation implies that $\mathcal{D}$ norms $\mathcal{C}$, although this is not true in general.
</p>projecteuclid.org/euclid.ijm/1462450709_20160505081831Thu, 05 May 2016 08:18 EDTWell-posedness of the martingale problem for superprocess with interactionhttp://projecteuclid.org/euclid.ijm/1462450710<strong>Leonid Mytnik</strong>, <strong>Jie Xiong</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 485--497.</p><p><strong>Abstract:</strong><br/>
We consider the martingale problem for superprocess with interactive immigration mechanism. The uniqueness of the solution to this martingale problem is established using the strong uniqueness of the solution to a corresponding SPDE, which is obtained by an extended version of the Yamada–Watanabe argument.
</p>projecteuclid.org/euclid.ijm/1462450710_20160505081831Thu, 05 May 2016 08:18 EDTOn a linearized p -Laplace equation with rapidly oscillating coefficientshttp://projecteuclid.org/euclid.ijm/1462450711<strong>Harri Varpanen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 2, 499--529.</p><p><strong>Abstract:</strong><br/>
Related to a conjecture of Tom Wolff, we solve a singular Neumann problem for a linearized $p$-Laplace equation in the unit disk.
</p>projecteuclid.org/euclid.ijm/1462450711_20160505081831Thu, 05 May 2016 08:18 EDTNormality preserving operations for Cantor series expansions and associated fractals, Ihttp://projecteuclid.org/euclid.ijm/1475266396<strong>Dylan Airey</strong>, <strong>Bill Mance</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 531--543.</p><p><strong>Abstract:</strong><br/>
It is well known that rational multiplication preserves normality in base $b$. We study related normality preserving operations for the $Q$-Cantor series expansions. In particular, we show that while integer multiplication preserves $Q$-distribution normality, it fails to preserve $Q$-normality in a particularly strong manner. We also show that $Q$-distribution normality is not preserved by non-integer rational multiplication on a set of zero measure and full Hausdorff dimension.
</p>projecteuclid.org/euclid.ijm/1475266396_20160930161325Fri, 30 Sep 2016 16:13 EDTExistence result for a class of quasilinear elliptic equations with ($p$-$q$)-Laplacian and vanishing potentialshttp://projecteuclid.org/euclid.ijm/1475266397<strong>M. J. Alves</strong>, <strong>R. B. Assunção</strong>, <strong>O. H. Miyagaki</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 545--575.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to establish the existence of positive solutions to a class of quasilinear elliptic equations involving the ($p$-$q$)-Laplacian operator. We consider a nonlinearity that can be subcritical at infinity and supercritical at the origin; we also consider potential functions that can vanish at infinity. The approach is based on variational arguments dealing with the mountain-pass lemma and an adaptation of the penalization method. In order to overcome the lack of compactness, we modify the original problem and the associated energy functional. Finally, to show that the solution of the modified problem is also a solution of the original problem we use an estimate obtained by the Moser iteration scheme.
</p>projecteuclid.org/euclid.ijm/1475266397_20160930161325Fri, 30 Sep 2016 16:13 EDTTensor products of measurable operatorshttp://projecteuclid.org/euclid.ijm/1475266398<strong>M. Anoussis</strong>, <strong>V. Felouzis</strong>, <strong>I. G. Todorov</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 577--595.</p><p><strong>Abstract:</strong><br/>
We introduce and study a stability property for submodules of measurable operators and Calkin spaces and characterize the tensor stable singly generated Calkin spaces. Given semifinite von Neumann algebras $(\mathcal{M},\tau)$, $(\mathcal{N},\sigma)$ and corresponding measurable operators $S$, $T$, we provide a necessary and sufficient condition for the operator $S\otimes T$ to be measurable with respect to $(\mathcal{M}\otimes\mathcal{N},\tau\otimes\sigma)$.
