47 research outputs found
From the hyperbolic 24-cell to the cuboctahedron
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by
deforming an infinite volume orbifold obtained from the ideal, hyperbolic
24-cell by removing two walls. This family provides an infinite number of
infinitesimally rigid, infinite covolume, geometrically finite discrete
subgroups of the isometry group of hyperbolic 4-space. It also leads to finite
covolume Coxeter groups which are the homomorphic image of the group of
reflections in the hyperbolic 24-cell. The examples are constructed very
explicitly, both from an algebraic and a geometric point of view. The method
used can be viewed as a 4-dimensional, but infinite volume, analog of
3-dimensional hyperbolic Dehn filling.Comment: The article has 78 pages and 37 figures. Many of the figures use
color in an essential way. If possible, use a color printe
K-theoretic Tutte polynomials of morphisms of matroids
We generalize the Tutte polynomial of a matroid to a morphism of matroids via
the K-theory of flag varieties. We introduce two different generalizations, and
demonstrate that each has its own merits, where the trade-off is between the
ease of combinatorics and geometry. One generalization recovers the Las Vergnas
Tutte polynomial of a morphism of matroids, which admits a corank-nullity
formula and a deletion-contraction recursion. The other generalization does
not, but better reflects the geometry of flag varieties.Comment: 27 pages; minor revisions. To appear in JCT
Associated Primes of Spline Complexes
The spline complex whose top homology is
the algebra of mixed splines over the fan
was introduced by Schenck-Stillman in
[Schenck-Stillman 97] as a variant of a complex
of Billera [Billera 88]. In this paper we
analyze the associated primes of homology modules of this complex. In
particular, we show that all such primes are linear. We give two applications
to computations of dimensions. The first is a computation of the third
coefficient of the Hilbert polynomial of , including cases
where vanishing is imposed along arbitrary codimension one faces of the
boundary of , generalizing the computations in [Geramita-Schenck
98,McDonald-Schenck 09]. The second is a description of the fourth coefficient
of the Hilbert polynomial of for simplicial fans
. We use this to derive the result of Alfeld, Schumaker, and Whiteley
on the generic dimension of tetrahedral splines for
[Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description
may be used to give the fourth coefficient in particular nongeneric
configurations.Comment: 40 pages, 10 figure
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
Complex Algebraic Geometry
The Conference focused on several classical and novel theories in the realm of complex algebraic geometry, such as Algebraic surfaces, Moduli theory, Minimal Model Program, Abelian Varieties, Holomorphic Symplectic Varieties, Homological algebra, Kähler manifolds theory, Holomorphic dynamics, Quantum cohomology
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom