On the $p$-compact groups corresponding to the $p$-adic reflection groups $G(q,r,n)$
Natàlia
Castellana
2799-2819
Abstract: There exists an infinite family of $p$-compact groups whose Weyl groups correspond to the finite $p$-adic pseudoreflection groups $G(q,r,n)$ of family 2a in the Clark-Ewing list. In this paper we study these $p$-compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical $J$-homomorphism. Finally, we also describe a faithful complexification homomorphism from these $p$-compact groups to the $p$-completion of unitary compact Lie groups.
Entire majorants via Euler--Maclaurin summation
Friedrich
Littmann
2821-2836
Abstract: It is the aim of this article to give extremal majorants of type $2\pi\delta$ for the class of functions $f_n(x)=$sgn$(x)x^n$, where $n\in\mathbb{N}$. As applications we obtain positive definite extensions to $\mathbb{R}$ of $\pm(it)^{-m}$ defined on $\mathbb{R}\backslash[-1,1]$, where $m\in\mathbb{N}$, optimal bounds in Hilbert-type inequalities for the class of functions $(it)^{-m}$, and majorants of type $2\pi$ for functions whose graphs are trapezoids.
Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems
Gen
Nakamura;
Jenn-Nan
Wang
2837-2853
Abstract: Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.
The flat model structure on complexes of sheaves
James
Gillespie
2855-2874
Abstract: Let $\mathbf{Ch}(\mathcal{O})$ be the category of chain complexes of $\mathcal{O}$-modules on a topological space $T$ (where $\mathcal{O}$ is a sheaf of rings on $T$). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on $\mathbf{Ch}(\mathcal{O})$. As a corollary, we have a general framework for doing homological algebra in the category $\mathbf{Sh}(\mathcal{O})$ of $\mathcal{O}$-modules. I.e., we have a natural way to define the functors $\operatorname{Ext}$ and $\operatorname{Tor}$ in $\mathbf{Sh}(\mathcal{O})$.
The parameterized Steiner problem and the singular Plateau problem via energy
Chikako
Mese;
Sumio
Yamada
2875-2895
Abstract: The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.
Quillen stratification for Hochschild cohomology of blocks
Jonathan
Pakianathan;
Sarah
Witherspoon;
and with an appendix by
Stephen
F.
Siegel
2897-2916
Abstract: We decompose the maximal ideal spectrum of the Hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian $p$-subgroups of a defect group. These subvarieties are described in terms of group cohomological varieties and the Alperin-Broué correspondence on blocks. Our description leads in particular to a homeomorphism between the Hochschild variety of the principal block and the group cohomological variety. The proofs require a result of Stephen F. Siegel, given in the Appendix, which states that nilpotency in Hochschild cohomology is detected on elementary abelian $p$-subgroups.
On Hölder continuous Riemannian and Finsler metrics
Alexander
Lytchak;
Asli
Yaman
2917-2926
Abstract: We discuss smoothness of geodesics in Riemannian and Finsler metrics.
Quantum cohomology and $S^1$-actions with isolated fixed points
Eduardo
Gonzalez
2927-2948
Abstract: This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element, that the (small) quantum cohomology of a $2n$-dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of $n$ copies of $\mathbb{P}^1$. This generalizes a result due to Tolman and Weitsman.
Alexander polynomials of equivariant slice and ribbon knots in $S^3$
James
F.
Davis;
Swatee
Naik
2949-2964
Abstract: This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.
Besov spaces with non-doubling measures
Donggao
Deng;
Yongsheng
Han;
Dachun
Yang
2965-3001
Abstract: Suppose that $\mu$ is a Radon measure on ${\mathbb R}^d,$ which may be non-doubling. The only condition on $\mu$ is the growth condition, namely, there is a constant $C_0>0$ such that for all $x\in {\rm {\,supp\,}}(\mu)$ and $r>0,$ \begin{displaymath}\mu (B(x, r))\le C_0r^n,\end{displaymath} where $0<n\leq d.$ In this paper, the authors establish a theory of Besov spaces $\dot B^s_{pq}(\mu)$ for $1\le p, q\le\infty$ and $\vert s\vert<\theta$, where $\theta>0$ is a real number which depends on the non-doubling measure $\mu$, $C_0$, $n$ and $d$. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.
Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces
Leonardo
Biliotti
3003-3022
Abstract: We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.
Invariance in $\boldsymbol{\mathcal{E}^*}$ and $\boldsymbol{\mathcal{E}_\Pi}$
Rebecca
Weber
3023-3059
Abstract: We define $G$, a substructure of $\mathcal{E}_\Pi$ (the lattice of $\Pi^0_1$ classes), and show that a quotient structure of $G$, $G^\diamondsuit$, is isomorphic to $\mathcal{E}^*$. The result builds on the $\Delta^0_3$ isomorphism machinery, and allows us to transfer invariant classes from $\mathcal{E}^*$ to $\mathcal{E}_\Pi$, though not, in general, orbits. Further properties of $G^\diamondsuit$ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.
Finite Bruck loops
Michael
Aschbacher;
Michael
K.
