11,086 research outputs found
Geometry on surfaces, a source for mathematical developments
We present a variety of geometrical and combinatorial tools that are used in
the study of geometric structures on surfaces: volume, contact, symplectic,
complex and almost complex structures. We start with a series of local rigidity
results for such structures. Higher-dimensional analogues are also discussed.
Some constructions with Riemann surfaces lead, by analogy, to notions that hold
for arbitrary fields, and not only the field of complex numbers. The Riemann
sphere is also defined using surjective homomorphisms of real algebras from the
ring of real univariate polynomials to (arbitrary) fields, in which the field
with one element is interpreted as the point at infinity of the Gaussian plane
of complex numbers. Several models of the hyperbolic plane and hyperbolic
3-space appear, defined in terms of complex structures on surfaces, and in
particular also a rather elementary construction of the hyperbolic plane
usingreal monic univariate polynomials of degree two without real roots.
Several notions and problems connected with conformal structures in dimension 2
are discussed, including dessins d'enfants, the combinatorial characterization
of polynomials and rational maps of the sphere, the type problem,
uniformization, quasiconformal mappings, Thurston's characterization of Speiser
graphs, stratifications of spaces of monic polynomials, and others. Classical
methods and new techniques complement each other. The final version of this
paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A.
Papadopoulos), Springer Nature Switzerland, 2024
Sasakian Geometry, Holonomy, and Supersymmetry
In this expository article we discuss the relations between Sasakian
geometry, reduced holonomy and supersymmetry. It is well known that the
Riemannian manifolds other than the round spheres that admit real Killing
spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly
parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the
relations between the latter two and Sasaki-Einstein geometry.Comment: 40 pages, some minor corrections made, to appear in the Handbook of
pseudo-Riemannian Geometry and Supersymmetr
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Urban Juxtaposition: A Precedent Analysis of New Urbanism in Denver, Colorado
The New Urbanist movement embraces neotraditional design principles in an attempt to create a more sustainable urban form; however, some New Urbanist developments, and to some extent their principles, are not progressive enough to make legitimate claims of increasing environmental and social sustainability. In Denver, Colorado New Urbanist neighborhoods are inconsistent with New Urbanist principles; the Stapleton neighborhood, Highland Garden Village, and Riverfront Park are three New Urbanist neighborhoods used in my precedent analysis in order to illustrate these inconsistencies. Riverfront Park accomplishes many of New Urbanisms principles and goals while the Stapleton neighborhood, the largest greyfield development in the country, lacks many New Urbanist principles in its implementation; primarily related to land use patterns, residential density, and transportation. My research concludes that discrepancies between municipalities and private developers, as well as national transportation standards and policies, have resulted in a compromise that limits the implementation of New Urbanist principles in New Urbanist developments. Historical frameworks, lagging policy change, and our neoliberal free market have perpetuated a cycle of environmentally and socially unsustainable growth that needs to change in order for our built environment to continue growing into a more sustainable form
A Duality Exact Sequence for Legendrian Contact Homology
We establish a long exact sequence for Legendrian submanifolds L in P x R,
where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that
displaces the projection of L off of itself. In this sequence, the singular
homology H_* maps to linearized contact cohomology CH^* which maps to
linearized contact homology CH_* which maps to singular homology. In
particular, the sequence implies a duality between the kernel of the map
(CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this
duality is compatible with Poincare duality in L in the following sense: the
Poincare dual of a singular class which is the image of a in CH_* maps to a
class \alpha in CH^* such that \alpha(a)=1.
The exact sequence generalizes the duality for Legendrian knots in Euclidean
3-space [24] and leads to a refinement of the Arnold Conjecture for double
points of an exact Lagrangian admitting a Legendrian lift with linearizable
contact homology, first proved in [6].Comment: 57 pages, 10 figures. Improved exposition and expanded analytic
detai
The Mathematics of Phylogenomics
The grand challenges in biology today are being shaped by powerful
high-throughput technologies that have revealed the genomes of many organisms,
global expression patterns of genes and detailed information about variation
within populations. We are therefore able to ask, for the first time,
fundamental questions about the evolution of genomes, the structure of genes
and their regulation, and the connections between genotypes and phenotypes of
individuals. The answers to these questions are all predicated on progress in a
variety of computational, statistical, and mathematical fields.
The rapid growth in the characterization of genomes has led to the
advancement of a new discipline called Phylogenomics. This discipline results
from the combination of two major fields in the life sciences: Genomics, i.e.,
the study of the function and structure of genes and genomes; and Molecular
Phylogenetics, i.e., the study of the hierarchical evolutionary relationships
among organisms and their genomes. The objective of this article is to offer
mathematicians a first introduction to this emerging field, and to discuss
specific mathematical problems and developments arising from phylogenomics.Comment: 41 pages, 4 figure
Sasaki-Einstein Manifolds and Volume Minimisation
We study a variational problem whose critical point determines the Reeb
vector field for a Sasaki-Einstein manifold. This extends our previous work on
Sasakian geometry by lifting the condition that the manifolds are toric. We
show that the Einstein-Hilbert action, restricted to a space of Sasakian
metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in
fact is a function on the space of Reeb vector fields. We relate this function
both to the Duistermaat-Heckman formula and also to a limit of a certain
equivariant index on X that counts holomorphic functions. Both formulae may be
evaluated by localisation. This leads to a general formula for the volume
function in terms of topological fixed point data. As a result we prove that
the volume of a Sasaki-Einstein manifold, relative to that of the round sphere,
is always an algebraic number. In complex dimension n=3 these results provide,
via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional
superconformal field theories. We also show that our variational problem
dynamically sets to zero the Futaki invariant of the transverse space, the
latter being an obstruction to the existence of a Kahler-Einstein metric.Comment: 82 pages, 9 figures; homogeneity of the (n,0)-form now derived from
the Einstein-Hilbert action of the link, example of an orbifold resolution
added, various clarifications and references added; minor changes; published
versio
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