</p>projecteuclid.org/euclid.ijm/1475266398_20160930161325Fri, 30 Sep 2016 16:13 EDTExpansive actions of countable amenable groups, homoclinic pairs, and the Myhill propertyhttp://projecteuclid.org/euclid.ijm/1475266399<strong>Tullio Ceccherini-Silberstein</strong>, <strong>Michel Coornaert</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 597--621.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a compact metrizable space equipped with a continuous action of a countable amenable group $G$. Suppose that the dynamical system $(X,G)$ is expansive and is the quotient by a uniformly bounded-to-one factor map of a strongly irreducible subshift. Let $\tau\colon X\to X$ be a continuous map commuting with the action of $G$. We prove that if there is no pair of distinct $G$-homoclinic points in $X$ having the same image under $\tau$ then $\tau$ is surjective.
</p>projecteuclid.org/euclid.ijm/1475266399_20160930161325Fri, 30 Sep 2016 16:13 EDTLusternik–Schnirelmann category for cell complexes and posetshttp://projecteuclid.org/euclid.ijm/1475266400<strong>Kohei Tanaka</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 623--636.</p><p><strong>Abstract:</strong><br/>
This paper introduces two analogues of the Lusternik–Schnirelmann category from a combinatorial viewpoint. One analogue is defined for finite cell complexes using their subcomplexes and simple homotopy theory; the other is an invariant for finite posets with respect to simple equivalence based on the notion of weak beat points. We examine the relation between these two invariants by taking the face posets of complexes or order complexes of posets.
</p>projecteuclid.org/euclid.ijm/1475266400_20160930161325Fri, 30 Sep 2016 16:13 EDTNotes on the linearity defect and applicationshttp://projecteuclid.org/euclid.ijm/1475266401<strong>Hop D. Nguyen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 637--662.</p><p><strong>Abstract:</strong><br/>
The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to Şega, we study the behavior of linearity defect along short exact sequences. We point out two classes of short exact sequences involving Koszul modules, along which linearity defect behaves nicely. We also generalize the notion of Koszul filtrations from the graded case to the local setting. Among the applications, we prove that if $R\to S$ is a surjection of noetherian local rings such that $S$ is a Koszul $R$-module, and $N$ is a finitely generated $S$-module, then the linearity defect of $N$ as an $R$-module is the same as its linearity defect as an $S$-module. In particular, we confirm that specializations of absolutely Koszul algebras are again absolutely Koszul, answering positively a question due to Conca, Iyengar, Nguyen and Römer.
</p>projecteuclid.org/euclid.ijm/1475266401_20160930161325Fri, 30 Sep 2016 16:13 EDTA random pointwise ergodic theorem with Hardy field weightshttp://projecteuclid.org/euclid.ijm/1475266402<strong>Ben Krause</strong>, <strong>Pavel Zorin-Kranich</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 663--674.</p><p><strong>Abstract:</strong><br/>
Let $a_{n}$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0<a<1/2$, and let $p(n)=n^{1+\varepsilon}$, $0<\varepsilon<1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f\in L^{1}(X)$ the modulated, random averages
\[\frac{1}{N}\sum_{n=1}^{N}e(p(n))T^{a_{n}(\omega)}f\] converge to 0 pointwise almost everywhere.
</p>projecteuclid.org/euclid.ijm/1475266402_20160930161325Fri, 30 Sep 2016 16:13 EDTIdeals generated by principal minorshttp://projecteuclid.org/euclid.ijm/1475266403<strong>Ashley K. Wheeler</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 675--689.</p><p><strong>Abstract:</strong><br/>
A minor is principal means it is defined by the same row and column indices. We study ideals generated by principal minors of size $t\leq n$ of a generic $n\times n$ matrix $X$, in the polynomial ring generated over an algebraically closed field by the entries of $X$. When $t=2$ the resulting quotient ring is a normal complete intersection domain. We show for any $t$, upon inverting $\det X$ the ideals given respectively by the size $t$ and the size $n-t$ principal minors become isomorphic. From that we show the algebraic set given by the size $n-1$ principal minors has a codimension $4$ component defined by the determinantal ideal, plus a codimension $n$ component. When $n=4$ the two components are linked, and in fact, geometrically linked.