Kinyon;
J.
D.
Phillips
3061-3075
Abstract: Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops $X$, showing that $X$ is essentially the direct product of a Bruck loop of odd order with a $2$-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite $2$-element Bruck loops are $2$-loops, leaving open the question of whether such obstructions actually exist.
Projective Fraïssé limits and the pseudo-arc
Trevor
Irwin;
Slawomir
Solecki
3077-3096
Abstract: The aim of the present work is to develop a dualization of the Fraïssé limit construction from model theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Fraïssé limits, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem on projective homogeneity of the pseudo-arc (which generalizes a result of Lewis and Smith on density of homeomorphisms of the pseudo-arc among surjective continuous maps from the pseudo-arc to itself). We also get a new characterization of the pseudo-arc via the projective homogeneity property.
A structure theorem for the elementary unimodular vector group
Selby
Jose;
Ravi
A.
Rao
3097-3112
Abstract: Given a pair of vectors $v,w\in R^{r+1}$ with $\langle v,w\rangle=v\cdot w^T=1$, A. Suslin constructed a matrix $S_r(v,w)\in Sl_{2^r}(R)$. We study the subgroup $SUm_r(R)$ generated by these matrices, and its (elementary) subgroup $EUm_r(R)$ generated by the matrices $S_r(e_1\varepsilon,e_1\varepsilon^{T^{-1}})$, for $\varepsilon\in E_{r+1}(R)$. The basic calculus for $EUm_r(R)$ is developed via a key lemma, and a fundamental property of Suslin matrices is derived.
Commutative ideal theory without finiteness conditions: Completely irreducible ideals
Laszlo
Fuchs;
William
Heinzer;
Bruce
Olberding
3113-3131
Abstract: An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.
Maximal families of Gorenstein algebras
Jan
O.
Kleppe
3133-3167
Abstract: The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring $R$. Let $\operatorname{GradAlg}^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. We prove there is a close relationship between the irreducible components of $\operatorname{GradAlg}^H(R)$, whose general member is a Gorenstein codimension $(c+1)$ quotient, and the irreducible components of $\operatorname{GradAlg}^{H'}(R)$ to ``Gorenstein'' components of $\operatorname{GradAlg}^{H}(R)$, in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein'' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay'' component and a sum of two invariants of $B$. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.
Layers and spikes in non-homogeneous bistable reaction-diffusion equations
Shangbing
Ai;
Xinfu
Chen;
Stuart
P.
Hastings
3169-3206
Abstract: We study $\varepsilon^2\ddot{u}=f(u,x)=A\, u\, (1-u)\,(\phi-u)$, where $A=A(u,x)>0$, $\phi=\phi(x)\in(0,1)$, and $\varepsilon>0$ is sufficiently small, on an interval $[0,L]$ with boundary conditions $\dot{u}=0$ at $x=0,L$. All solutions with an $\varepsilon$ independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to $f=A(u,x)\; (u-\phi_-)\,(u-\phi)\,(u-\phi_+)$ with $\phi_-(x)<\phi(x)<\phi_+(x)$ and also to an infinite interval.
Moduli of curves and spin structures via algebraic geometry
Gilberto
Bini;
Claudio
Fontanari
3207-3217
Abstract: Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type $(p,0)$ and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.
Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero
William
F.
Rulla
3219-3237
Abstract: Let $X_n := \overline{M}_{0,n}$, the moduli space of $n$-pointed stable genus zero curves, and let $X_{n,m}$ be the quotient of $X_n$ by the action of $\mathcal{S}_{n-m}$ on the last $n-m$ marked points. The cones of effective divisors $\overline{NE}^1(X_{n,m})$, $m = 0,1,2$, are calculated. Using this, upper bounds for the cones $Mov(X_{n,m})$ generated by divisors with moving linear systems are calculated, $m = 0,1$, along with the induced bounds on the cones of ample divisors of $\overline{M}_g$ and $\overline{M}_{g,1}$. As an application, the cone $\overline{NE}^1(\overline{M}_{2,1})$ is analyzed in detail.
An infinitary extension of the Graham--Rothschild Parameter Sets Theorem
Timothy
J.
Carlson;
Neil
Hindman;
Dona
Strauss
3239-3262
Abstract: The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of $\beta S$, the Stone-Cech compactification of a discrete semigroup, we derive an infinitary extension of the Graham-Rothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extension of the original. The original theorem says that whenever $k<m$ in $\mathbb{N}$ and the $k$-parameter words are colored with finitely many colors, there exist a color and an $m$-parameter word $w$ with the property that whenever a $k$-parameter word of length $m$ is substituted in $w$, the result is in the specified color. The ``simplest finite instance'' referred to above is that, given finite colorings of the $k$-parameter words for each $k<m$, there is one $m$-parameter word which works for each $k$. Some additional Ramsey Theoretic consequences are derived. We also observe that, unlike any other Ramsey Theoretic result of which we are aware, central sets are not necessarily good enough for even the $k=1$ and $m=2$ version of the Graham-Rothschild Parameter Sets Theorem.