Homotopical variations and high-dimensional Zariski-van Kampen theorems
D.
Chéniot;
C.
Eyral
1-10
Abstract: We give a new definition of the homotopical variation operators occurring in a recent high-dimensional Zariski-van Kampen theorem, a definition which opens the way to further generalizations of theorems of this kind.
Polar sets on metric spaces
Juha
Kinnunen;
Nageswari
Shanmugalingam
11-37
Abstract: We show that if $X$ is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of $X$ with zero $p$-capacity are precisely the $p$-polar sets; that is, a relatively compact subset of a domain in $X$ is of zero $p$-capacity if and only if there exists a $p$-superharmonic function whose set of singularities contains the given set. In addition, we prove that if $X$ is a $p$-hyperbolic metric space, then the $p$-superharmonic function can be required to be $p$-superharmonic on the entire space $X$. We also study the the following question: If a set is of zero $p$-capacity, does there exist a $p$-superharmonic function whose set of singularities is precisely the given set?
A generalization of Euler's hypergeometric transformation
Robert
S.
Maier
39-57
Abstract: Euler's transformation formula for the Gauss hypergeometric function ${}_2F_1$ is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new one-term evaluations of ${}_2F_1(-1)$ and ${}_3F_2(1)$ are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of ${}_2F_1(-1)$ with linearly constrained parameters are derived as well.
The complexity of recursion theoretic games
Martin
Kummer
59-86
Abstract: We show that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured. Several consequences are pointed out; for instance, the set of all $\Pi_2$-sentences that are uniformly valid in the lattice of recursively enumerable sets is undecidable. Furthermore we show that these games are equivalent to natural subclasses of effectively presented Borel games.
Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds
Yasushi
Homma
87-114
Abstract: Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.
Morse theory from an algebraic viewpoint
Emil
Sköldberg
115-129
Abstract: Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.
On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring
Thomas
Geisser;
Lars
Hesselholt
131-145
Abstract: We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic $(0,p)$, the $p$-adic étale $K$-theory and $p$-adic topological cyclic homology agree.
The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system
Gen
Nakamura;
Jenn-Nan
Wang
147-165
Abstract: In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous aniso- tropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.
Quantifier elimination for algebraic $D$-groups
Piotr
Kowalski;
Anand
Pillay
167-181
Abstract: We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial)$ of characteristic $0$, then the first order structure consisting of $G$ together with the algebraic $D$-subvarieties of $G, G\times G,\dots$, has quantifier-elimination. In other words, the projection on $G^{n}$ of a $D$-constructible subset of $G^{n+1}$ is $D$-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.
A new approach to the theory of classical hypergeometric polynomials
José
Manuel
Marco;
Javier
Parcet
183-214
Abstract: In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.
Symmetric functions in noncommuting variables
Mercedes
H.
Rosas;
Bruce
E.
Sagan
215-232
Abstract: Consider the algebra $\mathbb{Q}\langle \langle x_1,x_2,\ldots\rangle \rangle$ of formal power series in countably many noncommuting variables over the rationals. The subalgebra $\Pi(x_1,x_2,\ldots)$of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.
Real and complex earthquakes
Dragomir
Saric
233-249
Abstract: We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane $\mathbb{H} ^2$. We show that an earthquake restricted to the boundary $S^1$ of $\mathbb{H} ^2$ is a quasisymmetric map if and only if its earthquake measure is bounded. Multiplying an earthquake measure by a positive parameter we obtain an earthquake path. Consequently, an earthquake path with a bounded measure is a path in the universal Teichmüller space. We extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part. Such obtained complex earthquake (or bending) is holomorphic in the parameter. Moreover, the restrictions to $S^1$ of a bending with complex parameter of small imaginary part is a holomorphic motion of $S^1$in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.
Some counterexamples to a generalized Saari's conjecture
Gareth
E.
Roberts
251-265
Abstract: For the Newtonian $n$-body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.
On the Castelnuovo-Mumford regularity of connected curves
Daniel
Giaimo
267-284
Abstract: In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in $\mathbb{P} ^4$ having maximal regularity, but no extremal secants. We also show that any connected curve in $\mathbb{P} ^3$ of degree at least 5 with maximal regularity and no linear components has an extremal secant.
Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics
B.
M.
Hambly;
Michel
L.
Lapidus
285-314
Abstract: In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.
An invariant of tangle cobordisms
Mikhail
Khovanov
315-327
Abstract: We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the cobordism.
The automorphism tower of groups acting on rooted trees
Laurent
Bartholdi;
Said
N.
Sidki
329-358
Abstract: The group of isometries $\operatorname{Aut}(\mathcal{T}_n)$ of a rooted $n$-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in $\operatorname{Aut}(\mathcal{T}_n)$. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group $\mathfrak{G}$ studied by R. Grigorchuk, and the group $\ddot\Gamma$ studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as $\mathfrak{G}$ and $\ddot\Gamma$. We describe this tower for all subgroups of $\operatorname{Aut}(\mathcal{T}_2)$ which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of $\mathfrak{G}$ and $\ddot\Gamma$. More precisely, the tower of $\mathfrak{G}$ is infinite countable, and the terms of the tower are $2$-groups. Quotients of successive terms are infinite elementary abelian $2$-groups. In contrast, the tower of $\ddot\Gamma$ has length $2$, and its terms are $\{2,3\}$-groups. We show that $\operatorname{Aut}^2(\ddot\Gamma) /\operatorname{Aut}(\ddot\Gamma)$ is an elementary abelian $3$-group of countably infinite rank, while $\operatorname{Aut}^3(\ddot\Gamma)=\operatorname{Aut}^2(\ddot\Gamma)$.
Unramified cohomology of classifying varieties for exceptional simply connected groups
Skip
Garibaldi
359-371
Abstract: Let $BG$ be a classifying variety for an exceptional simple simply connected algebraic group $G$. We compute the degree 3 unramified Galois cohomology of $BG$ with values in
Polygonal invariant curves for a planar piecewise isometry
Peter
Ashwin;
Arek
Goetz
373-390
Abstract: We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.
A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal
Volker
Runde
391-402
Abstract: Let $G$ be a locally compact group, and let $\mathcal{WAP}(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[\operatorname{SIN}]$-group, but not compact, then the dual Banach algebra $\mathcal{WAP}(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[\operatorname{SIN}]$-group, $\mathcal{WAP}(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups $G$ such that $\mathcal{WAP}(G)^\ast$ does have a normal, virtual diagonal.
Quasi-finite modules for Lie superalgebras of infinite rank
Ngau
Lam;
R.
B.
Zhang
403-439
Abstract: We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\widehat{\rm gl}_{\infty\vert\infty}$, $\widehat{\mathcal{C}}$and $\widehat{\mathcal{ D}}$, and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.
A density theorem on automorphic $L$-functions and some applications
Yuk-Kam
Lau;
Jie
Wu
441-472
Abstract: We establish a density theorem on automorphic $L$-functions and give some applications on the extreme values of these $L$-functions at $s=1$ and the distribution of the Hecke eigenvalue of holomorphic cusp forms.