Rational $S^1$-equivariant homotopy theory
Laura
Scull
1-45
Abstract: We give an algebraicization of rational $S^1$-equivariant homotopy theory. There is an algebraic category of `` $\mathbb{T}$-systems'' which is equivalent to the homotopy category of rational $S^1$-simply connected $S^1$-spaces. There is also a theory of ``minimal models'' for $\mathbb{T}$-systems, analogous to Sullivan's minimal algebras. Each $S^1$-space has an associated minimal $\mathbb{T}$-system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.
Cohomology of buildings and finiteness properties of $\widetilde{A}_n$-groups
Jacqui
Ramagge;
Wayne
W.
Wheeler
47-61
Abstract: Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free $S$-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre's approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented $\widetilde{A}_n$-group is a virtual duality group. A number of other finiteness conditions for $\widetilde{A}_n$-groups are also established.
Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes
Frédéric
Chapoton
63-74
Abstract: Stasheff polytopes, introduced by Stasheff in his study of $H$-spaces, are linked to associativity. The direct sum of their cellular complexes is the underlying complex of the $A_{\infty}$ operad which describes homotopy associative algebras. In particular, there exists a quasi-isomorphism $A_{\infty}\rightarrow \operatorname{As}$. Here, we define on the direct sum of their dual cellular complexes the structure of a differential graded operad. This construction extends the dendriform operad of Loday, which corresponds to the vertices of the polytopes. We also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes. We define a quasi-isomorphism from $\operatorname{As}$ to each of these operads. We also define non-differential variants of the two preceding operads and a morphism from $\operatorname{As}$ to each of these operads. We show that the free algebras have a coproduct which turns them into bialgebras. RÉSUMÉ. Les polytopes de Stasheff, introduits pour l'étude des $H$-espaces, sont liés à l'associativité. La somme directe de leurs complexes cellulaires forme le complexe sous-jacent à l'opérade $A_\infty$ qui décrit les algèbres associatives à homotopie près. En particulier, il existe un quasi-isomorphisme $A_\infty\to \operatorname{As}$. Ici, on munit la somme directe des duaux de leurs complexes cellulaires d'une structure d'opérade différentielle graduée. Cette construction généralise l'opérade des algèbres dendriformes de Loday, qui correspond aux sommets des polytopes. On munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d'une structure d'opérade différentielle graduée. On définit un quasi-isomorphisme de $\operatorname{As}$ dans chacune de ces deux opérades. On construit également des variantes non différentielles des deux opérades précédentes. On définit un morphisme de $\operatorname{As}$ dans chacune de ces opérades et on montre que les algèbres libres sont munies d'un coproduit coassociatif qui en fait des bigèbres.
Localisation homotopique et tour de Taylor pour une catégorie abélienne
Olivier
Renaudin
75-89
Abstract: On indique comment une sous-catégorie colocalisante d'une catégorie abélienne induit une localisation dans la catégorie dérivé. Ceci permet une nouvelle construction de la tour de Taylor d'un foncteur à valeur dans une catégorie de module.
The best constant in the Davis inequality for the expectation of the martingale square function
Donald
L.
Burkholder
91-105
Abstract: A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than $\sqrt3$ times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales $M$ indexed by $[0,\infty)$. Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.
APS boundary conditions, eta invariants and adiabatic limits
Xianzhe
Dai
107-122
Abstract: We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.
On arithmetic Macaulayfication of Noetherian rings
Takesi
Kawasaki
123-149
Abstract: The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
Weakly defective varieties
L.
Chiantini;
C.
Ciliberto
151-178
Abstract: A projective variety $X$ is `$k$-weakly defective' when its intersection with a general $(k+1)$-tangent hyperplane has no isolated singularities at the $k+1$ points of tangency. If $X$ is $k$-defective, i.e. if the $k$-secant variety of $X$ has dimension smaller than expected, then $X$ is also $k$-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of $k$-defective surfaces.
Polytopal linear retractions
Winfried
Bruns;
Joseph
Gubeladze
179-203
Abstract: We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension $1$ retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable `polytopal' generalization of linear algebra (and, subsequently, that of algebraic $K$-theory).
The multiplier ideals of a sum of ideals
Mircea
Mustata
205-217
Abstract: We prove that if $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}\subseteq\mathcal{O}_X$ are nonzero sheaves of ideals on a complex smooth variety $X$, then for every $\gamma\in{\mathbb Q}_+$we have the following relation between the multiplier ideals of $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}$ and $\underline{\mathbf{a}}+\underline{\mathbf{b}}$: \begin{displaymath}\mathcal{I}\left(X,\gamma\cdot(\underline{\mathbf{a}}+ \under... ...thbf{a}})\cdot\mathcal{I}(X,\beta\cdot \underline{\mathbf{b}}).\end{displaymath} A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals. We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.
