63 research outputs found
Lagrangian 3-torus fibrations
We prove that Mark Gross' topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual
Lagrangian pairs of pants
We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of an exact one form defined on the real blow up of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of which lift tropical subvarieties in . As an example we explain how to lift tropical curves in to Lagrangian submanifolds of . We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone
Balanced superprojective varieties
We first review the definition of superprojective spaces from the functor-of-points perspective. We derive the relation between superprojective spaces and supercosets in the framework of the theory of sheaves. As an application of the geometry of superprojective spaces, we extend Donaldson\u2019s definition of balanced manifolds to supermanifolds and we derive the new conditions of a balanced supermanifold. We apply the construction to superpoints viewed as submanifolds of superprojective spaces. We conclude with a list of open issues and interesting problems that can be addressed in the present context
Cech and de Rham Cohomology of Integral Forms
We present a study on the integral forms and their Cech/de Rham cohomology.
We analyze the problem from a general perspective of sheaf theory and we
explore examples in superprojective manifolds. Integral forms are fundamental
in the theory of integration in supermanifolds. One can define the integral
forms introducing a new sheaf containing, among other objects, the new basic
forms delta(dtheta) where the symbol delta has the usual formal properties of
Dirac's delta distribution and acts on functions and forms as a Dirac measure.
They satisfy in addition some new relations on the sheaf. It turns out that the
enlarged sheaf of integral and "ordinary" superforms contains also forms of
"negative degree" and, moreover, due to the additional relations introduced,
its cohomology is, in a non trivial way, different from the usual superform
cohomology.Comment: 20 pages, LaTeX, we expanded the introduction, we add a complete
analysis of the cohomology and we derive a new duality between cohomology
group
20 questions on Adaptive Dynamics
Abstract Adaptive Dynamics is an approach to studying evolutionary change when fitness is density or frequency dependent. Modern papers identifying themselves as using this approach first appeared in the 1990s, and have greatly increased up to the present. However, because of the rather technical nature of many of the papers, the approach is not widely known or understood by evolutionary biologists. In this review we aim to remedy this situation by outlining the methodology and then examining its strengths and weaknesses. We carry this out by posing and answering 20 key questions on Adaptive Dynamics. We conclude that Adaptive Dynamics provides a set of useful approximations for studying various evolutionary questions. However, as with any approximate method, conclusions based on Adaptive Dynamics are valid only under some restrictions that we discuss
Altruism can proliferate through group/kin selection despite high random gene flow
The ways in which natural selection can allow the proliferation of
cooperative behavior have long been seen as a central problem in evolutionary
biology. Most of the literature has focused on interactions between pairs of
individuals and on linear public goods games. This emphasis led to the
conclusion that even modest levels of migration would pose a serious problem to
the spread of altruism in group structured populations. Here we challenge this
conclusion, by analyzing evolution in a framework which allows for complex
group interactions and random migration among groups. We conclude that
contingent forms of strong altruism can spread when rare under realistic group
sizes and levels of migration. Our analysis combines group-centric and
gene-centric perspectives, allows for arbitrary strength of selection, and
leads to extensions of Hamilton's rule for the spread of altruistic alleles,
applicable under broad conditions.Comment: 5 pages, 2 figures. Supplementary material with 50 pages and 26
figure
Evolution of Assortative Mating in a Population Expressing Dominance
In this article, we study the influence of dominance on the evolution of
assortative mating. We perform a population-genetic analysis of a two-locus
two-allele model. We consider a quantitative trait that is under a mixture
of frequency-independent stabilizing selection and density- and frequency-dependent
selection caused by intraspecific competition for a continuum of resources.
The trait is determined by a single (ecological) locus and expresses intermediate
dominance. The second (modifier) locus determines the degree of assortative
mating, which is expressed in females only. Assortative mating is based on
similarities in the quantitative trait (‘magic trait’ model).
Analytical conditions for the invasion of assortment modifiers are derived
in the limit of weak selection and weak assortment. For the full model, extensive
numerical iterations are performed to study the global dynamics. This allows
us to gain a better understanding of the interaction of the different selective
forces. Remarkably, depending on the size of modifier effects, dominance can
have different effects on the evolution of assortment. We show that dominance
hinders the evolution of assortment if modifier effects are small, but promotes
it if modifier effects are large. These findings differ from those in previous
work based on adaptive dynamics
Rapid Transition towards the Division of Labor via Evolution of Developmental Plasticity
A crucial step in several major evolutionary transitions is the division of labor between components of the emerging higher-level evolutionary unit. Examples include the separation of germ and soma in simple multicellular organisms, appearance of multiple cell types and organs in more complex organisms, and emergence of casts in eusocial insects. How the division of labor was achieved in the face of selfishness of lower-level units is controversial. I present a simple mathematical model describing the evolutionary emergence of the division of labor via developmental plasticity starting with a colony of undifferentiated cells and ending with completely differentiated multicellular organisms. I explore how the plausibility and the dynamics of the division of labor depend on its fitness advantage, mutation rate, costs of developmental plasticity, and the colony size. The model shows that the transition to differentiated multicellularity, which has happened many times in the history of life, can be achieved relatively easily. My approach is expandable in a number of directions including the emergence of multiple cell types, complex organs, or casts of eusocial insects
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