</p>projecteuclid.org/euclid.ijm/1475266403_20160930161325Fri, 30 Sep 2016 16:13 EDTTwisted Reidemeister torsion and the Thurston norm: Graph manifolds and finite representationshttp://projecteuclid.org/euclid.ijm/1475266404<strong>Stefan Friedl</strong>, <strong>Matthias Nagel</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 691--705.</p><p><strong>Abstract:</strong><br/>
We show that the Thurston norm of any irreducible 3-manifold can be detected using twisted Reidemeister torsions corresponding to integral representations and also corresponding to representations over finite fields. In particular, our result holds for all graph manifolds, these are not covered by the earlier work of the first author and Vidussi.
</p>projecteuclid.org/euclid.ijm/1475266404_20160930161325Fri, 30 Sep 2016 16:13 EDTCompact composition operators with symbol a universal covering map onto a multiply connected domainhttp://projecteuclid.org/euclid.ijm/1475266405<strong>Matthew M. Jones</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 707--715.</p><p><strong>Abstract:</strong><br/>
We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.
</p>projecteuclid.org/euclid.ijm/1475266405_20160930161325Fri, 30 Sep 2016 16:13 EDTA refinement of analytic characterizations of gaugeability for generalized Feynman–Kac functionalshttp://projecteuclid.org/euclid.ijm/1475266406<strong>Daehong Kim</strong>, <strong>Mila Kurniawaty</strong>, <strong>Kazuhiro Kuwae</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 717--771.</p><p><strong>Abstract:</strong><br/>
We relax the conditions for measures in our previous paper [Analytic characterizations of gaugeability for generalized Feynman–Kac functionals (2016) Preprint] on analytic characterizations of (conditional) gaugeability for generalized Feynman–Kac functionals in the framework of symmetric Markov processes. The analytic characterization is also equivalent to the maximum principle for generalized Feynman–Kac semigroups, extending the result by Takeda [The bottom of the spectrum of time-changed processes and the maximum principle of Schrödinger operators (2015) Preprint].
</p>projecteuclid.org/euclid.ijm/1475266406_20160930161325Fri, 30 Sep 2016 16:13 EDTAn isoperimetric inequality for an integral operator on flat torihttp://projecteuclid.org/euclid.ijm/1475266407<strong>Braxton Osting</strong>, <strong>Jeremy Marzuola</strong>, <strong>Elena Cherkaev</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 773--793.</p><p><strong>Abstract:</strong><br/>
We consider a class of Hilbert–Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all unit-volume flat tori, the equilateral torus maximizes the operator norm and the Hilbert–Schmidt norm.
</p>projecteuclid.org/euclid.ijm/1475266407_20160930161325Fri, 30 Sep 2016 16:13 EDTModel theory and the QWEP conjecturehttp://projecteuclid.org/euclid.ijm/1475266408<strong>Isaac Goldbring</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 795--799.</p><p><strong>Abstract:</strong><br/>
We observe that Kirchberg’s QWEP conjecture is equivalent to the statement that $C^{*}(\mathbb{F})$ is elementarily equivalent to a QWEP C$^{*}$ algebra. We also make a few other model-theoretic remarks about WEP and LLP C$^{*}$ algebras.
</p>projecteuclid.org/euclid.ijm/1475266408_20160930161325Fri, 30 Sep 2016 16:13 EDTA note on reduced and von Neumann algebraic free wreath productshttp://projecteuclid.org/euclid.ijm/1475266409<strong>Jonas Wahl</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 801--817.</p><p><strong>Abstract:</strong><br/>
We study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb{G}\wr_{*}S_{N}^{+}$, where $\mathbb{G}$ is a compact matrix quantum group. Based on recent results on their corepresentation theory by Lemeux and Tarrago in [Lemeux and Tarrago (2014)], we prove that $\mathbb{G}\wr_{*}S_{N}^{+}$ is of Kac type whenever $\mathbb{G}$ is, and that the reduced version of $\mathbb{G}\wr_{*}S_{N}^{+}$ is simple with unique trace state whenever $N\geq8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G}\wr_{*}S_{N}^{+}$ does not have property $\Gamma$.
