29,933 research outputs found
Bunches of cones in the divisor class group -- A new combinatorial language for toric varieties
As an alternative to the description of a toric variety by a fan in the
lattice of one parameter subgroups, we present a new language in terms of what
we call bunches -- these are certain collections of cones in the vector space
of rational divisor classes. The correspondence between these bunches and fans
is based on classical Gale duality. The new combinatorial language allows a
much more natural description of geometric phenomena around divisors of toric
varieties than the usual method by fans does. For example, the numerically
effective cone and the ample cone of a toric variety can be read off
immediately from its bunch. Moreover, the language of bunches appears to be
useful for classification problems.Comment: Minor changes, to appear in Int. Math. Res. No
A survey on spectral multiplicities of ergodic actions
Given a transformation of a standard measure space , let \Cal
M(T) denote the set of spectral multiplicities of the Koopman operator
defined in by . It is discussed in
this survey paper which subsets of are realizable as
\Cal M(T) for various : ergodic, weakly mixing, mixing, Gaussian, Poisson,
ergodic infinite measure preserving, etc. The corresponding constructions are
considered in detail. Generalizations to actions of Abelian locally compact
second countable groups are also discussed
On spectral disjointness of powers for rank-one transformations and M\"obius orthogonality
We study the spectral disjointness of the powers of a rank-one
transformation. For a large class of rank-one constructions, including those
for which the cutting and stacking parameters are bounded, and other examples
such as rigid generalized Chacon's maps and Katok's map, we prove that
different positive powers of the transformation are pairwise spectrally
disjoint on the continuous part of the spectrum. Our proof involves the
existence, in the weak closure of {U_T^k: k in Z}, of "sufficiently many"
analytic functions of the operator U_T. Then we apply these disjointness
results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic)
symbolic models associated to these rank-one constructions: All sequences
realized in these models are orthogonal to the M\"obius function
Arithmetic lattices and weak spectral geometry
This note is an expansion of three lectures given at the workshop "Topology,
Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University
in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and
Arithmetic of Hyperbolic Spaces". Comments welcom
Geometric Invariant Theory via Cox Rings
We consider actions of reductive groups on a varieties with finitely
generated Cox ring, e.g., the classical case of a diagonal action on a product
of projective spaces. Given such an action, we construct via combinatorial data
in the Cox ring all maximal open subsets such that the quotient is
quasiprojective or embeddable into a toric variety. As applications, we obtain
an explicit description of the chamber structure of the linearized ample cone
and several Gelfand-MacPherson type correspondences relating quotients of
reductive groups to quotients of torus actions. Moreover, our approach provides
information on the geometry of many of the resulting quotient spaces.Comment: 27 pages, minor changes, Example 8.8 replaced, to appear in Journal
of Pure and Applied Algebr
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
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