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Structures on Manifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best ... More

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best described as a G-structure which may or may not be (partially) integrable. Some selected topics include: Riemannian metrics, complex structures, symplectic structures, contact structures, quaternionic structures, group actions, pseudogroups, sheaves, bundles, connections, holonomy, curvature and integrability.Less

Kähler Manifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge ... More

This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge theory, Chern classes, positivity and Fano varieties, line bundles and divisors. Of particular interest is Yau's famous proof of the Calabi conjecture which ends this chapter.Less

The Human Prefrontal Cortex Stores Structured Event Complexes

Frank Krueger and Jordan Grafman

in Understanding Events: From Perception to Action

Event sequence knowledge is necessary for learning, planning, and performing activities of daily living. Clinical observations suggest that the prefrontal cortex (PFC) is crucial for goal-directed ... More

Event sequence knowledge is necessary for learning, planning, and performing activities of daily living. Clinical observations suggest that the prefrontal cortex (PFC) is crucial for goal-directed behavior such as carrying out plans, controlling a course of actions, or organizing everyday life routines. This chapter proposes a “representational” approach to PFC function, which assumes that the PFC (a) stores long-term memories of goal-oriented event sequence knowledge and (b) seeks to establish the format and categories according to which such information is stored. It argues that the human PFC stores a unique type of knowledge in the form of structured event complexes (SECs). SECs are representations composed of higher-order goal-oriented sequences of events that are involved in the planning and monitoring of complex behavior.Less

Hodge Structures with Complex Multiplication

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining ... More

This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.Less

Classification of Mumford-Tate Subdomains

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

This chapter develops an algorithm for determining all Mumford-Tate subdomains of a given period domain. The result is applied to the classification of all complex multiplication Hodge structures (CM ... More

This chapter develops an algorithm for determining all Mumford-Tate subdomains of a given period domain. The result is applied to the classification of all complex multiplication Hodge structures (CM Hodge structures) of rank 4 and when the weight n = 1 and n = 3, to an analysis of their Hodge tensors and endomorphism algebras, and the number of components of the Noether-Lefschetz locus. The result is that one has a complex but very rich arithmetic story. Of particular note is the intricate structure of the components of the Noether-Lefschetz loci in D and in its compact dual, and the two interesting cases where the Hodge tensors are generated in degrees 2 and 4. One application is that a particular class of period maps appearing in mirror symmetry never has image in a proper subdomain of D.Less

Arithmetic of Period Maps of Geometric Origin

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family X → S of smooth, projective ... More

This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family X → S of smooth, projective varieties defined over a number field k. The chapter recalls the notion of absolute Hodge classes (AH) and strongly absolute Hodge classes (SAH). The particular case when the Noether-Lefschetz locus consists of isolated points is alluded to in the discussion of complex multiplication Hodge structures (CM Hodge structures). A related observation is that one may formulate a variant of the “Grothendieck conjecture” in the setting of period maps and period domains. The chapter also describes a behavior of fields of definition under the period map, along with the existence and density of CM points in a motivic variation of Hodge structure.Less

Conclusions

Alwyn Lishman

in Patient Participation in Palliative Care: A voice for the voiceless

This concluding chapter discusses the personal experience of the author in dealing with palliative care and the concept of the user's voice, which is widely welcomed in seeking to refine the way ... More

This concluding chapter discusses the personal experience of the author in dealing with palliative care and the concept of the user's voice, which is widely welcomed in seeking to refine the way forward. It reveals that the common thread that unites the chapters in this book is a deep humanity, which is focused on improving the lot of the terminally ill. The chapter also shows that palliative care needs to operate within complex structures: organizational and even political. Alliances between diverse groups of people must be formed and financial limitations faced.Less

Executive Functions

Jordan Grafman

in Neuroergonomics: The brain at work

This chapter shows that key higher-level cognitive functions known as the executive functions are strongly associated with the human prefrontal cortex (HPFC). It argues that an important way to ... More