</p>projecteuclid.org/euclid.ijm/1475266409_20160930161325Fri, 30 Sep 2016 16:13 EDTHolomorphic functional calculus on upper triangular forms in finite von Neumann algebrashttp://projecteuclid.org/euclid.ijm/1475266410<strong>K. Dykema</strong>, <strong>F. Sukochev</strong>, <strong>D. Zanin</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 3, 819--824.</p><p><strong>Abstract:</strong><br/>
The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup–Schultz hyperinvariant projections, behave well with respect to holomorphic functional calculus.
</p>projecteuclid.org/euclid.ijm/1475266410_20160930161325Fri, 30 Sep 2016 16:13 EDTDecompositions of rational functions over real and complex numbers and a question about invariant curveshttp://projecteuclid.org/euclid.ijm/1488186011<strong>Peter Müller</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 825--838.</p><p><strong>Abstract:</strong><br/>
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves in the complex plane which are invariant under a rational function.
</p>projecteuclid.org/euclid.ijm/1488186011_20170227040113Mon, 27 Feb 2017 04:01 ESTOn a quantum version of Ellis joint continuity theoremhttp://projecteuclid.org/euclid.ijm/1488186012<strong>Biswarup Das</strong>, <strong>Colin Mrozinski</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 839--858.</p><p><strong>Abstract:</strong><br/>
We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. We give two applications of our results: a “noncommutative” version of Ellis joint continuity theorem for semitopological groups, a corollary to which is a new C∗-algebraic proof of the theorem for classical semitopological semigroup; we also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a “noncommutative” version of the converse Haar’s theorem.
</p>projecteuclid.org/euclid.ijm/1488186012_20170227040113Mon, 27 Feb 2017 04:01 ESTStretching factors, metrics and train tracks for free productshttp://projecteuclid.org/euclid.ijm/1488186013<strong>Stefano Francaviglia</strong>, <strong>Armando Martino</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 859--899.</p><p><strong>Abstract:</strong><br/>
In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.
We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.
We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).
Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.
A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.
</p>projecteuclid.org/euclid.ijm/1488186013_20170227040113Mon, 27 Feb 2017 04:01 ESTExtending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groupshttp://projecteuclid.org/euclid.ijm/1488186014<strong>Nguyen Ngoc Hung</strong>, <strong>Philani R. Majozi</strong>, <strong>Hung P. Tong-Viet</strong>, <strong>Thomas P. Wakefield</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 901--924.</p><p><strong>Abstract:</strong><br/>
We propose to extend a conjecture of Bertram Huppert [ Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group $G$ and a finite quasi-simple group $H$ with ${\mathrm{Mult}}(H/\mathbf{Z}(H))$ cyclic have the same set of irreducible character degrees (not counting multiplicity), then $G$ is isomorphic to a central product of $H$ and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions $2$ and $3$.
</p>projecteuclid.org/euclid.ijm/1488186014_20170227040113Mon, 27 Feb 2017 04:01 ESTOn metrics of curvature $1$ with four conic singularities on tori and on the spherehttp://projecteuclid.org/euclid.ijm/1488186015<strong>Alexandre Eremenko</strong>, <strong>Andrei Gabrielov</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 925--947.</p><p><strong>Abstract:</strong><br/>
We discuss conformal metrics of curvature $1$ on tori and on the sphere, with four conic singularities whose angles are multiples of $\pi$. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles $(\pi,3\pi,\pi,3\pi)$. As a consequence we find new Heun’s equations whose general solution is algebraic.
</p>projecteuclid.org/euclid.ijm/1488186015_20170227040113Mon, 27 Feb 2017 04:01 ESTBoundedness of a family of Hilbert-type operators and its Bergman-type analoguehttp://projecteuclid.org/euclid.ijm/1488186016<strong>Justice S. Bansah</strong>, <strong>Benoît F. Sehba</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 949--977.</p><p><strong>Abstract:</strong><br/>
In this paper, we first consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. Second, we consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.
