On the subgroup structure of exceptional groups of Lie type
Martin
W.
Liebeck;
Gary
M.
Seitz
3409-3482
Abstract: We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle $X(q)$ of Lie type in the natural characteristic. Our approach is to show that for sufficiently large $q$ (usually $q>9$ suffices), $X(q)$ is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.
Connected finite loop spaces with maximal tori
J.
M.
Møller;
D.
Notbohm
3483-3504
Abstract: Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.
A growth dichotomy for o-minimal expansions of ordered groups
Chris
Miller;
Sergei
Starchenko
3505-3521
Abstract: Let $\mathfrak{R}$ be an o-minimal expansion of a divisible ordered abelian group $(R,<,+,0,1)$ with a distinguished positive element $1$. Then the following dichotomy holds: Either there is a $0$-definable binary operation $\cdot$ such that $(R,<,+,\cdot ,0,1)$ is an ordered real closed field; or, for every definable function $f:R\to R$ there exists a $0$-definable $\lambda \in \{0\}\cup \operatorname{Aut}(R,+)$ with $\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R$. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure $\mathfrak{M}:=(M,<,\dots )$ there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) $\mathfrak{M}$-definable groups with underlying set $M$.
Periodic billiard orbits are dense in rational polygons
M.
Boshernitzan;
G.
Galperin;
T.
Krüger;
S.
Troubetzkoy
3523-3535
Abstract: We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of $\pi.$
Poincaré embedding of the diagonal
Yanghyun
Byun
3537-3553
Abstract: There is a Poincaré embedding structure on the diagonal $X\rightarrow X\times X$ under the conditions: i) $X$ is formed by gluing two compact smooth manifolds along their boundaries using a homotopy equivalence and ii) a square-root closed condition is satisfied by the fundamental groupoid of the boundary.
Copies of $c_0$ and $\ell_\infin$ in topological Riesz spaces
Lech
Drewnowski;
Iwo
Labuda
3555-3570
Abstract: The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of $c_{0}$ or $\ell _{\infty }$.
Induction theorems on the stable rationality of the center of the ring of generic matrices
Esther
Beneish
3571-3585
Abstract: Following Procesi and Formanek, the center of the division ring of $n\times n$ generic matrices over the complex numbers $\mathbf C$ is stably equivalent to the fixed field under the action of $S_n$, of the function field of the group algebra of a $ZS_n$-lattice, denoted by $G_n$. We study the question of the stable rationality of the center $C_n$ over the complex numbers when $n$ is a prime, in this module theoretic setting. Let $N$ be the normalizer of an $n$-sylow subgroup of $S_n$. Let $M$ be a $ZS_n$-lattice. We show that under certain conditions on $M$, induction-restriction from $N$ to $S_n$ does not affect the stable type of the corresponding field. In particular, $\mathbf C (G_n)$ and $\mathbf C(ZG\otimes _{ZN}G_n)$ are stably isomorphic and the isomorphism preserves the $S_n$-action. We further reduce the problem to the study of the localization of $G_n$ at the prime $n$; all other primes behave well. We also present new simple proofs for the stable rationality of $C_n$ over $\mathbf C$, in the cases $n=5$ and $n=7$.
Subvarieties of $\mathcal{SU}_C(2)$ and $2\theta$-divisors in the Jacobian
W.
M.
Oxbury;
C.
Pauly;
E.
Previato
3587-3614
Abstract: We explore some of the interplay between Brill-Noether subvarieties of the moduli space ${\mathcal{SU}}_C(2,K)$ of rank 2 bundles with canonical determinant on a smooth projective curve and $2\theta$-divisors, via the inclusion of the moduli space into $|2\Theta|$, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus $<6$, though no higher, bundles with $>2$ sections are cut out by $\Gamma _{00}$; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.
Quantum cohomology of projective bundles over $\mathbb P^n$
Zhenbo
Qin;
Yongbin
Ruan
3615-3638
Abstract: In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space $\mathbb{P}^{n}$. Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over $\mathbb{P}^{n}$ is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.
On zeta functions and Iwasawa modules
Jangheon
Oh
3639-3655
Abstract: We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules $X^{-}_{k({\zeta }_{p})}$ for almost all $p$ determine the zeta function ${\zeta }_{k}$ when $k$ is a totally real field. Conversely, we prove that the $\lambda$-part of the Iwasawa module $X_{k}$ is determined by its zeta-function ${\zeta }_{k}$ up to pseudo-isomorphism for any number field $k.$ Moreover, we prove that arithmetically equivalent CM fields have also the same ${\mu }^{-}$-invariant.
On the conjectures of J. Thompson and O. Ore
Erich
W.
Ellers;
Nikolai
Gordeev
3657-3671
Abstract: If $G$ is a finite simple group of Lie type over a field containing more than $8$ elements (for twisted groups $^{l} X_{n} (q^{l})$ we require $q > 8$, except for $^{2} B_{2} (q^{2})$, $^{2} G_{2} (q^{2})$, and $^{2} F_{4} (q^{2})$, where we assume $q^{2} > 8$), then $G$ is the square of some conjugacy class and consequently every element in $G$ is a commutator.
Self-duality operators on odd dimensional manifolds
Houhong
Fan
3673-3706
Abstract: In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free $S^{1}$-actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.
Hypercyclicity in the scattering theory for linear transport equation
H.
Emamirad
3707-3716
Abstract: We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted $L^{1}$ spaces.
Convergence of random walks on the circle generated by an irrational rotation
Francis
Edward
Su
3717-3741
Abstract: Fix $\alpha \in [0,1)$. Consider the random walk on the circle $S^1$ which proceeds by repeatedly rotating points forward or backward, with probability $\frac 12$, by an angle $2\pi\alpha$. This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number $\xi=2\alpha$. We obtain bounds for rates when $\xi$ is any irrational, and a sharp rate when $\xi$ is a quadratic irrational. In that case the discrepancy falls as $k^{-\frac 12}$ (up to constant factors), where $k$ is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of $\xi$ which allows for tighter bounds on terms which appear in the Erdös-Turán inequality.
Curve-straightening and the Palais-Smale condition
Anders
Linnér
3743-3765
Abstract: This paper considers the negative gradient trajectories associated with the modified total squared curvature functional $\int k^{2} +\nu ds$. The focus is on the limiting behavior as $\nu$ tends to zero from the positive side. It is shown that when $\nu =0$ spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. As $\nu$ tends to zero the critical curves in the first subset tend to the critical points present when $\nu =0$. Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are `dragged' all the way to `infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a `critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. There are examples where the length of the curves along the same trajectory is not a monotone function of the flow-time. It is shown how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus $p$ must satisfy $2E(p)/K(p)=1\pm |g|/\widetilde L$ when the space is limited to curves of fixed length $\widetilde L$ and the endpoints are separated by the vector $g$.
Tessellations of solvmanifolds
Dave
Witte
3767-3796
Abstract: Let $A$ be a closed subgroup of a connected, solvable Lie group $G$, such that the homogeneous space $A\backslash G$ is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation $A\backslash G/\Gamma$ of $A\backslash G$ is finitely covered by a compact homogeneous space $G'/\Gamma'$. We prove that the covering map can be taken to be very well behaved - a ``crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on $A\backslash G/\Gamma$ that has a dense orbit is covered by a natural flow on $G'/\Gamma'$.
Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations
Joseph
Gruendler
3797-3814
Abstract: Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.
Operations and Spectral Sequences. I
James
M.
Turner
3815-3835
Abstract: Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer's result for Steenrod operations on Serre spectral sequence and extend A. Bahri's action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.