Explicit merit factor formulae for Fekete and Turyn polynomials
Peter
Borwein;
Kwok-Kwong
Stephen
Choi
219-234
Abstract: We give explicit formulas for the $L_{4}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials \begin{displaymath}f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac{k}{q}\right ) z^{k} \end{displaymath} where $\left (\frac{\cdot }{q}\right )$ is the Legendre symbol. For example for $q$ an odd prime, \begin{displaymath}\Vert f_{q}\Vert _{4}^{4} : = \frac{5q^{2}}{3}-3q+ \frac{4}{3} - 12 (h(-q))^{2} \end{displaymath} where $h(-q)$ is the class number of $\mathbb{Q}(\sqrt {-q})$. Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of $f_{q}$. This is the sequence that has the largest known asymptotic merit factor. Explicitly, \begin{displaymath}R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+[q/4] }{q}\right ) z^{k} \end{displaymath} where $[\cdot ]$ denotes the nearest integer, satisfies \begin{displaymath}\Vert R_{q}\Vert _{4}^{4} = \frac{7q^{2}}{6}- {q} - \frac{1}{6} - \gamma _{q} \end{displaymath} where \begin{displaymath}\gamma _{q}: = \begin{cases} h(-q) (h(-q)-4) & \text{if} \qu... ...pmod 8, 0 & \text{if} \quad q \equiv 7 \pmod 8. \end{cases}\end{displaymath} Indeed we derive a closed form for the $L_{4}$ norm of all shifted Fekete polynomials \begin{displaymath}f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+t}{q}\right ) z^{k}. \end{displaymath} Namely \begin{align*}\Vert f_{q}^{t} \Vert _{4}^{4} &= \frac{1}{3}(5q^{2}+3q+4)+8t^{2}-... ...tyle \sum _{n=1}^{q-1}n\left(\frac{n+t}{q}\right)}\right \vert^{2}, \end{align*} and $\Vert f_{q}^{q-t+1} \Vert _{4}^{4}= \Vert f_{q}^{t} \Vert _{4}^{4}$ if $1 \le t \le (q+1)/2$.
Topological mixing in $CAT\left(-1\right)$-spaces
Charalambos
Charitos;
Georgios
Tsapogas
235-264
Abstract: If $X$ is a proper $CAT\left( -1\right)$-space and $\Gamma$ a non-elementary discrete group of isometries acting properly discontinuously on $X,$ it is shown that the geodesic flow on the quotient space $Y=X/\Gamma$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wandering set of the flow equals the whole quotient space of geodesics $GY:=GX/\,\Gamma$ (the latter being redundant when $Y$ is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete $CAT\left( -1\right)$-spaces by a one-ended group of isometries and (C) finite $n$-dimensional ideal polyhedra.
On nonlinear oscillations in a suspension bridge system
Zhonghai
Ding
265-274
Abstract: In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.
Representation type of Hecke algebras of type $A$
Karin
Erdmann;
Daniel
K.
Nakano
275-285
Abstract: In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type $A$, stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.
The extraspecial case of the $k(GV)\;$ problem
David
Gluck;
Kay
Magaard
287-333
Abstract: Let $E$ be an extraspecial-type group and $V$ a faithful, absolutely irreducible $k[E]$-module, where $k$ is a finite field. Let $G$ be the normalizer in $GL(V)$ of $E$. We show that, with few exceptions, there exists a $v\in V$such that the restriction of $V$ to $C_H(v)$ is self-dual whenever $H\le G$and $(\vert H\vert, \vert V\vert)=1$.
Subgroup properties of fully residually free groups
Ilya
Kapovich
335-362
Abstract: We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.
Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II
Yi-Zhi
Huang;
Antun
Milas
363-385
Abstract: We construct the intertwining operator superalgebras and vertex tensor categories for the $N=2$ superconformal unitary minimal models and other related models.
Dynamics of implicit operations and tameness of pseudovarieties of groups
Jorge
Almeida
387-411
Abstract: This work gives a new approach to the construction of implicit operations. By considering ``higher-dimensional'' spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesski{\u{\i}}\kern.15em, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of $p$-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups $\mathbf{V}$, if it can be decided whether a finitely generated subgroup of the free group with the pro- $\mathbf{V}$ topology is dense, then $\mathbf{V}$ is tame.
Shellability in reductive monoids
Mohan
S.
Putcha
413-426
Abstract: The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let $M$ be a reductive monoid with unit group $G$, Borel subgroup $B$ and Weyl group $W$. We study the partially ordered set of $B\times B$-orbits (with respect to Zariski closure inclusion) within a $G\times G$-orbit of $M$. This is the same as studying a $W\times W$-orbit in the Renner monoid $R$. Such an orbit is the retract of a `universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.