</p>projecteuclid.org/euclid.ijm/1488186016_20170227040113Mon, 27 Feb 2017 04:01 ESTSpectrally unstable domainshttp://projecteuclid.org/euclid.ijm/1488186017<strong>Gerardo A. Mendoza</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 979--997.</p><p><strong>Abstract:</strong><br/>
Let $H$ be a separable Hilbert space, $A_{c}:\mathcal{D}_{c}\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal{D}_{\min}$ be the domain of the closure of $A_{c}$ and $\mathcal{D}_{\max}$ that of the adjoint. Assume that $\mathcal{D}_{\max}$ with the graph norm is compactly contained in $H$ and that $\mathcal{D}_{\min}$ has finite positive codimension in $\mathcal{D}_{\max}$. Then the set of domains of selfadjoint extensions of $A_{c}$ has the structure of a finite-dimensional manifold $\mathfrak{SA}$ and the spectrum of each of its selfadjoint extensions is bounded from below. If $\zeta$ is strictly below the spectrum of $A$ with a given domain $\mathcal{D}_{0}\in\mathfrak{SA}$, then $\zeta$ is not in the spectrum of $A$ with domain $\mathcal{D}\in\mathfrak{SA}$ near $\mathcal{D}_{0}$. But $\mathfrak{SA}$ contains elements $\mathcal{D}_{0}$ with the property that for every neighborhood $U$ of $\mathcal{D}_{0}$ and every $\zeta\in\mathbb{R}$ there is $\mathcal{D}\in U$ such that $\operatorname{spec}(A_{\mathcal{D} })\cap(-\infty,\zeta)\neq\emptyset$. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_{c}$.
</p>projecteuclid.org/euclid.ijm/1488186017_20170227040113Mon, 27 Feb 2017 04:01 ESTThue equations and latticeshttp://projecteuclid.org/euclid.ijm/1488186018<strong>Jeffrey Lin Thunder</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 999--1023.</p><p><strong>Abstract:</strong><br/>
We consider Diophantine equations of the kind $|F(x,y)|=m$, where $F(X,Y)\in\mathbb{Z}[X,Y]$ is a homogeneous polynomial of degree at least 3 that has non-zero discriminant, $m$ is a fixed positive integer and $x,y$ are relatively prime integer solutions. Our results improve upon previous theorems due to Bombieri and Schmidt and also Stewart. We further provide reasonable heuristics for conjectures of Schmidt and Stewart regarding such equations.
</p>projecteuclid.org/euclid.ijm/1488186018_20170227040113Mon, 27 Feb 2017 04:01 ESTAbstract convolution function algebras over homogeneous spaces of compact groupshttp://projecteuclid.org/euclid.ijm/1488186019<strong>Arash Ghaani Farashahi</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1025--1042.</p><p><strong>Abstract:</strong><br/>
This paper presents a systematic study for structure of abstract Banach function $*$-algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the homogeneous space $G/H$ associated to the Weil’s formula and $1\leq p<\infty$. Then we introduce the notions of convolution and involution for the Banach function spaces $L^{p}(G/H,\mu)$.
</p>projecteuclid.org/euclid.ijm/1488186019_20170227040113Mon, 27 Feb 2017 04:01 ESTPreservation of $p$-Poincaré inequality for large $p$ under sphericalization and flatteninghttp://projecteuclid.org/euclid.ijm/1488186020<strong>Estibalitz Durand-Cartagena</strong>, <strong>Xining Li</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1043--1069.</p><p><strong>Abstract:</strong><br/>
Li and Shanmugalingam showed that annularly quasiconvex metric spaces endowed with a doubling measure preserve the property of supporting a $p$-Poincaré inequality under the sphericalization and flattening procedures. Because natural examples such as the real line or a broad class of metric trees are not annularly quasiconvex, our aim in the present paper is to study, under weaker hypotheses on the metric space, the preservation of $p$-Poincaré inequalites under those conformal deformations for sufficiently large $p$. We propose the hypotheses used in a previous paper by the same authors, where the preservation of $\infty$-Poincaré inequality has been studied under the assumption of radially star-like quasiconvexity (for sphericalization) and meridian-like quasiconvexity (for flattening). To finish, using the sphericalization procedure, we exhibit an example of a Cheeger differentiability space whose blow up at a particular point is not a PI space.
