In a 2012 paper, the second author showed that a tunnel of a tunnel number one, fibered link in ${{S}^{3}}$ can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper we observe that this is true for fibered links in any 3-manifold, we analyze how the arc behaves under the monodromy action, and we show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${{\mathbb{F}}_{p}}$, we study the frequency to which a given group $G$ arises as a group of points $E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form ${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group ${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when $m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when $m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer $N$ arises as a group order $\#E\left( {{\mathbb{F}}_{p}} \right)$.

We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet–Langlands correspondence relating modular forms on $\text{G}{{\text{L}}_{\text{2}}}/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

This article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of ${{\mathbf{R}}^{n}}$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in ${{\mathbf{R}}^{n}}$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

In this paper, we extend the definition of the $SL\left( 2,\,\mathbb{C} \right)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$ -degree of the $\widehat{A}$ -polynomial of $K$ . We prove a product formula for the $\widehat{A}$ -polynomial of the connected sum ${{K}_{1}}\#{{K}_{2}}$ of two knots in ${{S}^{3}}$ and deduce additivity of the $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant under connected sums for a large class of knots in ${{S}^{3}}$ . We also present an example of a nontrivial knot $K$ in ${{S}^{3}}$ with trivial $\widehat{A}$ -polynomial and trivial $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant, showing that neither of these invariants detect the unknot.

We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field ${{\mathbb{F}}_{q}}$ is equal to

$${{\left( q-1 \right)}^{n+1}}_{{}}{{q}^{\frac{\left( n+1 \right)\left( n-2 \right)}{2}}}\sum\limits_{\theta }{{{q}^{inv\left( \theta \right)}}}$$

where the sum is over all indecomposable permutations in ${{S}_{n+1}}$ and where inv $\left( \theta \right)$ stands for the number of inversions of $\theta $ .

Let $p\,>\,2$ be a prime and let $X$ be a compactified PEL Shimura variety of type $\left( \text{A} \right)$ or $\left( \text{C} \right)$ such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens, we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

We construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}\left[ \sqrt{2} \right]$ .

If $f,\,g:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}_{\ge 0}}$ are non-negative measurable functions, then the Prékopa–Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater than or equal to the 0-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa–Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

We study unitary quotients of the free product unitary pivotal category ${{A}_{2}}\,*\,{{T}_{2}}$. We show that such quotients are parametrized by an integer $n\,\ge \,1$ and an $2n$–th root of unity $\omega $. We show that for $n\,=\,1,\,2,\,3$, there is exactly one quotient and $\omega \,=\,1$. For $4\,\le \,n\,\le \,10$, we show that there are no such quotients. Our methods also apply to quotients of ${{T}_{2}}\,*\,{{T}_{2}}$, where we have a similar result.

The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of ${{A}_{2}}\,*\,{{T}_{2}}$ and ${{T}_{2}}\,*\,{{T}_{2}}$, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.

During the preparation of this manuscript, we learnt of Liu's independent result on composites of ${{A}_{3}}$ and ${{A}_{4}}$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch–Haagerup showed that the principal graph of a composite of ${{A}_{3}}$ and ${{A}_{4}}$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\,\ge \,4$.

This is an abridged version of arxiv:1308.5723.

An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma $ acting on $X$ and $\mathfrak{q}$ . In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma $ is abelian and acts freely on $X$ . We show that such representations are parameterized by a certain set of $\Gamma $ -equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$ . In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

We collect evidence in support of a conjecture of Griffiths, Green, and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi–Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\,\le \,d\,\le \,6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is ${{G}_{2}}$ ) of the family of 6-folds, and the theory of boundary components of Mumford–Tate domains.

We formulate and prove a “winding number” index theorem for certain “Toeplitz” operators in the same spirit as Gohberg–Krein, Lesch and others. The “number” is replaced by a self-adjoint operator in a subalgebra $Z\,\subseteq \,Z\left( A \right)$ of a unital $C*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau $ on $A$ left invariant under an action $\alpha :\,\text{R}\,\to \,\text{Aut}\left( A \right)$ leaving $Z$ pointwise fixed. If $\delta $ is the infinitesimal generator of $\alpha $ and $u$ is invertible in $\text{dom}\left( \delta \right)$, then the “winding operator” of $u$ is $\frac{1}{2\pi i}\tau \left( \delta \left( u \right){{u}^{-1}} \right)\,\in \,{{Z}_{sa}}$. By a careful choice of representations we extend $\left( A,\,Z,\,\tau ,\,\alpha \right)$ to a von Neumann setting $\left( \mathfrak{A},\,\mathfrak{Z},\,\bar{\tau },\,\bar{\alpha } \right)$ where $\mathfrak{A}\,=\,{A}''$ and $\mathfrak{Z}\,=\,{Z}''$. Then $A\,\subset \,\mathfrak{A}\,\subset \,\mathfrak{A}\,\rtimes \,\mathbf{R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$. If $P$ is the projection in $\mathfrak{A}\,\rtimes \,\mathbf{R}$ corresponding to the non-negative spectrum of the generator of $\mathbf{R}$ inside $\mathfrak{A}\,\rtimes \,\mathbf{R}$ and $\tilde{\pi }:\,A\,\to \,\mathfrak{A}\,\rtimes \,\mathbf{R}$ is the embedding, then we define ${{T}_{u}}\,=\,P\tilde{\pi }\left( u \right)P\,\text{for}\,u\,\in \,{{A}^{-1}}$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of ${{T}_{u}}$ is the negative of the winding operator. In outline the proof follows that of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$ requires the nontrivial development of a $\mathfrak{Z}$-Hilbert algebra theory. We show that certain of these Fredholm operators fiber as a “section” of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.

In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton, we are able to show that both problems are solvable and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions.The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type that is stable. Some numerical results are included to conûrm the theory.

We answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., $\operatorname{cov}\left( \text{M} \right)=\mathfrak{c})$, a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as corresponding Laurent polynomials.

We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so–called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi–Yau varieties.

Let $G$ be a finite group. A faithful $G$ -variety $X$ is called strongly incompressible if every dominant $G$ -equivariant rationalmap of $X$ onto another faithful $G$ -variety $Y$ is birational. We settle the problem of existence of strongly incompressible $G$ -curves for any finite group $G$ and any base field $k$ of characteristic zero.

The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on “square roots” of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.