This chapter shows that key higher-level cognitive functions known as the executive functions are strongly associated with the human prefrontal cortex (HPFC). It argues that an important way to understand the functions of the HPFC is to adapt the representational model that has been the predominant approach to understanding the neuropsychological aspects of, for example, language processing and object recognition. The representational approach developed is based on the structured event complex framework. This framework claims that there are multiple subcomponents of higher-level knowledge that are stored throughout the HPFC as distinctive domains of memory. The chapter also argues that there are topographical distinctions in where these different aspects of knowledge are stored in the HPFC.Less

Solution of flexible molecular structures by simulated annealing

Peter G. Bruce and Yuri G. Andreev

in Structure Determination from Powder Diffraction Data

The chapter describes a simulated annealing approach to solving highly flexible molecular crystal structures. Particular attention is paid to the description of the models whose position, ... More

The chapter describes a simulated annealing approach to solving highly flexible molecular crystal structures. Particular attention is paid to the description of the models whose position, orientation, and conformation are to be optimised against the observed powder diffraction data. The mathematics underlying the construction and positioning of the models is given. The approach is illustrated with the solution of various polymer: salt complex crystal structures. The chapter concludes with a discussion of the likely limits of structural complexity that might be addressed by such an approach.Less

Architecture of Counterfactual Thought in the Prefrontal Cortex

Aron K. Barbey, Frank Krueger, and Jordan Grafman

in Predictions in the Brain: Using Our Past to Generate a Future

This chapter develops an integrative cognitive neuroscience framework for understanding counterfactual reasoning on the basis of structured event complexes (SECs) in the human prefrontal cortex ... More

This chapter develops an integrative cognitive neuroscience framework for understanding counterfactual reasoning on the basis of structured event complexes (SECs) in the human prefrontal cortex (PFC). The reviewed evidence in support of the SEC framework confirms the importance and uniqueness of the human PFC for representing knowledge in the form of cognitive events and action sequences. It is argued that SECs are the key to understanding the human ability to represent mental models of events, which guide the selection of goal-directed action sequences and the on-line updating of behavior based on past outcomes or anticipated future events.Less

Generalized Kähler Metrics from Hamiltonian Deformations

Marco Gualtieri

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

This chapter provides a new characterization of generalized Kähler structures in terms of the corresponding complex Dirac structures. It then gives an alternative proof of Hitchin’s partial ... More

This chapter provides a new characterization of generalized Kähler structures in terms of the corresponding complex Dirac structures. It then gives an alternative proof of Hitchin’s partial unobstructedness for holomorphic Poisson structures. Its main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kähler structures. That is, it shows that any generalized Kähler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin’s deformation. Finally, it indicates a close relationship between this deformation and the notion of a Hamiltonian family of Poisson structures.Less

Almost complex structures

Dusa McDuff and Dietmar Salamon

in Introduction to Symplectic Topology

The chapter begins with a general discussion of almost complex structures on symplectic manifolds and then addresses the problem of integrability. Subsequent sections discuss a variety of examples of ... More

The chapter begins with a general discussion of almost complex structures on symplectic manifolds and then addresses the problem of integrability. Subsequent sections discuss a variety of examples of Kähler manifolds, in particular those of complex dimension two, and show how to compute the Chern classes and Betti numbers of hypersurfaces in complex projective space. The last section is a brief introduction to the theory of J-holomorphic curves.Less

Moduli and deformations

Simon Donaldson

in Riemann Surfaces

This chapter first examines almost-complex structures, Beltrami differentials, and the integrability theorem. It then discusses deformations and cohomology.