</p>projecteuclid.org/euclid.ijm/1488186020_20170227040113Mon, 27 Feb 2017 04:01 ESTSufficient conditions for Strassen’s additivity conjecturehttp://projecteuclid.org/euclid.ijm/1488186021<strong>Zach Teitler</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1071--1085.</p><p><strong>Abstract:</strong><br/>
We give a sufficient condition for the strong symmetric version of Strassen’s additivity conjecture: the Waring rank of a sum of forms in independent variables is the sum of their ranks, and every Waring decomposition of the sum is a sum of decompositions of the summands. We give additional sufficient criteria for the additivity of Waring ranks and a sufficient criterion for additivity of cactus ranks and decompositions.
</p>projecteuclid.org/euclid.ijm/1488186021_20170227040113Mon, 27 Feb 2017 04:01 ESTLong turns, INP’s and indices for free group automorphismshttp://projecteuclid.org/euclid.ijm/1488186022<strong>Thierry Coulbois</strong>, <strong>Martin Lustig</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1087--1109.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to introduce a new tool, called long turns , which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.
</p>projecteuclid.org/euclid.ijm/1488186022_20170227040113Mon, 27 Feb 2017 04:01 ESTIndex realization for automorphisms of free groupshttp://projecteuclid.org/euclid.ijm/1488186023<strong>Thierry Coulbois</strong>, <strong>Martin Lustig</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1111--1128.</p><p><strong>Abstract:</strong><br/>
For any surface $\Sigma$ of genus $g\geq1$ and (essentially) any collection of positive integers $i_{1},i_{2},\ldots,i_{\ell}$ with $i_{1}+\cdots+i_{\ell}=4g-4$ Masur and Smillie ( Comment. Math. Helv. 68 (1993) 289–307) have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma\to\Sigma$ with precisely $\ell$ singularities $S_{1},\ldots,S_{\ell}$ in its stable foliation $\mathcal{L}$, such that $\mathcal{L}$ has precisely $i_{k}+2$ separatrices raying out from each $S_{k}$.
In this paper, we prove the analogue of this result for automorphisms of a free group ${F}_{N}$, where “pseudo-Anosov homeomorphism” is replaced by “fully irreducible automorphism” and the Gauss–Bonnet equality $i_{1}+\cdots+i_{\ell}=4g-4$ is replaced by the index inequality $i_{1}+\cdots+i_{\ell}\leq2N-2$ from ( Duke Math. J. 93 (1998) 425–452).
</p>projecteuclid.org/euclid.ijm/1488186023_20170227040113Mon, 27 Feb 2017 04:01 ESTWolfgang Haken: A biographical sketchhttp://projecteuclid.org/euclid.ijm/1498032019<strong>Ilya Kapovich</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 3--9.</p>projecteuclid.org/euclid.ijm/1498032019_20170621040059Wed, 21 Jun 2017 04:00 EDTOn homotopy 3-sphereshttp://projecteuclid.org/euclid.ijm/1498032020<strong>Wolfgang Haken</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 11--30.</p>projecteuclid.org/euclid.ijm/1498032020_20170621040059Wed, 21 Jun 2017 04:00 EDTFour-dimensional Haken cobordism theoryhttp://projecteuclid.org/euclid.ijm/1498032021<strong>Bell Foozwell</strong>, <strong>Hyam Rubinstein</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 1--17.</p><p><strong>Abstract:</strong><br/>
Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_{1}(W)$ is injective. In this paper, we prove that there are $4$-dimensional Haken cobordisms whose boundary consists of any two closed Haken $3$-manifolds. In particular, each closed Haken $3$-manifold is the $\pi_{1}$-injective boundary of some Haken $4$-manifold.
</p>projecteuclid.org/euclid.ijm/1498032021_20170621040059Wed, 21 Jun 2017 04:00 EDTStrip maps of small surfaces are convexhttp://projecteuclid.org/euclid.ijm/1498032022<strong>François Guéritaud</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 19--37.</p><p><strong>Abstract:</strong><br/>
The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.
</p>projecteuclid.org/euclid.ijm/1498032022_20170621040059Wed, 21 Jun 2017 04:00 EDTThin surface subgroups in cocompact lattices in $\operatorname{SL}(3,\mathbf{R})$http://projecteuclid.org/euclid.ijm/1498032023<strong>D. D. Long</strong>, <strong>A. W. Reid</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 39--53.</p><p><strong>Abstract:</strong><br/>
We show certain cocompact lattices in $\operatorname{SL}(3,\mathbf{R})$ contain closed surface groups. With further restrictions, we exhibit such lattices containing infinitely many commensurability classes of closed surface groups.
</p>projecteuclid.org/euclid.ijm/1498032023_20170621040059Wed, 21 Jun 2017 04:00 EDTThe $SL(3,\mathbb{C})$-character variety of the figure eight knothttp://projecteuclid.org/euclid.ijm/1498032024<strong>Michael Heusener</strong>, <strong>Vicente Muñoz</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 55--98.</p><p><strong>Abstract:</strong><br/>
We give explicit equations that describe the character variety of the figure eight knot for the groups $\mathrm{SL}(3,\mathbb{C})$, $\mathrm{GL}(3,\mathbb{C})$ and $\mathrm{PGL}(3,\mathbb{C})$. For any of these $G$, it has five components of dimension $2$, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $\mathrm{SL}(2,\mathbb{C})$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.
</p>projecteuclid.org/euclid.ijm/1498032024_20170621040059Wed, 21 Jun 2017 04:00 EDTEmbedding of groups and quadratic equations over groupshttp://projecteuclid.org/euclid.ijm/1498032025<strong>D. F. Cummins</strong>, <strong>S. V. Ivanov</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 99--115.</p><p><strong>Abstract:</strong><br/>
We prove that, for every integer $n\ge2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, that is, every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.
</p>projecteuclid.org/euclid.ijm/1498032025_20170621040059Wed, 21 Jun 2017 04:00 EDTOne-domination of knotshttp://projecteuclid.org/euclid.ijm/1498032026<strong>M. Boileau</strong>, <strong>S. Boyer</strong>, <strong>D. Rolfsen</strong>, <strong>S. C. Wang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 117--139.</p><p><strong>Abstract:</strong><br/>
We say that a knot $k_{1}$ in the $3$-sphere $1$-dominates another $k_{2}$ if there is a proper degree 1 map $E(k_{1})\to E(k_{2})$ between their exteriors, and write $k_{1}\ge k_{2}$. When $k_{1}\ge k_{2}$ but $k_{1}\ne k_{2}$ we write $k_{1}>k_{2}$. One expects in the latter eventuality that $k_{1}$ is more complicated . In this paper, we produce various sorts of evidence to support this philosophy.
</p>projecteuclid.org/euclid.ijm/1498032026_20170621040059Wed, 21 Jun 2017 04:00 EDTTori and Heegaard splittingshttp://projecteuclid.org/euclid.ijm/1498032027<strong>Abigail Thompson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 141--148.</p><p><strong>Abstract:</strong><br/>
In Studies in modern topology (1968) 39–98 Prentice Hall, Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the “interesting” surface increases from zero, more complicated phenomena occur. Kobayashi ( Osaka J. Math. 24 (1987) 173–215) showed that if a 3-manifold $M^{3}$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a (strongly irreducible) Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general, there is no global bound on the number of curves in this collection. We show that given a 3-manifold $M$, a minimal genus, strongly irreducible Heegaard surface $F$ for $M$, and an essential torus $T$, we can either restrict the number of curves of intersection of $T$ with $F$ (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of $M$, or produce a small Seifert-fibered piece of $M$.
</p>projecteuclid.org/euclid.ijm/1498032027_20170621040059Wed, 21 Jun 2017 04:00 EDTWolfgang Haken and the four-color problemhttp://projecteuclid.org/euclid.ijm/1498032028<strong>Robin Wilson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 149--178.</p><p><strong>Abstract:</strong><br/>
In 1852, Augustus De Morgan, Professor of Mathematics at University College, London, was asked: Can every map be colored with just four colors in such a way that neighboring countries are colored differently? Over a century later, in a controversial proof that made substantial use of a computer, Wolfgang Haken and Kenneth Appel of the University of Illinois answered the question in the affirmative. But how did Haken come to be involved with the problem, and what was his role in its solution?