Review

Tim Lenton and Andrew Watson

in Revolutions that made the Earth

This chapter endeavours to review the similarities between what is happening now and the past revolutions. There are several common features to the great revolutions that have made the present Earth. ... More

This chapter endeavours to review the similarities between what is happening now and the past revolutions. There are several common features to the great revolutions that have made the present Earth. These can be broadly categorised as the main characteristics of change, and the features which must emerge for it to be ‘successful’ – in the sense that permanent change persists and includes a thriving biosphere. Two out of the three past revolutions have been underlain by a step change in information transmission between living organisms. An increase in the amount of information that can be passed on allows life to build more complex structures. Complex structures can, in turn, improve the information-transmission mechanisms, allowing more information to be passed on, so these two processes mutually reinforce one another.Less

Bleibt Alles Anders: Modeling Invention

Markus Eberl

in War Owl Falling: Innovation, Creativity, and Culture Change in Ancient Maya Society

Unlike earlier conceptions of society as homogeneous, modern models emphasize its factious nature. Correspondingly, it cannot be reduced to a structure; instead, it contains multiple, coexisting, and ... More

Unlike earlier conceptions of society as homogeneous, modern models emphasize its factious nature. Correspondingly, it cannot be reduced to a structure; instead, it contains multiple, coexisting, and even overlapping structures. Chapter 4 visualizes these complex structures as a Garden of Forking Paths in which worlds—some real, others imaginary—coexist. To plan their actions, individuals trace a coherent path between worlds. Their actions adhere to cultural, although not necessarily practical, logic. They deal with worlds differently and apply different modalities. The latter distinguishes innovators from others. The material nature of inventions exhibits interaction and adoption publicly. Therefore, individual decision-making interweaves, Gardens of Forking Paths overlap, and the resulting interferences map the structural changes in society. Less

Classification of Boundary Lefschetz Fibrations over the Disc

Stefan Behrens, Gil R. Cavalcanti, and Ralph L. Klaasse

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

This chapter shows that a 4-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S1×S3#nCP¯2,#mCP2#nCP¯2 or #m(S2×S2). Given the relation between ... More

This chapter shows that a 4-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S1×S3#nCP¯2,#mCP2#nCP¯2 or #m(S2×S2). Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, the chapter concludes that the 4-manifolds S1×S3#nCP¯2,#(2m+1)CP2#nCP¯2 and #(2m+1)S2×S2 admit stable generalized complex structures whose type change locus has a single component and are the only 4-manifolds whose stable structure arises from boundary Lefschetz fibrations over the disc.Less

Transformations to multiaged stand structures

Kevin L. O’Hara

in Multiaged Silviculture: Managing for Complex Forest Stand Structures

There is a growing, international interest in transforming even-aged to multiaged stands. There are many possible pathways for transformation or conversion of stands, and many different possible ... More

There is a growing, international interest in transforming even-aged to multiaged stands. There are many possible pathways for transformation or conversion of stands, and many different possible multiaged stand structures. These can include relatively simple stand structures, such as those with only two age classes, or more complex stand structure. The transformation process typically will take longer with greater complexity in the desired target structure. Transformation typically begins with a treatment that regenerates a new age class of trees. Constraints to transformation include having a sufficiently stable even-aged stand to treat. Many even-aged stands will be too dense to transform. In other cases, trees may be too old to survive a long transformation process. The silviculture of transformation should also be highly adaptive rather than assuming preestablished plans will suffice without adjustments later in the transformation process. Transformation targets should be flexible, as should the treatment regimes that are used to achieve these targets.Less

Aggregation

Paul Charbonneau

in Natural Complexity: A Modeling Handbook

This chapter explores how naturally occurring inanimate structures grow by accretion of smaller-sized components, focusing on one specific accretion process: diffusion-limited aggregation (DLA). In ... More

This chapter explores how naturally occurring inanimate structures grow by accretion of smaller-sized components, focusing on one specific accretion process: diffusion-limited aggregation (DLA). In DLA, particles move about in random fashion, but stick together when they come into contact. Clumps of particles then form and grow further by colliding with other individual particles, or clumps of particles. Over time, one or more aggregates of individual particles will grow. After providing an overview of how DLA works, the chapter describes its numerical implementation and shows a representative simulation of a two-dimensional DLA aggregate. It then considers two peculiar geometrical properties of aggregates resulting from the DLA process, namely self-similarity and scale invariance, and shows that rule based growth through DLA can lead to the buildup of complex structures, sometimes exhibiting fractal geometry. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.Less