</p>projecteuclid.org/euclid.ijm/1498032028_20170621040059Wed, 21 Jun 2017 04:00 EDTA knot without a nonorientable essential spanning surfacehttp://projecteuclid.org/euclid.ijm/1498032029<strong>Nathan M. Dunfield</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 179--184.</p><p><strong>Abstract:</strong><br/>
This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the same authors. The proof is a rigorous calculation using Thurston’s spun-normal surfaces in the spirit of Haken’s original normal surface algorithms.
</p>projecteuclid.org/euclid.ijm/1498032029_20170621040059Wed, 21 Jun 2017 04:00 EDTFour color theorem from three points of viewhttp://projecteuclid.org/euclid.ijm/1498032030<strong>Yuri Matiyasevich</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 185--205.</p><p><strong>Abstract:</strong><br/>
The Four Color Conjecture, which in 1977 became the Four Color Theorem of Kenneth Appel and Wolfgang Haken, is famous for the number of its reformulations. Three of them found by the author at different time are discussed in this paper.
</p>projecteuclid.org/euclid.ijm/1498032030_20170621040059Wed, 21 Jun 2017 04:00 EDTProposed Property 2R counterexamples examinedhttp://projecteuclid.org/euclid.ijm/1498032031<strong>Martin Scharlemann</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 207--250.</p><p><strong>Abstract:</strong><br/>
In 1985, Akbulut and Kirby analyzed a homotopy $4$-sphere $\Sigma$ that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysi showed that $\Sigma$ was one of an infinite collection of examples, all of which were (sadly) the standard $S^{4}$, but with an unusual handle structure.
Recent work with Gompf and Thompson, showed that the construction gives rise to a family $L_{n}$ of $2$-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each $L_{n}$, one component is the simple square knot $Q$, and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the $L_{n}$. In view of the continuing interest and potential importance of the Cappell-Shaneson-Akbulut-Kirby-Gompf examples (e.g., the original $\Sigma$ is known to embed very efficiently in $S^{4}$ and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.
</p>projecteuclid.org/euclid.ijm/1498032031_20170621040059Wed, 21 Jun 2017 04:00 EDTA state calculus for graph coloringhttp://projecteuclid.org/euclid.ijm/1498032032<strong>Louis H. Kauffman</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 251--271.</p><p><strong>Abstract:</strong><br/>
This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and we give a simple extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs presented as immersions in the plane.
</p>projecteuclid.org/euclid.ijm/1498032032_20170621040059Wed, 21 Jun 2017 04:00 EDTTutte relations, TQFT, and planarity of cubic graphshttp://projecteuclid.org/euclid.ijm/1498032033<strong>Ian Agol</strong>, <strong>Vyacheslav Krushkal</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 273--288.</p><p><strong>Abstract:</strong><br/>
It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.
</p>projecteuclid.org/euclid.ijm/1498032033_20170621040059Wed, 21 Jun 2017 04:00 EDTThe 3D-index and normal surfaceshttp://projecteuclid.org/euclid.ijm/1498032034<strong>Stavros Garoufalidis</strong>, <strong>Craig D. Hodgson</strong>, <strong>Neil R. Hoffman</strong>, <strong>J. Hyam Rubinstein</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 289--352.</p><p><strong>Abstract:</strong><br/>
Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.
</p>projecteuclid.org/euclid.ijm/1498032034_20170621040059Wed, 21 Jun 2017 04:00 EDTBoundaries of Kleinian groupshttp://projecteuclid.org/euclid.ijm/1498032035<strong>Peter Haïssinsky</strong>, <strong>Luisa Paoluzzi</strong>, <strong>Genevieve Walsh</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 353--364.</p><p><strong>Abstract:</strong><br/>
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
</p>projecteuclid.org/euclid.ijm/1498032035_20170621040059Wed, 21 Jun 2017 04:00 EDT